diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzatdy" "b/data_all_eng_slimpj/shuffled/split2/finalzzatdy"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzatdy"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe observation of a superfluid (SF) to Mott insulator (MI) transition in an \noptical lattice \\cite{greiner_02} have opened a new paradigm to explore the\nphysics of quantum many-body systems. Optical lattices are clean and highly\ncontrollable; in contrast, the condensed matter systems of interest are never\ndevoid of impurities. Thus, some of the fundamental questions in condensed \nmatter physics are related to quantum phase transitions in the presence of \ndisorder. The presence of disorder constrains the evolution of a quantum \nsystem in the Hilbert space and gives rise to quantum glassy phases \nlike Bose glass (BG) \\cite{fisher_89,lin_12} and phenomena like Anderson \nlocalization \\cite{anderson_58, schulte_05, billy_08, roati_08}. The early\ntheoretical investigations of disordered Bose Hubbard model (DBHM) \n\\cite{fisher_89, giamarchi_88} showed that there is no MI-SF transition in \npresence of diagonal disorder as the BG phase always occurs as the intermediate\nphase. The theorem of inclusion \\cite{pollet_09, gurarie_09} agrees well with \nthis prediction while identifying BG phase as a Griffiths phase containing \nthe rare regions. In these rare-regions, the energy gap of adding \nanother boson to the system vanishes and thus can be identified as SF islands.\n\n  The DBHM have been studied with diverse techniques: \nmean field \\cite{krutitsky_06}, projected Gutzwiller method \\cite{lin_12}, \nsite independent and multisite mean-field method \n\\cite{buonsante_07, pisarski_11}, stochastic mean field \\cite{bissbort_10},\nquantum Monte Carlo \\cite{gimperlein_05, soyler_11, sengupta_07}, \ndensity matrix renormalisation group (DMRG) \\cite{rapsch_99, gerster_16} for \n1D system and numerous others \n\\cite{pai_96, nikolay_04, kruger_09, kruger_11,carrasquilla_10}.\nIn all the cases the introduction of disorder leads to the emergence of BG \nphase which is characterized by finite compressibility and zero \nsuperfluid stiffness. In the present work, we study 2D DBHM at finite \ntemperatures using single site Gutzwiller and cluster Gutzwiller mean field \ntheories. More importantly, we examine the effect of the artificial gauge\nfields in DBHM. Here, it must be emphasized that most of the theoretical \ninvestigations of DBHM are at zero temperatures, but the experimental \nrealizations are at finite temperatures. This gap is addressed in the present \nwork by examining the consequent effects of thermal fluctuations to the BG \nphase. One key finding is the presence of normal fluid (NF) phase at finite \ntemperatures and melting of Bose glass phase. The latter is consistent with \nthe findings reported in ref. \\cite{thomson_16}. \n\n  In optical lattices it is possible to create an equivalent of Lorentz force \nwith artificial gauge fields\n\\cite{lin_09,lin_11,dalibard_11,hof_76, garcia_12, aidelsburger_11} and\nis associated with what is referred to as synthetic magnetic field.\nThe introduction of the artificial gauge field breaks time reversal symmetry\nand modify the band structure. Through the introduction of tunable artificial\ngauge field it has been possible to observe the single particle mobility \nedge \\cite{gadway_18} in zig-zag chains. Apart from the transport properties, \nthe localization effect of the artificial gauge field can enhance the glassy \nfeatures of DBHM. Indeed, our study reveals that localization in DBHM\ncan be controlled through the artificial gauge field. For this we use\nEdward Anderson order parameter (EAOP) as a measure of localization \nwhile tuning the strength of artificial gauge field. The EAOP is a measure of \nnumber fluctuation over disorder realizations and it is finite for the BG \nphase, but zero and close to zero for the MI and SF\nphases, respectively. From the values of EAOP we find that\nthere is enhancement of the BG phase in the presence of artificial gauge\nfield. From the experimental point of view this is important as it could\nfacilitate detailed studies of the BG phase.\n\n  Experimentally, DBHM can be realized by the addition of speckle type of \ndisorder \\cite{clement_05, clement_08, white_09}, or by the generation of \nincommensurate multichromatic lattice \\cite{damski_03,fallani_07}. Indirect\nmeasurements on SF-BG transition have been reported in 1D \\cite{gadway_11} and \n3D \\cite{pasienski_10,meldgin_16} systems through transport and coherence \nmeasurements. In 2D, the observation of center of mass dynamics \\cite{yan_17} \nhas been theoretically proposed as a method to detect the BG phase \nwhile ref. \\cite{delande_09} suggests measuring the radius of the atomic cloud.\nReplica symmetry breaking \\cite{thomson_14, morrison_08} also has been \nproposed as a possible detection scheme. In spite of these various \nproposals and progresses towards the realization of a Bose glass, a clear \nand unambiguous experimental evidence of BG phase is yet to be achieved. \nIn future studies, quantum gas microscopes \\cite{bakr_09} could probe the\nproperties of the BG phase as it can study the SF islands in BG phase.\nAnd, recent work has proposed  it as an experimental tool to detect \nBG phases \\cite{thomson_16}. \n  \n  This paper is organized as follows. In the Section II we give an account of \nthe single site and cluster Gutzwiller mean field theories. This is then \nfollowed by a description of the artificial gauge field and observable \nmeasures to distinguish different phases in Section III and IV. Then, in \nSection V we provide detailed description of the results obtained from our\nstudies and discuss our observations. And, we then conclude in Section VI.\n\n\n\n\\section{Model and Gutzwiller mean field theory}\n\\label{model}\n\nThe DBHM for a square lattice with nearest neighbour hopping is defined by the \nHamiltonian\n\\begin{eqnarray}\n  \\hat{H} &= &-\\sum_{p,q}\\left [ J_x\\left( \\hat{b}_{p+1, q}^{\\dagger}\n              \\hat{b}_{p,q} + {\\rm H.c.} \\right ) \n              + J_y\\left( \\hat{b}_{p, q+1}^{\\dagger}\n              \\hat{b}_{p,q} + {\\rm H.c.}\\right ) \\right ]\n                          \\nonumber \\\\\n          && + \\sum_{p,q}\\hat{n}_{p,q} \\left [\\frac{U}{2}(\\hat{n}_{p,q}\n            -1) -\\tilde\\mu_{p,q} \\right],\n  \\label{dbhm}\n\\end{eqnarray}\nwhere $p$ ($q$) is the lattice index along $x$ ($y$) axis, \n$\\hat{b}_{p,q}^{\\dagger}$ ($\\hat{b}_{p,q}$) is the creation (annihilation) \noperator for a a boson at the $(p,q)$ lattice site, and $\\hat{n}_{p,q}$ is the boson \ndensity operator; $J_x$ ($J_y$) is the hopping strength between two nearest \nneighbour sites along $x$ ($y$) axis,  $U>0$ is the on-site inter-atomic \ninteraction strength, and $\\tilde\\mu_{p,q} = \\mu - \\epsilon_{p,q}$ is the local\nchemical potential. The disorder is introduced through the random energy \noffset $\\epsilon_{p,q}$ which are uniformly distributed independent random \nnumbers $r_{p,q} \\in [-D, D]$ and $D$ is bound of \nrandom numbers. Depending on the ratio of $J$ and $U$ the above Hamiltonian \ncan describe three possible phases of the system --- MI, BG and \nSF~\\cite{fisher_89}. In the strong on-site interaction limit \n$(J\/U\\rightarrow 0)$ the system is either in the MI phase (gapped phase), or \nin the BG phase. Whereas the system is in SF phase when the tunneling overcomes \nrepulsive interaction. \n \n\n\n\n\\subsection{Zero temperature Gutzwiller mean-field theory}\n\\label{gmf}\n\n  In the present work we employ the Gutzwiller mean-field theory to compute the \nproperties of the DBHM. In this section we describe two variants of the \nGutzwiller mean field theory: First is the single site Gutzwiller mean-field \n(SGMF) method, where the lattice sites are correlated through a scalar mean \nfield $\\phi$ and cannot describe entangled states such as the quantum Hall state. \nAnd, the second is the cluster Gutzwiller mean field (CGMF) method, which \nincorporates the correlation within a cluster of neighbouring lattice sites \nexactly and inter-cluster correlation through $\\phi$. A larger cluster captures the\ncorrelation effects better but at the cost of higher computation.\n\n\n\n\\subsubsection{SGMF method}\n\\label{sgmf}\n\nIn the SGMF method, $\\hat{b}_{p, q}$ ( $\\hat{b}^\\dagger_{p, q}$) at a \nparticular lattice site $(p,q)$ is decomposed into mean field \n$\\phi_{p, q}$ ($\\phi^{*}_{p, q}$) and fluctuation $\\delta \\hat{b}_{p, q}$\n($\\delta \\hat{b}^{\\dagger}_{p, q}$) parts as\n\\begin{subequations}\n\\begin{eqnarray}\n  \\hat{b}_{p, q}           &=& \\phi_{p,q} + \\delta \\hat{b}_{p, q},       \\\\\n  \\hat{b}^{\\dagger}_{p, q} &=& \\phi^{*}_{p, q} + \\delta \\hat{b}^{\\dagger}_{p, q}\n \\label{decompose} \n\\end{eqnarray}\n\\end{subequations}\nwhere, $\\phi_{p,q} = \\langle\\hat{b}_{p,q}\\rangle$, and $\\phi^{*}_{p, q} = \n\\langle\\hat{b}^{\\dagger}_{p,q}\\rangle$ are the mean field and its complex\nconjugate, respectively. The expectations are defined with respect to the \nground state of the system. Employing this decomposition, the Hamiltonian\nin Eq. (\\ref{dbhm}) is reduced to the SGMF Hamiltonian\n\\begin{eqnarray}\n  \\hat{H}^{\\rm MF} &=& \\sum_{p, q}\\Biggr\\{-J_x\n                        \\bigg [ \\Big(\\hat{b}_{p + 1, q}^{\\dagger}\\phi_{p, q} \n                        + \\phi_{p + 1, q}^{*}\\hat{b}_{p, q}  \n                        - \\phi_{p+1,q}^{*}\\phi_{p, q}\\Big) \n                   \\nonumber\\\\   \n                   && + {\\rm H.c.}\\bigg ] \n                      - J_y\\bigg [ \\Big(\\hat{b}_{p, q+1}^{\\dagger} \\phi_{p, q} \n                      + \\phi_{p, q+1}^{*}\\hat{b}_{p, q} \n                      - \\phi_{p, q+1}^{*}\\phi_{p, q}\\Big)\n                   \\nonumber\\\\   \n                   && + {\\rm H.c.}\\bigg ] \n                      + \\biggr[\\frac{U}{2}\\hat{n}_{p, q}\n                      (\\hat{n}_{p, q}-1) - \\tilde{\\mu}_{p, q}\n                      \\hat{n}_{p, q}\\biggr] \\Bigg \\},\n\\label{mf_hamil}\n\\end{eqnarray}\nwhere terms up to linear in fluctuation operators are considered and those\nquadratic in fluctuation operators are neglected. The total Hamiltonian \nin the above expression can be rewritten as \n$\\hat{H}^{\\rm MF} = \\sum_{p,q}\\hat{H}^{\\rm MF}_{p,q}$, where \n$\\hat{H}^{\\rm MF}_{p,q}$ is the single site mean field Hamiltonian. The mean \nfield $\\phi_{p, q}$ can be identified as the SF order parameter which\ndefines the MI to BG phase-transition in DBHM. Thus, $\\phi_{p, q}$ is\nzero, when the  system is in MI phase, and finite in BG as well as in the SF phase. \n\nTo compute the ground state of the system the Hamiltonian matrix of \n$\\hat{H}^{\\rm MF}_{p,q}$ can be diagonalized for each lattice site $(p, q)$ separately. \nAnd, then the ground state of the system is direct product of the single\nsite ground states $\\ket{\\psi}_{p,q}$. Using the Gutzwiller approximation, the \nground state of the system is\n\\begin{eqnarray}\n \\ket{\\Psi_{\\rm GW}} = \\prod_{p, q}\\ket{\\psi}_{p, q}\n                       = \\prod_{p, q} \\sum_{n = 0}^{N_{\\rm b}}c^{(p,q)}_n\n                         \\ket{n}_{p, q},\n \\label{gw_state}\n\\end{eqnarray}\nwhere $N_b$ is the maximum allowed occupation number basis (Fock space basis),\nand $c^{(p,q)}_n$ are the coefficients of the occupation number state $\\ket{n}$\nat the lattice site $(p,q)$. From $\\ket{\\Psi_{\\rm GW}}$  we can calculate\n$\\phi_{p, q}$, the SF order parameter, as\n\\begin{equation}\n\\phi_{p, q} = \\langle\\Psi_{\\rm GW}|\\hat{b}_{p, q}|\\Psi_{\\rm GW}\\rangle \n            = \\sum_{n = 0}^{N_{\\rm b}}\\sqrt{n} \n              {c^{(p,q)}_{n-1}}^{*}c^{(p,q)}_{n}.\n\\label{gw_phi}              \n\\end{equation}\nFrom the above expression it is evident that $\\phi_{p, q}$ is zero in the MI \nphase as only one occupation number state $\\ket{n}$ contributes to \n$\\ket{\\psi}_{p,q}$ and hence only one $c^{(p,q)}_n$ has nonzero value. \nSimilarly, the occupancy and number fluctuation at a lattice site are\n\\begin{eqnarray}\n   \\langle \\hat{n}_{p,q}\\rangle &=& \n    \\sum_{n = 0}^{N_{\\rm b}} | c_n^{(p,q})|^2 n_{p,q},\\label{number} \\\\\n      \\delta n _{p,q} &=& \\sqrt{\\langle \\hat{n}_{p,q}^2\\rangle \n                       - \\langle \\hat{n}_{p,q}\\rangle ^2 }\n  \\label{deltan} \n\\end{eqnarray}\nIn the MI phase $\\delta n _{p,q}$ is zero, which makes MI phase incoherent. In \nthe BG and SF phase $\\delta n _{p,q}$ has nonzero value, but the value of \n$\\delta n _{p,q}$ in the BG phase is very small which arises due to\nthe presence of SF islands in the BG phase. The nonzero and \nrelatively large $\\delta n _{p,q}$ in the SF phase indicates strong phase \ncoherence. Thus $\\delta n _{p,q}$ can also be considered as the\norder parameter for MI-BG phase transition.  \n\n\n\n\n\\subsubsection{CGMF method}\n\\label{cgmf}\n In the CGMF method, to incorporate the hopping term exactly and hence improve\nthe correlation effects, the total lattice considered is partitioned into \nclusters. That is, for an optical lattice of dimension $K\\times L$, we can \nseparate it into $W$ clusters ($C$) of size $M\\times N$, that is \n$W=(K\\times L)\/(M\\times N)$. Thus, the case of CGMF with $M = N = 1$ is \nequivalent to SGMF.  In CGMF, the kinetic energy or the hopping term is decomposed\ninto two types. First is the intra-cluster or hopping within the lattice sites\nin a cluster, and second is the inter-cluster which is between neighbouring \nlattice sites which lie on the boundary of different clusters. The details of\nthe present implementation of the CGMF method is reported in ref. \\cite{bai_18} \nand the Hamiltonian of a single cluster is\n\\begin{eqnarray}\n \\hat{H}_C & =& -{\\sum_{p, q \\in C}}'\\biggr[J_x \n              \\hat{b}_{p+1, q}^{\\dagger}\\hat{b}_{p, q} \n              + J_y \\hat{b}_{p, q+1}^{\\dagger}\\hat{b}_{p, q}\n              + {\\rm H.c.}\\biggr]\\nonumber\\\\\n              &&-\\sum_{p, q\\in \\delta C}\n              \\biggr[J_x (\\phi^c_{p+1,q})^{\\ast}\\hat{b}_{p, q} \n              + J_y (\\phi^c_{p,q+1})^{\\ast}\\hat{b}_{p, q}\n              + {\\rm H.c.}\\biggr]\\nonumber\\\\\n           && +\\sum_{p, q \\in C}\n              \\biggr[\\frac{U}{2}\\hat{n}_{p, q}(\\hat{n}_{p, q}-1) - \n              \\tilde{\\mu}_{p, q}\\hat{n}_{p, q}\\biggr] \n\\label{cg_hamil}         \n\\end{eqnarray}\nwhere $(\\phi^c_{p,q})^{\\ast} = \\sum_{p^{'},q^{'}\\not\\in  C} \\langle \nb_{p^{'},q^{'}}\\rangle$ is the SF order parameter at the lattice site $(p, q)$ which lies\nat the boundary of neighbouring cluster. The prime in the summation of the \nfirst term is to indicate that the $(p+1,q)$ and $(p,q+1)$ lattice points are\nalso within the cluster. And, in the second term $\\delta C$ denotes the lattice\nsites at the boundary of the cluster. The matrix element of $\\hat{H}_C$ is\ndefined in terms of the cluster basis states\n\\begin{equation}\n  \\ket{\\Phi_c}_\\ell = \\prod_{q=0}^{N-1}\\prod_{p=0}^{M-1} \\ket{n_p^q},\n\\end{equation}\nwhere $\\ket{n_p^q}$ is the occupation number basis at the $(p,q)$ lattice\nsite, and $\\ell \\equiv \\{n_0^0, n_1^0, \\ldots, n_{M-1}^0, n_0^1, n_1^1,\\ldots\nn_{M-1}^1, \\ldots, n_{M-1}^{N-1}\\}$ is \nthe index quantum number to identify the cluster state. After diagonalizing \nthe Hamiltonian, we can get the ground state of the cluster as\n\\begin{equation}\n   |\\Psi_c\\rangle = \\sum_{\\ell} C_\\ell\\ket{\\Phi_c}_\\ell.\n\\end{equation}\nwhere $C_\\ell$ is the coefficient of the cluster state.\nThe ground state of the entire $K\\times L$ lattice, like in SGMF, is the direct\nproduct of the cluster ground states\n\\begin{equation}\n \\ket{\\Psi^c_{\\rm GW}} = \\prod_k\\ket{\\Psi_c}_k\n \\label{cgw_state}\n\\end{equation}\nwhere, $k$ is the cluster index and varies from 1 to \n$W$. The SF order parameter $\\phi$ is computed similar \nto Eq.~(\\ref{gw_phi}) as\n\\begin{equation}\n   \\phi_{p,q} = \\bra{\\Psi^c_{\\rm GW}}\\hat{b}_{p,q}\\ket{\\Psi^c_{\\rm GW}}.\n\\label{cgw_phi}              \n\\end{equation}\nWith respect to cluster basis, the average occupancy and number fluctuation \nof lattice sites in the $k$th cluster are\n\\begin{eqnarray}\n   \\langle \\hat{n}\\rangle _k &=& \\frac{\\sum_{p,q\\in C} \\langle\\hat{n}_{p,q}\n                                 \\rangle _k}{MN} \\label{cnumber}  \\\\  \n   (\\delta n)_k &=& \\sqrt{\\langle\\hat{n}^2\\rangle _k - \n                 \\langle\\hat{n}\\rangle^2_k}.\n   \\label{cdeltan} \n\\end{eqnarray}\nFor the entire lattice, the average density can be defined as the mean of the\naverage occupancy of the clusters.\n\n\n\n\n\\subsection{Finite temperature Gutzwiller mean field theory}\n\\label{gutz_t}\n\n To incorporate finite temperature effects we require the entire set of \neigenvalues and eigenfunctions obtained from the diagonalization of the mean\nfield Hamiltonian. So, in the case of SGMF we use all the single site \neigenvectors $\\ket{\\psi}^l_{p,q}$ and corresponding eigenvalues $E^l_{p,q}$ to\ndefine the single site partition function\n\\begin{eqnarray}\n  Z = \\sum_{l=1}^{N_b}e^{-\\beta E_l},\n  \\label{pf}\n\\end{eqnarray}\nwhere $\\beta = 1\/k_BT$, $T$ being the temperature of the system. Since the \nenergy $E_l$ is scaled with respect to $U$, $T$ is in units of $U\/k_{\\rm B}$ or \nin other words in the rest of the paper temperature is defined in terms\nof the dimensionless unit $k_{\\rm B}T\/U$. In a similar way, for the CGMF we \ncan define the cluster partition function in terms of the eigenfunctions \n$\\ket{\\Psi_c}^l$ and the corresponding eigenvalues. \nUsing the above description, the thermal average of the SF order parameter at \nthe $(p,q)$ lattice site is\n\\begin{equation}\n   \\langle \\phi_{p,q}\\rangle = \\frac{1}{Z}\\sum_{l}{_k^l\\bra{\\Psi_c}}\n                \\hat{b}_{p,q} e^{-\\beta E_l} \\ket{\\Psi_c}_k^l,\n\\label{phi_t}\n\\end{equation}\nwhere $\\langle\\ldots\\rangle$ is used to represent thermal averaging and \n$\\ket{\\Psi_c}_k^l$ is the $l$th excited state of the $k$th cluster within \nwhich the $(p,q)$ lattice site lies. Similarly, the occupancy or the density \ncan be computed as \n\\begin{equation}\n   \\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle =  \\frac{1}{Z}\\sum_{l}\n                      {_k^l\\bra{\\Psi_c}}\\hat{n}_{p,q}e^{-\\beta E_l}\n                      \\ket{\\Psi_c}^l_k,\n   \\label{number_t}\n\\end{equation}\nwhere, following the notations in Eq. (\\ref{number}) and (\\ref{cnumber}), the \nadditional $\\langle\\ldots\\rangle$ represents thermal averaging. Once we \nobtain $\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$, the average density or \noccupancy is \n$\\langle\\rho\\rangle=\\langle n \\rangle \n= \\sum_{p,q}\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle \/(K\\times L)$. Then, \nlike defined earlier, the number fluctuation is \n\\begin{equation}\n   \\delta n_{p,q} = \\sqrt{\\langle\\langle \\hat{n}^2_{p,q}\\rangle\\rangle\n                 -\\langle\\langle\\hat{n}_{p,q}\\rangle\\rangle^2}.\n\\end{equation}\nA new feature of considering finite temperature effects is, it is possible to \nhave vanishing $\\langle \\phi_{p,q}\\rangle$ but with  non-integer \n$\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$. This heralds a new \nphase in the phase diagram and is referred to as the normal fluid (NF). \nThus, at finite temperatures SF order parameter can act as the order parameter \nfor the NF-BG transition. \nCompared to the NF phase, the MI on the other hand has vanishing \n$\\langle \\phi_{p,q}\\rangle$ and integer \n$\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$. So, with vanishing \n$\\langle \\phi_{p,q}\\rangle$  the change from \ninteger value to non-integer $\\langle\\langle \\hat{n}_{p,q} \\rangle\\rangle$\ncan be identified as MI-NF transition.\n\n\n\n\n\\section{Artificial gauge field}\n\\label{art_gauge}\n  Artificial gauge fields \\cite{lin_09,lin_11,dalibard_11} engineered through \noptical fields can create synthetic magnetic fields for charge neutral \nultracold atoms in optical lattices. This generates an equivalent of Lorentz \nforce for these atoms, and optical lattice is, then, endowed with properties\nanalogous to the quantum Hall system. Such a system is an excellent model \nsystem to study the physics of strongly correlated states like quantum Hall\nstates and their properties. The same logic also applies to the DBHM.\nIn the Hamiltonian description, the presence of an artificial gauge \nfield induces a complex hopping parameter $J \\rightarrow J\\exp(i\\Phi)$ and\naccordingly the SGMF Hamiltonian in Eq.~(\\ref{mf_hamil}) is modified to\n\\begin{eqnarray}\n  \\hat{H}^{\\rm MF} &=& \\sum_{p, q}\\Biggr\\{-J_x e^{i\\Phi}\n                        \\bigg [ \\Big(\\hat{b}_{p + 1, q}^{\\dagger}\\phi_{p, q}  \n                        + \\phi_{p + 1, q}^{*}\\hat{b}_{p, q}  \n                                  \\nonumber\\\\   \n                   && - \\phi_{p+1,q}^{*}\\phi_{p, q}\\Big) \n                      + {\\rm H.c.}\\bigg ] \n                      - J_y\\bigg [ \\Big(\\hat{b}_{p, q+1}^{\\dagger} \\phi_{p, q} \n                                  \\nonumber\\\\   \n                   && + \\phi_{p, q+1}^{*}\\hat{b}_{p, q} \n                      - \\phi_{p, q+1}^{*}\\phi_{p, q}\\Big)\n                      + {\\rm H.c.}\\bigg ] \n                                  \\nonumber\\\\   \n                   && + \\biggr[\\frac{U}{2}\\hat{n}_{p, q}\n                      (\\hat{n}_{p, q}-1) - \\tilde{\\mu}_{p, q}\n                      \\hat{n}_{p, q}\\biggr] \\Bigg \\},\n\\label{mf_hamil_gauge}\n\\end{eqnarray}\nwhere, $\\Phi$ is the phase an atom acquires when it traverses around a unit\ncell or plaquette. The artificial gauge field is considered in the Landau gauge \nand the phase for hopping along $x$ direction arises via the Peierl's\nsubstitution \\cite{hof_76, garcia_12}. The artificial gauge field, then,\ncreates a staggered synthetic magnetic flux \\cite{aidelsburger_11} along \n$y$ direction.  The phase can also be defined in terms of the $\\alpha$, the \nflux quanta per plaquette, through the relation $\\Phi = 2\\pi\\alpha q$, and \nthe flux quanta is restricted in the domain $0\\le \\alpha\\le 1\/2$.  In the present work, we \nexamine the properties of bosons in presence of artificial gauge field while \nexperiencing a random local chemical potential. Although, the effect of an \nartificial gauge field on BHM is quite well studied, the same is not true of \nDBHM. \n\n    \n\n\n\\section{Characterization of states}\n\n  Each of the low temperature phases supported by DBHM has special properties \nand this leads to unique combinations of order parameters as signatures of \neach phase. The values of these order parameters also determine\nthe phase boundaries. In Table.~\\ref{table:tab}, we list the order parameters\ncorresponding to each phase.  \n\n\n\n\n\\subsection{Superfluid stiffness and compressibility}\n\nPhase coherence is a characteristic property of the SF phase, and it is \nabsent in the other phases (MI, NF and BG) supported by DBHM. Thus in the \nSF phase it requires finite amount of energy to alter the phase coherence, or \nin other words, it acquires stiffness towards phase change. This property is \nreferred to as the superfluid stiffness $\\rho_s$, and hence plays an important\nrole in determining the phase boundary between BG and SF phase. To \ncompute $\\rho_s$, a twisted boundary condition (TBC) is imposed on the state. \nIf the TBC is applied along the $x$ direction, the hopping term in the \nDBHM is transformed as \n\\begin{equation}\n\\!\\!\\!\\!J_x(\\hat{b}_{p+1,q}^{\\dagger}\\hat{b}_{p,q} + {\\rm H.c})\\rightarrow \n             J_x(\\hat{b}_{p+1,q}^{\\dagger}\\hat{b}_{p,q}e^{i2\\pi\\varphi\/L} \n                              + {\\rm H.c}),\n\\label{twist}\n\\end{equation}\nwhere, $\\varphi$ is the phase shift or twist applied to the periodic boundary\ncondition, $L$ is the size of the lattice along $x$ direction, and \n$2\\pi\\varphi\/L$ is phase shift of an atom when it hops between nearest \nneighbours. Accordingly, $\\rho_s$ is computed employing the following \nexpression \\cite{gerster_16}\n\\begin{equation}\n  \\rho_s=\\frac{L}{8\\pi^2}\\frac{\\partial^2E_0}{\\partial\\varphi^2}|_{\\varphi = 0}.\n\\label{stiff}\n\\end{equation}\nwhere $E_0$ is the ground state energy with TBC.\nThe SF phase is a compressible state as $\\delta n$ is finite. However, MI phase\nand strongly correlated phase like quantum Hall states are incompressible. \nThus, the compressibility $\\kappa$ is a property of the system which can be\nemployed a diagnostic to support the phases determined through the order\nparameters. By definition, $\\kappa$ is given by\n\\begin{equation}\n   \\kappa=\\frac{\\partial\\langle\\hat{n}\\rangle}{\\partial\\mu}.\n   \\label{sfactor}\n\\end{equation}\nThat is, $\\kappa$ is the sensitivity of $n$ to the change of $\\mu$.\n\n\n\n\\begin{figure}\n  \\includegraphics[width=8cm]{eaop_dord.jpg}\n  \\caption{$q_{\\text{\\tiny{EA}}}$ as a function of $\\mu\/U$ and $J\/U$ at zero \n          temperature. (a)-(d) show $q_{\\text{\\tiny{EA}}}$ using SGMF method\n          and (e)-(h) are obtained employing the CGMF method with 2$\\times$ 2\n          cluster. (c)-(d) show the enhancement of the BG phase region in the \n          presence of an artificial gauge field with $\\alpha = 1\/4$ compared to \n          (a)-(b) corresponding to $\\alpha = 0$ with disorder strengths\n           $D\/U= 0.2$ and $0.6$ respectively. This enhancement is also\n           captured in (g)-(h) for $\\alpha = 1\/4$ compared to (e)-(f) using the\n           CGMF method.The increase of BG regions with an increase of $D\/U$ is\n           also notable for both in the presence and in the absence of an artificial\n           gauge field. In all the above figures, $q_{\\text{\\tiny{EA}}}$ is\n           obtained by averaging over 50 different disorder distributions. }\n    \\label{eaop-t0}\n\\end{figure}\n\n\n\\subsection{Edwards-Anderson order parameter}\n\nFor a disordered system the natural and hence, more appropriate order parameter\nis the Edwards-Anderson order parameter (EAOP). It can distinguish the \nGriffiths phase by its non zero value and can describe the effect of disorder \nbetter than other properties like  $\\rho_s$, $\\kappa$, structure factor, etc.\nIn the studies with mean field theory, EAOP was first introduced to detect the \nnon trivial breaking of ergodicity. Since then various type of EAOP have been \nproposed in literature \\cite{morrison_08,graham_09,thomson_14,khellil_16}. In \nour study we consider the EAOP of the following form \\cite{thomson_14}\n\\begin{eqnarray}\n   q_{\\text{\\tiny{EA}}} = \\overline{\\langle{\\hat{n}}_{p,q}\\rangle^2}\n       -\\overline{\\langle{\\hat{n}}_{p,q}\\rangle}^2,\n\\label{eaop}\n\\end{eqnarray}  \nwhere, $n_{p,q}$ is the number of atoms at the $(p,q)$ lattice site. The above \nexpression involves two types of averages: $\\langle\\cdots\\rangle$ represents\nthermal; and $\\overline{\\cdots}$ indicates average over disorder distribution.\nFor the $\\langle\\cdots\\rangle$ we consider all the excited states.\nFrom the definition, as the MI phase is identified by integer \nvalues of $\\langle{\\hat{n}}_{p,q}\\rangle$ $q_{\\text{\\tiny{EA}}}$ is zero. In \nthe SF phase $\\langle{\\hat{n}}_{p,q}\\rangle$ is real and $\\delta n_{p,q}$ is \nfinite, however, for the clean system $q_{\\text{\\tiny{EA}}}$ is \nzero as $\\langle{\\hat{n}}_{p,q}\\rangle$ is homogeneous. With disorder, \n$\\langle{\\hat{n}}_{p,q}\\rangle$ is inhomogeneous in the SF phase and hence, \n$q_{\\text{\\tiny{EA}}}$ is finite but small $O(10^{-3})$ \\cite{thomson_16}. In \nthe BG phase $q_{\\text{\\tiny{EA}}}$ is relatively large due to correlation \nbetween number density and disorder. Thus using $q_{\\text{\\tiny{EA}}}$ the BG \nphase is distinguishable from MI and NF phases in the present in the system. In \nzero temperature limit we define $q_{\\text{\\tiny{EA}}}$  as\n\\begin{eqnarray}\n   q_{\\text{\\tiny{EA}}}|_{(T=0)} = \\overline{\\langle{\\hat{n}}_{p,q}\\rangle^2}\n       -\\overline{\\langle{\\hat{n}}_{p,q}\\rangle}^2,\n   \\label{eaop0}\n\\end{eqnarray}  \nwhere we consider expectations only for the ground state.\n\\begin{table}[h!]\n  \\begin{ruledtabular}\n    \\begin{tabular}{lr} \n      \\textbf{Quantum phase} & \\textbf{Order parameter} \\\\\n      \\colrule\n      Superfluid (SF) &  $q_{\\text{\\tiny{EA}}} > 0$, $\\rho_s > 0$, $\\kappa > 0$, $\\phi\\ne 0$ \\\\\n      Mott insulator (MI) & $q_{\\text{\\tiny{EA}}} = 0$, $\\rho_s = 0$, $\\kappa = 0$, $\\phi = 0$ \\\\\n      Bose glass (BG) & $q_{\\text{\\tiny{EA}}} > 0$, $\\rho_s = 0$, $\\kappa > 0$, $\\phi\\ne 0$ \\\\\n      Normal fluid (NF) & $q_{\\text{\\tiny{EA}}} > 0$, $\\rho_s = 0$, $\\kappa > 0$, $\\phi= 0$\\\\\n    \\end{tabular}\n    \\caption{ Classification of quantum phases and the order parameters \n              supported by DBHM at zero and finite temperatures.}\n      \\label{table:tab}\n  \\end{ruledtabular}\n\\end{table}\n\n\n\n\\section{Results and Discussions}\n\\label{results}\n\n  To compute the ground state of the system and determine the phase diagram,\nwe scale the parameters of the DBHM Hamiltonian with respect to the\ninteraction strength $U$. So, the relevant parameters of the model are\n$J\/U$, $\\mu\/U$ and $D\/U$. We, then, determine the phase diagram of the DBHM\nin the $J\/U-\\mu\/U$ plane for different values of $D\/U$, and one unique feature\nof the model is the emergence of the BG phase. The local glassy nature of\nthe BG phase leads to very different properties from the incompressible and\ngapped MI phase, and compressible and gapless SF phase. Thus as mentioned\nearlier, one of the key issues in the study of DBHM is to identify appropriate\norder parameters to distinguish different phases. And, in particular, to\ndetermine the BG phase without ambiguity based on its local properties. To\nconstruct the phase diagram, we consider a $12\\times 12$ square\nlattice superimposed with a homogeneous disorder distribution. \n\nIn DBHM, depending on the magnitude of $D\/U$, the phase diagrams can be\nclassified into three broad categories. First, at low disorder strength\n$D\/U \\leqslant 0.1$, BG phase emerge in the phase diagram. Second, at moderate\ndisorder strengths $0.2\\leqslant D\/U \\leqslant 1$, the domain of BG phase is \nenhanced.  This is the most important regime to explore the physics of BG \nphase. The distinctive features in this range consist of shrinking of MI phase\nand enhancement of the BG phase. Finally, at very high disorder strengths \n$D\/U > 1$, the MI phase disappears and DBHM supports only two phases, BG and SF.\nFor reference the selected zero temperature results are shown in the \nAppendix.\n\n\n\n\n\\subsection{Zero temperature results}\n\\label{t0}\n\n  The synthetic magnetic field arising from the introduction of the artificial \ngauge field localizes the bosons and suppresses their itinerant property. \nThis manifests as a larger MI lobe in the presence of artificial gauge field.  \nHowever, locally the combined effect of disorder and artificial gauge field \nfavours the formation of SF islands. This synergy, then, creates a larger domain \nof BG phase in the phase diagram. In terms of identifying the phase boundaries, \nunlike in the $\\alpha=0$ where  $\\rho_s$ has linear dependence\non $J\/U$ in the SF domain, $\\rho_s$ cannot be used here as it exhibits no dependence \non $J\/U$. The two possible causes of this are: the TBC required to compute \n$\\rho_s$ modifies the magnetic unit cell associated with the chosen value of \n$\\alpha$; and with $\\alpha\\neq 0$ the SF phase contains vortices which reduce \nthe SF phase coherence. So, we use $q_{\\text{\\tiny{EA}}}$ as the order \nparameter to distinguish BG phase from the MI and SF phases. For consistency \nwe compute  $q_{\\text{\\tiny{EA}}}$ both for $\\alpha = 0$ and $\\alpha = 1\/4$ \nemploying SGMF and the results are shown in Fig.~\\ref{eaop-t0}(a)-(d), \nwhere  $q_{\\text{\\tiny{EA}}}$ is shown as a function of $\\mu\/U$ and $J\/U$. \nThe general trend is that  $q_{\\text{\\tiny{EA}}}$ is \nzero in MI and  O$(10^{-3})$ in the SF phase, and O$(10^{-1})$ in BG phase.  From \nthe figure, the presence of the BG phase between different MI lobes is \ndiscernible from the finite values of $q_{\\text{\\tiny{EA}}}$ and it is \nconsistent with the phase diagram determined from $\\rho_s$ shown in \nFig.~\\ref{ph-dia-al0}(g)-(j) in Appendix. We can define sharp MI-BG and SF-BG \nboundaries in the phase diagram by defining a threshold value \nof $q_{\\text{\\tiny{EA}}}$ between the Mott lobes, however, this is \nnon-trivial for the patina of BG phase present at the tip of Mott lobes. \nThis is the domain where the MI-SF quantum phase transition is driven by phase \nfluctuations and consequently, the number fluctuation is highly suppressed. As \na result the value of $q_{\\text{\\tiny EA}}$ is negligible and it cannot be \nused to distinguish BG and SF phases \\cite{buonsante_07,bissbort_10}. Thus, to \nidentify the BG domain it is essential to complement the results from \n$q_{\\text{\\tiny EA}}$ with those of other quantities.\n\n  For $\\alpha = 1\/4$, the region with finite values of $q_{\\text{\\tiny{EA}}}$ \nincreases significantly. This is discernible from the plot of \n$q_{\\text{\\tiny{EA}}}$ in Fig.~\\ref{eaop-t0}(d). For the case of $D\/U = 0.6$, \nwhen $\\alpha = 1\/4$, the $q_{\\text{\\tiny{EA}}}$ is finite with a value \nof $\\approx 0.2$ upto $J\/U \\approx 0.03$. Whereas, with $\\alpha=0$ as shown in \nFig.~\\ref{eaop-t0}(b), $q_{\\text{\\tiny{EA}}}$ has similar value only \nupto $J\/U \\approx 0.02$. This indicates the enhancement of BG region in the\npresence of the artificial gauge field. Employing CGMF method with \n$2 \\times 2$ cluster, the values of the $q_{\\text{\\tiny{EA}}}$ obtained are \nshown in Fig.~\\ref{eaop-t0}(e)-(h). One important change is that, \n$q_{\\text{\\tiny EA}}$ is no longer zero in the MI phase, but it is of \nO$(10^{-6})$. This is due to the presence of particle-hole \nexcitations in the cluster states. And, the non-zero value of \n$q_{\\text{\\tiny EA}}$ is consistent with the results reported in a previous \nwork \\cite{morrison_08}. The figures show similar trends of artificial \ngauge field induced enhancement of the BG region in the phase diagram. The\nincrease of BG regions with the increase of $D\/U$ is also notable for both \n$\\alpha = 0$ and $\\alpha = 1\/4$. Another observation is that, \n$q_{\\text{\\tiny{EA}}}$ obtained \nfrom the CGMF method contains less fluctuations and thus describes the boundary \nof SF-BG transition better compared to the SGMF method. Increasing the cluster \nsize CGMF can describe the BG-SF boundary more accurately but at the cost of\nmuch higher computational resources. \n\\begin{figure}[H]\n\t\\centering\n \\includegraphics[width=7.5cm]{phd-finite-t}\\\\\n\\vskip 0.1cm\n \\includegraphics[width=8cm]{eaop-al0}\n    \\caption{Finite temperature phase diagram using SGMF method\n             in absence of artificial gauge\n             field for six different temperatures (a) $T = 0.005 U\/k_B$, \n             (b)$T = 0.01U\/k_B$, (c) $T = 0.02U\/k_B$, (d) $T = 0.03U\/k_B$,\n             (e)$T = 0.05U\/k_B$ and (f) $T = 0.1U\/k_B$ .\n             Disorder strength is fixed at $D = 0.6U$ and each data in the plot\n             is obtained by averaging over 500 different disorder\n             distributions. (g) shows finite temperature effects on \n             Edward-Anderson order parameter ( $q_{\\text{\\tiny{EA}}}$) \n             with $D\/U = 0.6$ and $J\/U$ being fixed at 0.01. The magnitude \n             of $q_{\\text{\\tiny{EA}}}$ gradually decreases with increase of \n             temperature.}\n    \\label{ph-dia-t}\n\\end{figure}\n\n\n\n\n\n\\subsection{Finite temperature results}\n\\label{ftemp}\n The important outcome of finite temperature is the emergence of a new phase,\nthe NF phase. This new phase, like the SF phase, has real commensurate \nnumber of particles per site. But, unlike SF $\\phi$ is zero. So, the NF phase \nhas some features common to both the MI and SF phases. Previous works reported\nthe appearance of the NF phase at finite temperatures in the case of the \ncanonical Bose-Hubbard model \\cite{gerbier_07}, and extended\nBose-Hubbard model with nearest neighbour interactions \\cite{ng_10, lin_17}.\n\n\n\n\n\\subsubsection{$\\alpha=0$}\n\n The effect of the thermal fluctuations to the $q_{\\text{\\tiny{EA}}}$, in\nabsence of artificial gauge field ( $\\alpha=0$), is shown in \nFig.~\\ref{ph-dia-t}(g).  The \nresults presented in the figure correspond to $D\/U=0.6$  and \neach plot is an average over 500 realizations of disorder distributions. With \nincreasing temperature there is a monotonic decrease in $q_{\\text{\\tiny{EA}}}$,\nwhich indicates the {\\em melting} of BG phase. Along with the BG phase the MI \nphase also melts, however, this is not apparent from the values of \n$q_{\\text{\\tiny{EA}}}$. And, the extent of melting can be inferred from the \nphase diagram. To illustrate this point the phase diagram of DBHM at different \ntemperatures are shown in Fig~\\ref{ph-dia-t}(a-f). As mentioned earlier, \nprevious studies have also reported the melting of MI phase due to thermal \nfluctuations \\cite{gerbier_07}. But a clear theoretical description and phase \ndiagram incorporating finite temperature effects are lacking. Our present work \nshows that the BG phase also melts due to thermal fluctuations. Here, the key \npoint is the SF islands, which are hallmark of the BG phase, melts into NF.\nThis arises from \nthe local nature of the SF islands in BG phase, which as a result is affected \nby the local nature of the thermal fluctuations. The bulk SF phase, on the \nother hand, has long range phase correlations and is more robust against \nlocal fluctuations stemming from finite temperatures. \n\n\\begin{figure}[H]\n\t\\centering\n \\includegraphics[width=8cm]{phd-ctemp}\n    \\caption{Finite temperature phase diagram using CGMF for $2\\times 2$\n             cluster in absence of artificial gauge\n             field for two different temperatures (a) $T = 0.01 U\/k_B$, \n             (b)$T = 0.02U\/k_B$;\n             Disorder strength is fixed at $D = 0.6U$ and each data in the plot\n             is obtained by averaging over 50 different disorder distributions.\n             }\n    \\label{cphd-ft}\n\\end{figure}\n\n \n   In the plots the region within the black line is MI phase, whereas, the \nregion bounded by the black and green lines is the NF phase, where $\\phi$ is \nclose to zero  $\\phi \\leqslant 10^{-6}$. The BG phase lies in the region \nbounded by the green and orange lines, and the area right of the orange line \nis the SF phase. As the temperature is increased, due to the increased thermal \nfluctuations, the phase diagrams undergo several changes. First, the  MI \nlobes shrink and at $k_{\\rm B}T\/U = 0.02$, MI lobes disappear from the phase \ndiagram. This is due to the melting of MI phase and conversion into NF phase. \nSo, as discernible from the comparison of Fig.~\\ref{ph-dia-t}(a) and (b), the \nMI lobe with $\\rho=1$ is bounded and lies in the domain \n$0.40\\leqslant \\mu\/U\\leqslant 0.6$ at $k_{\\rm B}T\/U = 0.005$,\nbut it shrinks to 0.47 $\\leqslant \\mu\/U\\leqslant 0.53 $ at $k_{\\rm B}T\/U = 0.01$.\nSecond, the region of the BG phase is reduced with increasing temperature. The \nchange is more prominent in the regions which lie between the MI lobes. For \nexample, at $\\mu=0$ the BG phase exists in the domain \n$0.004\\leqslant J\/U\\leqslant 0.014$ for $k_{\\rm B}T\/U=0.005$. But, it is \nreduced to  $0.008\\leqslant J\/U\\leqslant 0.015$ when the temperature is \nincreased to $k_{\\rm B}T\/U = 0.01$. As discernible from Fig. ~\\ref{ph-dia-t}(f)\nat $k_{\\rm B}T\/U = 0.1$  the domain is reduced to \n$0.04\\leqslant J\/U\\leqslant 0.043$.  And, third, at finite temperatures the \nMI lobes are bounded from top and bottom by straight lines in the SGMF \nresults. But, as visible from Fig. \\ref{cphd-ft}, the MI boundary is not a \nstraight line with CGMF results. This is on account of the better correlation \neffects in CGMF, in contrast, SGMF tends to support sharp NF-MI boundaries \nas a function of $\\mu\/U$ due to short range coupling through $\\phi$.\n\\begin{figure}[H]\n\t\\centering\n \\includegraphics[width=8cm]{eaop-mu-jp02}\n    \\caption{ $q_{\\text{\\tiny{EA}}}$ as a function of $\\mu\/U$ for four\n             different values of $\\alpha$ at (a) $T = 0$ (b) $T = 0.03U\/k_B$; \n             with fixed disorder strength $D = 0.6U$ and\n             hopping strength $J = 0.02U$.\n             In each subfigure  $q_{\\text{\\tiny{EA}}}$ are calculated for \n             $\\alpha = 0, 1\/12, 1\/6$ and $1\/2$\n             and averaged over 500 different disorder distributions. \n             }\n    \\label{eaop-al}\n\\end{figure}\n\n\n Based on the above observations of the phase diagrams at different \ntemperatures, the NF-BG and BG-SF phase boundaries shift toward higher \n$J\/U$ with increasing temperature. This is due to higher hopping energy \nrequired to prevail over thermal fluctuations. So that the SF phase is present \nas islands or homogeneous network in BG and SF phases, respectively. The \nother important point is that, the SF phase does not melt directly to NF phase.\nIn other words, the BG phase advances into the SF phase with ever decreasing \nwidth with increasing temperature. Thus, the BG phase is an intermediate \nphase between the NF and SF phases. This is the finite temperature equivalent\nof the zero temperature phase structure, where BG phase is an intermediate\nphase between the MI and SF phases.\n\n To improve the accuracy of the phase diagram by incorporating additional \ncorrelation effects, we compute the phase diagram with CGMF using \n$2\\times 2$ cluster, and the resulting phase diagram is shown in \nFig.~\\ref{cphd-ft}. The results are for the temperatures $k_{\\rm B}T\/U = 0.01$ \nand $0.02$, and for better illustration the phase diagrams of only upto \n$\\mu\/U = 1.0$ are shown in the figure. As to be expected the MI lobes are \nlarger in the CGMF results, but the one important change is that the \nenvelope of BG phase around the MI and NF phases is more pronounced. \nConsequent to the larger MI lobes, the NF and BG phases encompass\nregions with higher $J\/U$ compared with the SGMF results. In particular, \nat $\\mu =0$ the BG phase occurs in the domain \n$0.011\\leqslant J\/U \\leqslant 0.018$ and $0.018\\leqslant J\/U \\leqslant 0.022$  \nfor the $k_{\\rm B}T\/U = 0.01$ and $k_{\\rm B}T\/U = 0.02$, respectively.\n\n\n\n\n\\subsubsection{$\\alpha \\ne 0$}\n\n The thermal fluctuations delocalize the atoms through the entire lattice,\nand melt MI phase. This tends to reduce $\\phi$. Whereas, as mentioned earlier,\nartificial gauge field localizes the atoms, and thereby enhances the\nMI lobes. So, these two have opposing effects on the DBHM, and the combined\neffects of these two physical factors on the $q_{\\text{\\tiny{EA}}}$ are shown \nin Fig.~\\ref{eaop-al}. In the figure 4 the plots of $q_{\\text{\\tiny{EA}}}$ for \n$k_{\\rm B}T\/U =0$ and $0.03$ are shown for different $\\alpha$ as a function \n$\\mu\/U$ at $J\/U=0.02$. From the figures it is apparent that the effect of the\nartificial gauge field is negligible in the region between the $\\rho=0$ and\n$\\rho=1$ Mott lobes. However, in the regions between other Mott lobes there is \nan enhancement of the BG phase as indicated by the increase in \n$q_{\\text{\\tiny{EA}}}$. As discernible from Fig.~\\ref{eaop-al}(a) the \nvalue of  $q_{\\text{\\tiny{EA}}}$ increases from $0.13$ to $0.19$ for the \nregion between $\\rho=1$ and $\\rho=2$ corresponding to \n$0.65 \\leqslant \\mu\/U\\leqslant 1.36 $ for non-zero $\\alpha$ at \n$k_{\\rm B}T\/U =0$. From the figure it is also evident that \n$q_{\\text{\\tiny{EA}}}$ gradually increases with the increase of $\\alpha$.\nConsequently, the enhancement of BG phase region in DBHM depends on the\nstrength of artificial gauge field. As a quantitative measure of it, for\n$\\alpha = 0, 1\/12, 1\/6$ and $1\/2$ $q_{\\text{\\tiny{EA}}}$\ntakes the value $0.139$, $0.148$, $0.164$ and $0.187$ respectively around \n$\\mu = U$. To demonstrate the combined effect of finite \ntemperature and artificial gauge field, the phase diagram in terms of \n$q_{\\text{\\tiny{EA}}}$ is shown in Fig. \\ref{eaop-ft}. As the \nfigure is based on $50$ disorder realizations, the general trends of \n$q_{\\text{\\tiny{EA}}}$ observable in Fig.~\\ref{eaop-al} are not apparent. \nHowever, from the figure the enlargement of the BG phase region between the \nMI lobes is discernible. Thus, this implies that the enhancement of the BG \nphase in presence of artificial gauge field is stable against thermal \nfluctuations.\n\\begin{figure}\n\t\\centering\n  \\includegraphics[width=8cm]{eaop_al1b4-cl}\\\\\n    \\caption{ $q_{\\text{\\tiny{EA}}}$ as a function of $\\mu\/U$ and $J\/U$ for\n             $\\alpha =1\/4$ for two different values\n             of temperature $T = 0.01 U\/k_B$ (a) and $T = 0.03 U\/k_B$. (b) \n             Disorder strength is kept fixed at $D= 0.6 U$ and \n             $q_{\\text{\\tiny{EA}}}$ are averaged over 50 different disorder \n             distributions with CGMF method. }\n    \\label{eaop-ft}\n\\end{figure}\n\n\n\n\n\\section{Conclusions}\n\\label{conc}\nAt finite temperatures, the thermal fluctuations lead to melting of the BG \nphase and formation of NF phase. The emergence of the NF phase at finite \ntemperatures necessitates using a combination of order parameters and \nproperties to identify each phase without ambiguity. More importantly, \nthe BG phase is an intermediate phase between the NF and SF phases. \nThis is similar to the zero temperature phase where the BG phase is an \nintermediate phase between the MI and SF phases. At higher temperatures the \nmelting of MI phase is complete and\nonly three phases NF, BG and SF phases exist in the system. The addition of \nartificial gauge field brings about a significant change in the phase diagram \nby enhancing the BG phase domain, which is observed in the trends of \nthe $q_{\\text{\\tiny{EA}}}$ without any ambiguity. This implies that such \nenhancements would be observable in quantum gas microscope experiments. To get \naccurate results with mean field theories it is desirable to use the CGMF \ntheory. It incorporates correlation effects better and the phase diagrams \nobtained from CGMF are quantitatively different from those obtained from SGMF.\n\n\n\n\n\\begin{acknowledgments}\n\nThe results presented in the paper are based on the computations\nusing Vikram-100, the 100TFLOP HPC Cluster at Physical Research Laboratory, \nAhmedabad, India.\n\n\\end{acknowledgments}\n\n\n\n\n\\section*{Appendix}\n\nTo determine the MI-BG phase boundary, we consider number fluctuation \n($\\delta n$) as the property which distinguishes the two phases. In the \nMI phase $\\delta n$ is zero for $D\/U=0$, however, for $D\/U\\neq 0$, it is \nnon-zero but small due to the disorder. We set $\\delta n < 10^{-6}$ as the \ncriterion to identify the MI phase in our computations. On the other hand, to \ndefine the BG-SF boundary, we compute the superfluid stiffness ($\\rho_s$). \nIn BG phase as the SF phase exists as islands the phase coherence is limited\nto these, so the $\\rho_s$ small, and we consider $\\rho_s < 10^{-2}$ as the \nthreshold to distinguish the BG from SF phase. In the SF phase as there is \nphase coherence throughout the system $\\rho_s$ is large and it is $O(1)$.  \n\n\\begin{figure}[H]\n~ \\includegraphics[width=7.8cm]{gstate-t0}\\\\\n  \\includegraphics[width=7.5cm]{phd-ss}\n    \\caption{ Order parameter $\\phi$ of DBHM at zero temperature for \n              $J\/U = 0.01$ and $D\/U$ keeping fixed at 0.6 (a)-(c) without and \n             (d)-(f) \n              with ($\\alpha=1\/4$) artificial gauge field. (a) \\& (d) MI phase \n              with $\\mu\/U = 0.5$; (b) \\& (e) BG phase with  $\\mu\/U = 0.1$; \n              and  (c) \\& (f)SF phase with $\\mu\/U = 1.0$. \n              (g)-(j) equilibrium phase diagram of DBHM using SGMF method at \n              zero temperature in absence of artificial gauge field\n              ($\\alpha=0$) for disorder strengths $D\/U=0$, $0.2$, $0.6$ and\n              $1.2$, respectively for 500 different disorder realizations.\n              }\n    \\label{ph-dia-al0}\n\\end{figure}\n\n The phase diagrams of DBHM with $\\alpha=0$ at  different values of $D\/U$ have \ndistinctive features \\cite{lin_12}. As examples, the phase \ndiagrams for the case of $D\/U = 0$, $0.2$, $0.6$ and $1.2$ obtained from the\nSGMF method are shown in Fig.~\\ref{ph-dia-al0}(g)-(j).  With $D\/U=0$, the phase \ndiagram as shown in Fig.~\\ref{ph-dia-al0}(g) consists of only two phases MI \nand SF. With non-zero $D\/U$ BG appears in the phase diagram, and as shown in \nFig.~\\ref{ph-dia-al0}(h) for $D = 0.2$ the domain of the MI phase shrinks and an\nenvelope of BG phase emerges around the MI lobes. From \nFig.~\\ref{ph-dia-al0}(h), it is clear that the BG phase is most prominent in\nbetween the MI lobes.  These are the domains with large density \nfluctuations and small disorder is sufficient to render the bosons itinerant \nto create islands of SF phase. This, then, leads to the formation of BG phase. \nWhen the $D\/U$ is increased to $0.6$, as shown in Fig.~\\ref{ph-dia-al0}(i), \nthe MI lobes shrink further and the area of the BG phase is enlarged. At \nsufficiently high disorder strength, $D = 1.2U$, the MI phase disappears and \nphase diagram Fig.~\\ref{ph-dia-al0}(j) is composed of only SF and BG phases. \n \n\nThere is an improvement in the phase diagram, which is apparent from the \nenlarged MI lobes, when the phase diagram is computed using CGMF. In particular,\nwe consider $2\\times 2$ cluster and the phase diagrams so obtained are\nshown in  Fig.~\\ref{cph-dia-al0}. The overall structure of the phase diagram\nis qualitatively similar to the SGMF case. However, there are few quantitative\nchanges. For comparison, consider the case of $D\/U = 0.6$, based on our \nresults and as visible in Fig.~\\ref{ph-dia-al0}(i) and \nFig.~\\ref{cph-dia-al0}(b), there are three important difference due to \nbetter correlation effects encapsulated in the CGMF method. First, the tip of \nthe Mott lobe $\\rho=1$ extends upto $0.035$ while it was $0.032$ with SGMF.  \nSecond, at $\\mu\/U \\simeq 0$, the SF-BG transition occurs at $J\/U\\approx  0.022$, \nwhich in the case of SGMF is at $ J\/U\\approx 0.014$. This is due to the \nassociation of BG phase with islands of SF phase, and CGMF captures the phase \ncorrelations in these islands better. The SGMF, on the other hand, tends to \nfavour long range phase correlations through the $\\phi$ coupling between the \nlattices sites. And, finally, around the tip of the Mott lobes, the area of BG \nphase increases in CGMF method.  \n\\begin{figure}[H]\n \\includegraphics[width=8cm]{c-ph-dia}\n    \\caption{Equilibrium phase diagram of DBHM using CGMF method with cluster\n             size 2$\\times$2 at zero temperature in absence of artificial\n             gauge field for disorder strength (a) $D\/U = 0.2$, and  \n             (b) $D\/U = 0.6$ for 50 different disorder realizations.}\n    \\label{cph-dia-al0}\n\\end{figure}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nSoftware-Defined Networking (SDN) decouples the network control plane from the data plane via a well-defined programming interface (such as OpenFlow~\\cite{mckeown2008}).\nThis decoupling allows the control logic to be logically centralized, easing the implementation of network policies, enabling advanced forms of traffic engineering (e.g., \\\\Google's B4~\\cite{jain2013}), and facilitating innovation (network virtualization~\\cite{koponen2014} being a prominent example).\n\nThe controllers are the crucial enabler of the SDN paradigm: they maintain the logically centralized network state to be used by applications and act as a common intermediary with the data plane.\nFigure \\ref{fig:intro_sdn} shows the normal execution in an SDN environment.\nUpon receiving a packet it does not know how to handle, the switch sends an \\emph{event} to the controller.\nThe controller delivers the event to the applications, which afterwards apply their logic based on this event, and eventually instruct the controller to send \\emph{commands} to the switches (e.g., to install flow rules).\n\n\\begin{figure}[h!]\n    \\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/intro_sdn.png}\n    \\caption[SDN flow execution]{SDN flow execution. Switches send events to the controller as needed and the controller replies with one or more commands that modify the switch's tables.}\n    \\label{fig:intro_sdn}\n\\end{figure}\n\nA trivial implementation of SDN using a centralized controller would lead to an undesirable outcome: a single point of failure.\nTo guarantee the required availability of network control, it is necessary the controller platform to be made fault-tolerant.\nFault tolerance demands transparency: for a controller that claims having such ability -- in other words, for it to be logically centralised -- applications that run on it should operate correctly in the presence of faults.\nThis is a fundamental requirement, as in case of controller failures the network view needs to be maintained consistent, otherwise applications will operate in a stale network view, leading to network anomalies that can have undesirable consequences (e.g., security breaches)~\\cite{levin2012, katta2015}.\n\nTo address this problem, traditional replication techniques are usually employed~\\cite{oki1988, lamport1998, ongaro2014}.\nHowever, building a consistent network view in the controllers is not enough to offer consistent logically centralized control that prevents the above-mentioned anomalies.\nIn SDN, it is necessary to include switch state into the system model to achieve this goal~\\cite{katta2015}.\nSince switches are programmed by controllers (and controllers can fail), there must be mechanisms to ensure that the entire event-processing cycle of SDN is handled consistently.\n\nA correct, fault-tolerant SDN environment needs to ensure \\emph{observational indistinguishability}~\\cite{katta2015} between an ideal central controller and a replicated controller platform.\nInformally, to ensure observational indistinguishability the fault-tolerant system should behave the same way as a fault-free SDN for its users (end-hosts and network applications).\nFot this purpose, it is necessary the following properties to be met:\n\n\\begin{itemize}\n\\item \\textbf{Total Event Ordering:} Controller replicas should process events in the same order and subsequently all controller application instances should reach the same internal state\n\\item \\textbf{Exactly-Once Event Processing:} All the events are processed, and neither are lost nor processed repeatedly.\n\\item \\textbf{Exactly-Once Execution of Commands:} Any given series of commands are executed once, and only once on the switches.\n\\end{itemize}\n\nTo the best of our knowledge, the problem of correct, fault-tolerant SDN control has only been addressed in the work by Katta~\\emph{et al.}~\\cite{katta2015}.\nInstead of just keeping the controller state consistent, the authors propose Ravana, a fault-tolerant SDN controller platform that handles the entire event-processing cycle as a transaction -- either all or none of the components of this transaction are executed.\nThis enables Ravana to correctly handle switch state and thus guarantee SDN correctness even under fault.\n\nTo achieve these properties, however, Ravana requires modifications to the OpenFlow protocol and to existing switches.\nSpecifically, switches need to maintain two buffers, one for events and one for commands, and four new messages need to be added to the protocol.\nThese modifications preclude the adoption of Ravana on existing systems and hinder the possibility of it being used in the near future (as there are no guarantees these messages be added to OpenFlow anytime soon, for instance).\n\nFaced with this challenge, we propose Rama, a fault-tolerant SDN controller platform that, similar to Ravana, offers a transparent control plane that allows unmodified network applications to run in a consistent and fault-tolerant environment.\nThe novelty of the solution lies in Rama not requiring changes to OpenFlow nor to the underlying hardware, allowing immediate deployment.\nFor this purpose, Rama exploits existing mechanisms in OpenFlow and orchestrates them to achieve its goals.\n\nThe main contributions of this work can be summarized as follows:\n\n\\begin{itemize}\n\\item A protocol for fault-tolerant SDN that provides the correctness  guarantees of a logically centralised controller \\emph{without} requiring changes to OpenFlow or modifications to switches.\n\\item The implementation and evaluation of a prototype controller -- Rama -- that demonstrates the overhead of the solution to be modest.\n\\end{itemize}\n\n\\section{Fault tolerance in SDN}\n\\label{motivation}\n\nKatta~\\emph{et al.} have experimentally shown~\\cite{katta2015} that traditional techniques for replicating controllers do not ensure correct network behaviour in case of failures.\nThe reason is that these techniques address only part of the problem: maintaining consistent state in controller replicas.\nBy not considering switch state (and the interaction controller-switches) inconsistencies may arise, resulting in potentially severe network anomalies.\nIn this section we present a summary of the problems of using techniques that do not incorporate switches in the system model, which lead to the design requirements of a \\emph{correct} fault-tolerant SDN solution. \nWe also present Ravana~\\cite{katta2015}, the first fault-tolerant controller that achieves the required correctness guarantees for SDN.\n\n\\subsection{Inconsistent event ordering}\n\nSince OpenFlow 1.3, switches can maintain TCP connections with multiple controllers.\nIn a fault-tolerant configuration switches can be set to send all their events to all known controller replicas.\nAs replicas process events as they are received, each one may end up building a different internal state.\nAlthough TCP guarantees the order of events delivered by each switch, there are no ordering guarantees between events sent to controllers by the different switches, leading to the problem.\n\nConsider a simple scenario with two controller replicas (c1 and c2) and two switches (s1 and s2) that send all events to both controllers.\nSwitch s1 sends two events -- e1 and e2, in this order -- and switch s2 sends two other events --  e3 and e4, in this order.\nOne possible outcome where both controllers receive events in a different order while respecting the TCP FIFO property is c1 receiving events in the order e1, e3, e2, e4 and c2 receiving in the order e3, e4, e1, e2.\nUnfortunately, an inconsistent ordering of events can lead to incorrect packet-processing decisions~\\cite{katta2015}.\nAs a result of this consistency problem we derive the first design goal for a fault-tolerant and correct SDN controller:\n\n\\textbf{Total event ordering:} controllers replicas should process the same (total) order of events and subsequently all controller application instances should reach the same internal state.\n\n\\subsection{Unreliable event delivery}\n\nIn order to achieve a total ordering of events between controller replicas two approaches can be used:\n\n\\begin{enumerate}\n\\item The master (primary) replica can store controller state (including state from network applications) in an external consistent data-store (as in Onix \\cite{koponen2010}, ONOS \\cite{berde2014}, and SMaRtLight \\cite{botelho2014});\n\\item The controller state can be kept consistent using replicated state machine protocols.\n\\end{enumerate}\n\nAlthough both approaches ensure a consistent ordering of events between controller replicas, they are not fault-tolerant in the standard case where only the master controller receives all events.\n\nIf we consider \u2013- for the first approach -- that the master replica can fail between receiving an event and finishing persisting the controller state in the external data-store (which happens after processing the event through controller applications), that event will be lost and the new master (i.e., one of the other controller replicas) will never receive it.\nThe same can happen in the second approach: the master replica can fail right after receiving the event and before replicating it in the shared log (which in this case happens before processing the event through the controller applications).\nIn these cases, since only the crashed master received the event, the other controller replicas will not have an updated view of the network.\nAgain, this may cause severe network problems~\\cite{katta2015}. \nSimilar problems can occur in case of repetition of events.\nThese problems lead to the second design goal:\n\n\\textbf{Exactly-once event processing:} All the events sent by switches are processed, and are neither lost nor processed repeatedly.\n\n\\subsection{Repetition of commands}\n\nIn either traditional state machine replication or consistent storage approaches, if the master controller fails while sending a series of commands, the new elected master may send repeated commands.\nThis may happen when the old master fails before informing the slave replica of its progress.\nSince some commands are not idempotent~\\cite{katta2015}, its duplication can lead to undesirable network behaviour. \nThis problem leads to the third and final design goal:\n\n\\textbf{Exactly-once command execution:} any series of commands are executed only once on the switches.\n\n\\subsection{Ravana}\n\nRavana~\\cite{katta2015} is the first controller to provide correct fault-tolerant SDN control.\nTo achieve this, Ravana processes control messages transactionally and exactly once (at both the controllers and the switches) using a replicated state machine approach, but without involving the switches in an expensive consensus protocol.\n\nThe protocol used by Ravana is show in Figure \\ref{fig:ravana-protocol}.\nSwitches buffer events (as they may need to be retransmitted) and send them to the master controller that will replicate them in a shared log with the slaves.\nThe controller will then reply back to the switch acknowledging the reception of the events.\nThen, events are delivered to applications that may after processing require one or more commands to be sent to switches.\nSwitches reply back to acknowledge the reception of these commands and buffer them to filter possible duplicates.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/ravana-protocol.jpg}\n  \\caption[Ravana Protocol]{Ravana protocol. In Ravana switches maintain two buffers (displayed on the left) to re-transmit events and filter repeated commands in case of master failure. New acknowledge messages (\\texttt{ack\\_event} and \\texttt{ack\\_cmd}) are exchanged between the switch and the master to guarantee the consistency requirements.}\n  \\label{fig:ravana-protocol}\n\\end{figure}\n\nWhile Ravana allows unmodified applications to run in a fault-tolerant environment, it requires modifications to the OpenFlow protocol and to switch hardware.\nNamely, Ravana leverages on buffers implemented on switches to retransmit events and filter possible repeated commands received from the controllers.\nAlso, explicit acknowledgement messages must be added to the OpenFlow protocol so that the switch and the controller acknowledge received messages.\nUnfortunately, these requirements preclude immediate adoption of Ravana.\nFor instance, it is not antecipated OpenFlow to be extended to include the required messages anytime soon.\nThese limitations are the main motivation for our proposal, which we present next.\n\n\\section{Rama design}\n\\label{design}\n\nThe goal of our work is to build a strongly consistent and fault-tolerant control plane for SDN to be used transparently by unmodified applications.\nThis section describes the architecture and protocol for such control plane, which is driven by the following four requirements. First, reliability: the system should maintain a correct and consistent state even in the presence of failures (in both the controllers and switches). Second, transparency: the consistency and fault-tolerance properties should be completely transparent to applications. Third, performance: the performance of the system should not degrade as the number of network elements (events and switches) grows. Fourth, immediate deployability: the solution should work with existing switches and not require new additions to the OpenFlow protocol.\n\n\\subsection{Architecture}\n\nThe high-level architecture of our system, Rama\\footnote{In the Hindu epic Ramayana, Rama is the hero whose wife (Sita) is abducted by Ravana.}, is depicted in Figure \\ref{fig:arquitectura_alto_nivel}.\nIts main components are: (i) OpenFlow enabled switches (switches that are implemented according to the OpenFlow specification), (ii) controllers that manage the switches and (iii) a coordination service.\nIn our model, we consider only one network domain with one primary controller and one or more backup controllers, depending on the number of faults to tolerate.\nEach switch connects to one primary controller and multiple (\\textit{f} to be precise) backup controllers (to tolerate up to \\emph{f} crash controller faults).\nThis primary\/backup model is supported by OpenFlow in the form of master\/slave and allows the system to tolerate controller faults.\nWhen the master controller fails, the remaining controllers will elect a new leader to act as the new master for the switches managed by the crashed master.\nThis election is supported by the coordination service.\n\n\\begin{figure}[t]\n    \\includegraphics[width=0.2\\textwidth]{figures\/architecture.png}\n    \\centering\n    \\caption[High level architecture of the system]{High level architecture of the system.}\n    \\label{fig:arquitectura_alto_nivel}\n\\end{figure}\n\nThe coordination service offers strong consistency and abstracts controllers from complex primitives like fault detection and total order, making them simpler and more robust.\nNote that the coordination system requires a number of replicas equal to \\textit{2f+1}, with \\textit{f} being the number of faults to tolerate.\nThe strong consistency model assures that updates to the coordination service made by the master will only return when they are persistently stored.\nThis means that slaves will always have the fresh modifications available as soon as the master receives confirmation of the update.\nThis results in a consistent network view among all controllers even if some fail.\nThe need for agreement between several replicas make the coordination service the system bottleneck~\\cite{hunt2010}.\nIn addition to the controllers' state, the switches also maintain state that must be handled consistently in the presence of faults.\nFulfilling this request is the main goal of the protocol we present next.\n\n\\subsection{Rama protocol}\n\\label{sec:ft_protocol}\n\nIn an SDN setting, switches generate events (e.g., when they receive packets or when the status of a port changes) that are forwarded to controllers.\nThe controllers run multiple applications that process the received events and may send commands to one or more switches in reply to each event.\nThis cycle repeats itself in multiple switches across the network as needed.\n\nIn order to maintain a correct system in the presence of faults, one must handle the state in the controllers and the state in the switches consistently.\nTo ensure this, the entire cycle presented in Figure \\ref{fig:control_loop} is processed as a \\emph{transaction}: either all or none of the components of this transaction are executed.\nThis means that\n\\begin{inlinelist}\n\\item the events are processed exactly once at the controllers,\n\\item all controllers process events in the same (total) order to reach the same state, and\n\\item the commands are processed exactly once in the switches.\n\\end{inlinelist}\nBecause the standard operation in OpenFlow switches is to simply process commands as they are received, the controllers must coordinate to guarantee the required exactly-once semantics.\nRavana~\\cite{katta2015} does not need this coordination because the (modified) switches can simply buffer the commands received and discard repeated commands (i.e., those with the same identifier) sent by the new controller.\n\n\\begin{figure}[h]\n    \\includegraphics[width=0.45\\textwidth]{figures\/control_loop.png}\n    \\centering\n    \\caption[Control loop]{Control loop of (1) event delivery, (2) event ordering, (3) event processing, and (4) command execution. Events are delivered to the master controller, which decides a total order on the received events. The events are processed by applications in the same order in all controllers. Applications issue commands to be executed in the switches.}\n    \\label{fig:control_loop}\n\\end{figure}\n\nBy default, in OpenFlow a master controller receives all asynchronous messages (e.g., \\texttt{OFPT{\\_}PACKET{\\_}IN}), whe\\-re\\-as the slaves controllers only receive a subset (e.g., port modifications).\nWith this configuration only the master controller would receive the events generated by switches.\nThere are two options to solve this problem.\nOne is for slaves to change this behaviour by sending an \\texttt{OFPT{\\_}SET{\\_}ASYNC} message to each switch that modifies the asynchronous configuration.\nAs a result, switches send all required events to the slaves.\nAlternatively, all controllers can set their role to \\texttt{EQUAL}.\nThe OpenFlow protocol specifies that switches should send all events to every controller with this role.\nThen, controllers need to coordinate between themselves who the master is (i.e., the one that processes and sends commands).\nWe have opted for the second solution and use the coordination service for leader election amongst controllers.\n\nThe fault-free execution of the protocol is represented in Figure \\ref{fig:rama-protocol}.\nIn the figure we consider a switch to be connected with one master controller and a single slave controller.\nThe main idea is that switches must send messages to \\textit{all controllers}, so that they can coordinate themselves even if some fail at any given point.\nIn Ravana, because switches simply buffer events (so that they can be retransmitted to a new master if needed), switches can send events only to the current master, instead of to every controller.\n\n\\begin{figure}[ht!]\n    \\includegraphics[width=0.45\\textwidth]{figures\/rama-protocol.png}\n    \\centering\n    \\caption[Fault-free case of the protocol]{Fault-free case of the protocol. Switches send generated events to all controllers so that no event is lost. The master controller replicates the event in the shared log and then feeds its applications with the events in log order. Commands sent are buffered by the switch until the controller sends a Commit Request. The corresponding Commit Reply message is forwarded to all controllers.}\n    \\label{fig:rama-protocol}\n\\end{figure}\n\nThe master controller then replicates the event in a shared log with the other controllers, imposing a total order on the events received (to simplify, the coordination service is omitted from the figure).\nWhen the event is replicated to the shared log controllers, it is processed by the master controller applications, which will generate zero or more commands.\nTo guarantee exactly-once semantics, the commands are sent to the switches in bundles (a feature introduced in OpenFlow 1.4, see Figure \\ref{fig:openflow-bundles}).\nWith this feature a controller can open a bundle, add multiple commands to it and then instruct the switch to commit all commands present in the bundle in an atomic and ordered fashion.\n\n\\begin{figure}[ht!]\n    \\includegraphics[width=0.45\\textwidth]{figures\/openflow-bundles.png}\n    \\centering\n    \\caption[OpenFlow Bundles]{OpenFlow Bundles.}\n    \\label{fig:openflow-bundles}\n\\end{figure}\n\nRama uses bundles in the following way.\nWhen an event is processed by all modules, the required commands are added by the master controller to a bundle.\nThe master then sends an \\texttt{OFPBCT\\_COMMIT\\_REQUEST} message to each switch affected by the event.\nThe switch processes the request and tries to apply all the commands in the bundle in order.\nAfterwards, it then sends a reply message indicating if the Commit Request was successful or not.\nThis message is used by Rama as an acknowledgement.\n\nAgain, we need to make sure that this reply message is sent to all controllers.\nThis is a challenge, because Bundle Replies are Controller-to-Switch messages and hence are only sent to the controller that made the request (using the same TCP connection).\nTo overcome this challenge we introduce a new mechanism in Rama.\nThe way we inform other controllers if the bundle was committed or not (so that they can decide later if they need to resend specific commands) is by including one \\texttt{OFPT{\\_}PACKET{\\_}OUT} message in the end of the bundle with the action \\texttt{output=controller}.\nThe outcome is that the switch will send the information included in the \\texttt{OFPT{\\_}PACKET{\\_}OUT} message to all connected controllers in a \\texttt{OFPT{\\_}PACKET{\\_}IN} message.\nThis message is set by the master controller to inform slave controllers about the events that were fully processed by the switch (in this bundle).\nThis prevents a new master from sending repeated commands, thus guaranteeing exactly-once semantics.\nRavana does not need to rely on bundles since switches buffer all received commands so that they can discard possible duplicates from a new master.\n\nThe master finishes the transaction by replicating an \\texttt{event} \\texttt{processed} \\texttt{message} in the log, informing backup controllers that they can safely feed the corresponding event in the log to their applications.\nThis is done to simply bring the slaves to the same updated state as the master controller (the resulting commands sent by the applications are naturally discarded).\n\n\\subsubsection{Fault cases}\n\\label{subsec:fault-cases}\n\nWhen the master controller fails, the backup controllers will detect the failure (by timeout) and run a leader election algorithm to elect a new master for the switches.\nUpon election, the new master must send a Role Request message to each switch, to register as the new master.\nThere are three main cases where the master controller can fail:\n\n\\begin{enumerate}\n\\item Before replicating the received event in the distributed log (Figure \\ref{fig:rama-protocol-fault-case-1});\n\\item After replicating the event but before sending the Commit Request (Figure \\ref{fig:rama-protocol-fault-case-2});\n\\item After sending the Commit Request message.\n\\end{enumerate}\n\n\\begin{figure}[h]\n    \\includegraphics[width=0.45\\textwidth]{figures\/rama-protocol-fault-case-1.png}\n    \\centering\n    \\caption[Failure case 1 of the protocol]{Case of the protocol where the master fails before replicating the event received. Because the slaves buffer all events, the event is not lost and the new master can resume the execution of the failed controller.}\n    \\label{fig:rama-protocol-fault-case-1}\n\\end{figure}\n\nIn the first case, the master failed to replicate the received events to the shared log.\nAs slave controllers receive and buffer all events, no events are lost.\nFirst, the new master must finish processing any events logged by the older master.\nNote that events marked as processed have their resulting commands filtered.\nThis makes the new master reach the same internal state as the previous one before choosing the new order of events to append to the log (this is valid for all other fault cases).\nThe new elected master then appends the buffered events in order to the shared log and continues operation (feeding the new events to applications and sending commands to switches).\n\nIn the cases where the event was replicated in the log (cases 2 and 3), the master that crashed may or may not have issued the Commit Request message.\nTherefore, the new master must carefully verify if the switch has processed everything it has received before re-sending the commands and the Commit Request message.\nTo guarantee ordering, OpenFlow provides a Barrier message, to which a switch can only reply after processing everything it has received before.\nIf a new master receives a Barrier Reply message without receiving a Commit Reply message (in form of \\texttt{OFPT{\\_}PACKET{\\_}OUT}), it can safely assume that the switch did not receive nor execute a Commit Request for that event from the old master (case 2)\\footnote{This relies on the FIFO properties of the controller-switch TCP connection.}.\nEven if the old master sent all commands but did not send the Commit Request message, the bundle will never be committed and will eventually be discarded.\nTherefore, the new master can safely resend the commands.\nIn case 3, since the old master sent the Commit Request before crashing, the new master will receive the confirmation that the switch processed the respective commands for that event and will not resend them (guaranteeing exactly-once semantics for commands).\n\n\\begin{figure}[ht]\n    \\includegraphics[width=0.45\\textwidth]{figures\/rama-protocol-fault-case-2.png}\n    \\centering\n    \\caption[Failure case 2 of the protocol]{Case of the protocol where the master fails after replicating the event. The first part of the protocol is identical to the fault-free case and is omitted from the figure. In this case, the crashed master may have already sent some commands or even the Commit Request to the switch.}\n    \\label{fig:rama-protocol-fault-case-2}\n\\end{figure}\n\n\\renewcommand{\\arraystretch}{1.7}\n\\newcolumntype{A}{ >{\\centering\\arraybackslash} m{.28\\linewidth} }\n\\newcolumntype{B}{ >{\\centering\\arraybackslash} m{.32\\linewidth} }\n\\newcolumntype{C}{ >{\\centering\\arraybackslash} m{.32\\linewidth} }\n\\begin{table*}[t!]\n\\begin{tabular}{ABC}\n\\hline\n\\vspace{-4px}\\textbf{Property} & \\vspace{-4px}\\textbf{Ravana} & \\vspace{-4px}\\textbf{Rama}\\\\\n\\hline\n\\textit{At least once events} & Buffering and retransmission of switch events & Switches send events to every controller with role EQUAL\\\\\n\\hline\n\\textit{At most once events} & \\multicolumn{2}{c}{Event IDs and filtering in the log}\\\\\n\\hline\n\\textit{Total event order} & \\multicolumn{2}{c}{Master appends events to a shared log}\\\\\n\\hline\n\\textit{At least once commands} & RPC acknowledgments from switches & \\multirow{2}{0.3\\textwidth}[-0.1cm]{\\centering{Bundle commit is known by every controller by piggybacking PacketOut in OpenFlow Bundle}}\\\\\n\\cline{1-2}\n\\textit{At most once commands} & Command IDs and filtering at switches & \\\\\n\\hline\n\\end{tabular}\n\\caption{How Rama and Ravana achieve the same consistency properties using different mechanisms}\n\\label{tab:rama-properties}\n\\end{table*}\n\n\n\\section{Correctness}\n\nThe Rama protocol we propose in this paper was designed to guarantee correctness of fault-tolerant SDN control.\nWe define correctness as in~\\cite{katta2015}, where the authors introduce the concept of observational indistinguishability in the SDN context, defined as follows:\n\n\\emph{Observational indistinguishability:} If the trace of observations made by users in the fault-tolerant system is a possible trace in the fault-free system, then the fault-tolerant system is observationally indistinguishable from a fault-free system.\n\nFor observational observability, it is necessary to guarantee transactional semantics to the entire control loop, including (i) exactly-once event delivery, (ii) event ordering and processing, and (iii) exactly-once command execution.\nIn this section we summarize how the mechanisms employed by our protocol fulfil each of these necessary requirements.\nFor a brief comparison with Ravana, see Table \\ref{tab:rama-properties}.\n\n\\textbf{Exactly once event processing:} events cannot be lost (processed \\textit{at least once}) due to controller faults nor can they be processed repeatedly (they must be processed \\textit{at most once}).\nContrary to Ravana, Rama does not need switches to buffer events neither that controllers acknowledge each received event to achieve \\textit{at-least once event processing} semantics.\nInstead, Rama relies on switches sending the generated events to \\textit{all (f+1)} controllers (considering that the system tolerates up to \\emph{f} crash faults) so that at least one will known about the event.\nUpon receiving these events, the master replicates them in the shared log while the slaves add the events to a buffer.\nAs such, in case the master fails before replicating the events, the new elected master can append the buffered events to the log.\nIf the master fails after replicating the events, the slaves will filter the events in the buffer to avoid duplicate events in the log.\nThis ensures \\textit{at-most once event processing} since the new master only processes each event in the log once.\nTogether, sending events to all controllers and filtering buffered events ensures \\textit{exactly-once event processing}.\n\n\\textbf{Total event ordering:} to guarantee that all controller replicas reach the same internal state, they must process any sequence of events in the same order.\nFor this, both Rama and Ravana rely on a shared log across the controller replicas (implemented using the external coordination service) which allows the master to dictate the order of events to be followed by all replicas.\nEven if the master fails, the new elected master always preserves the order of events in the log and can only append new events to it.\n\n\\textbf{Exactly once command execution:} for any given event received from one switch, the resulting series of commands sent by the controller are processed by the affected switches exactly \\textit{once}.\nHere, Ravana relies on switches acknowledging and buffering the commands received from controllers (to filter duplicates).\nAs this requires changes to the OpenFlow protocol and to switches, Rama relies on OpenFlow Bundles to guarantee transactional processing of commands.\nAdditionally, the Commit Reply message, which is triggered after the bundle finishes, is sent to \\textit{all} controllers and thus acts as an acknowledgement that is independent of controller faults. If the master fails, the new master needs to know if it should resend the commands for the logged events or not.\nA Packet Out message at the end of the bundle acts as a Commit Reply message to the slave controllers.\nThis way, upon becoming the new master, the controller replica has the required information to know if the switch processed the commands inside the bundle or not, without relying on the crashed master.\nFurthermore, the new master sends a Barrier Request message to the switch.\nReceiving the corresponding Barrier Reply message guarantees that neither the switch nor the link are slow (because a message was received and TCP maintains FIFO order) and thus there is no possibility of the Packet Out being delayed.\nTherefore, the use of Bundles that include a Packet Out at the end, in addition to the Barrier message ensures that commands will be processed by the switches \\textit{exactly-once}.\n\nIt is important to note that we also consider the case where switches fail.\nHowever, this is not a special case of the protocol because it is already treated by the OpenFlow protocol under normal operation.\nA switch failure will generate an event in the controller which will be delivered to applications, for them to act accordingly (e.g., re-route traffic around the failed switch).\nA particularly relevant case is when a switch fails before sending the Commit Reply to the master and the slave controllers.\nImportantly, this event does not result in transaction failure.\nSince this is a normal event in SDN, the controller replicas simply mark pending events for the failed switch as processed and continue operation.\n\nWhile we detail our reasoning as to why our protocol meets the correctness requirements of observational indistinguishability in SDN, modelling the Rama protocol and giving a formal proof is left as future work and out of the scope of this paper.\n\n\\section{Implementation}\n\\label{implementation}\n\nWe have built Rama on top of Floodlight~\\cite{floodlight}.\nFor coordination, we opted for ZooKeeper~\\cite{hunt2010}.\nThis service abstracts controllers from fault detection, leader election, and event transmission and storage (for controller recovery).\nRama introduces two main modules into Floodlight: the \\textit{Event Replication} module (Section \\ref{sec:event-replication}) and the \\textit{Bundle Manager} module (Section \\ref{sec:bundle-manager}).\nAdditionally, the Floodlight architecture was optimised for performance by introducing parallel network event collection and logging (Rama's multi-thread architecture is shown in Figure \\ref{fig:rama-threads}) and by batching events (Section \\ref{sec:event-batching}).\nThe multi-thread paralelism is introduced carefully, not to break TCP FIFO order of event processing, as will be explained next.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/rama-architecture.png}\n  \\caption{Rama thread architecture}\n  \\label{fig:rama-threads}\n\\end{figure}\n\n\nIn the original Floodlight, worker threads are used to collect network events and to process the modules pipeline (in Floodlight network applications are called ``modules'').\nThis design precludes event batching and other optimisations.\nIdeally, we want to free the threads that collect network events as soon as possible so that they can keep collecting more events.\nFor this purpose, the worker threads' only job in Rama is to push events to the Replication Queue.\nEvents for a particular switch are collected always by the same thread (although each thread can be shared by several switches) and thus TCP FIFO order is guaranteed in the Replication Queue.\nNext, the Rama runtime imposes a total order on the events by giving them a monotonically increasing ID.\nAs such, several Replication threads can then take events from this queue and execute the logic in the Event Replication module, which will send the events to ZooKeeper in batches, without breaking the required total order for correctness.\nThis technique is equivalent to Ravana's parallel event logging~\\cite{katta2015}. \nWhen ZooKeeper replies to the request, the events are added to the Pipeline Queue to be processed by the Floodlight modules.\nA single thread is used in this step, to guarantee the total order.\nThe slave replicas also follow the total order from the IDs assigned by the master.\n\nOne of our requirements was to make the control plane transparent for applications to execute unmodified.\nThe Event Replication module is transparent to other modules as it acts before the pipeline.\nThe modules will continue to receive events as usual in Floodlight and process them by changing their internal structures and sending commands to switches.\nThe process of sending messages inside OpenFlow Bundles as required by Rama is also made completely transparent to Floodlight modules, as will be explained in Section \\ref{sec:bundle-manager}.\n\n\\subsection{Event Replication and ZK Manager}\n\\label{sec:event-replication}\n\nThe Event Replication module is the bridge between receiving events from the worker threads and pushing them into the pipeline queue to be processed by Floodlight modules.\nEvents are only added to the pipeline queue after being stored in ZooKeeper.\nTo separate tasks, Event Replication leverages on the ZK Manager, an auxiliary class that acts as ZooKeeper client (establishing connection, making requests and processing replies) and keeps state regarding the events (an event log and an event buffer in case of slaves) and switch leadership.\nEvent Replication and the ZK Manager work together to attain exactly-once event delivery and total order as follows.\n\nWhen an event arrives at the Event Replication module, we check whether the controller is in master or slave mode.\nIn master mode the event is replicated in ZooKeeper and added to its in-memory log.\nThis log is a collection of \\texttt{RamaEvent} objects which, apart from the switch information and message content, contains the unique event identifier explained before.\nThe events are replicated in ZooKeeper in batches (see Section \\ref{sec:event-batching}), so each replication thread simply adds an event to the current batch and becomes free to process a new event.\nEventually the batch will be sent to ZooKeeper containing one or more events to be stored.\nUpon receiving the reply, the events are pushed to the pipeline queue, ordered according to the identifier given by the master to guarantee total order.\n\nIn slave mode, the event is simply buffered in memory (to be used in the case where the master controller fails).\nA special case is when the event received is the Packet Out that the master controller included in the bundle.\nIn this case, the slave marks that this switch already processed all commands for this event.\nSlaves also keep an event log as the master, but only events that come from the master are added to it.\nEvents from the master arrive via \\textit{watches} set in ZooKeeper nodes.\nSlaves set watches and are notified when the master creates event nodes under that node. \nNew events are added to the in memory log (so it is kept up-to-date with the log maintained by the master) and the events are added to the pipeline queue in the same way as in the master controller.\nAn important detail is that event identifiers are set by the master controller, and when slaves deserialize the data obtained from nodes stored in ZooKeeper they get the same \\texttt{RamaEvent} objects created by the master.\nTherefore, the events will be queued in the same order as they were in the master controller replica.\n\n\\subsection{Bundle Manager}\n\\label{sec:bundle-manager}\n\\begin{sloppypar}\nThe Bundle Manager module keeps state related to the open bundles for each switch (as result of an event) and is responsible for adding messages to the bundle, closing and committing it.\nTo guarantee transparency to applications, we modified the write method in \\texttt{OFSwitch.java} (the class that is used by all modules to send commands to switches) to call the Bundle Manager.\nThis module will wrap the message sent by application modules in a \\texttt{OFPT{\\_}BUNDLE{\\_}ADD\\_MESSAGE} and send it to the switch.\nThis process is transparent because applications are unaware of the Bundle Manager module.\n\\end{sloppypar}\n\\begin{sloppypar}\nIn the end of the pipeline, the Bundle Manager module is thus called to prepare and commit the bundles containing the commands instructed by the modules as a response to this event.\nNote that one event may cause modules to send commands to multiple switches, so in this step the Bundle Manager may send \\texttt{OFPBCT\\_COMMIT\\_REQUEST} to one or more switches.\nBefore committing the bundle, the Bundle Manager also adds a \\texttt{OFPT\\_PACKET\\_OUT} message to it, so that slave controllers will know if the commands for an event were committed or not in the switch (as explained in Section \\ref{sec:ft_protocol}). \nThis message will be received by the slave controllers as a \\texttt{OFPT\\_PACKET\\_IN} message with the required information set by the master controller.\n\\end{sloppypar}\n\n\\subsection{Event batching}\n\\label{sec:event-batching}\n\nFloodlight thread architecture was modified to allow event batching, for performance reasons.\nConsidering that ZooKeeper is running on a separated machine from the master controller replica, sending one event at a time to ZooKeeper would significantly degrade performance.\nTherefore, the ZKManager groups events before sending them to ZooKeeper in batches.\nBatches are sent to ZooKeeper using a special request called \\texttt{multi}, which contains a list of operations to execute (e.g., create, delete, set data).\nFor event replication, the multi request will have a list with multiple create operations as parameter.\nThis request is sent after reaching the maximum configured amount of events (e.g., 1000) or some time after receiving the first event in the batch (e.g., 50ms).\nThis means that each event has a maximum delay bound (regarding event batching).\n\n\\section{Evaluation}\n\\label{evaluation}\n\nIn this section we evaluate Rama to understand its viability and the costs associated with the mechanisms used to achieve the desired consistency properties (without modifying the OpenFlow protocol or switches).\n\n\\subsection{Setup}\n\nFor the evaluation we used 3 machines connected to the same switch via 1Gbps links as shown in Figure \\ref{fig:setup}.\nEach machine has an Intel Xeon E5-2407 2.2GHz CPU and 32 GB (4x8GB) of memory.\nMachine 1 runs one or more Rama instances, machine 2 runs ZooKeeper 3.4.8, and machine 3 runs Cbench to evaluate controller performance.\nThis setup tries to emulate a scenario similar to a real one with ZooKeeper on a different machine for fault-tolerance purposes, and Cbench on a different machine to include network latency. \n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{figures\/setup.png}\n  \\caption{Experiment setup}\n  \\label{fig:setup}\n\\end{figure}\n\n\\begin{figure*}[h!]\n\t\\centering\n    \\begin{subfigure}[t]{0.25\\linewidth}\n        \\centering\n        \\resizebox{\\columnwidth}{!}{\n            \\begin{tikzpicture}\n\t\t\t\t\\begin{axis}[\n\t\t\t\t\t\tybar=10pt,\n\t\t\t\t\t\tbar width=12pt,\n\t\t\t\t\t\tx=1.8cm,\n\t\t\t\t\t\tymin=0,\n\t\t\t\t\t\taxis on top,\n\t\t\t\t\t\tymax=60,\n\t\t\t\t\t\tylabel=Throughput (K Responses\/s),\n\t\t\t\t\t\txtick=data,\n\t\t\t\t\t\tenlarge x limits=0.6,\n\t\t\t\t\t\n                        symbolic x coords={Ravana,Rama},\n\t\t\t\t\t\taxis lines*=left,\n\t\t\t\t\t\tclip=false,\n\t\t\t\t\t\ttranspose legend,\n\t\t\t\t\t\tlegend style={draw=none,at={(0.5,1.3)},anchor=north},\n                        nodes near coords,\n                        cycle list name=black white,\n                        every axis plot\/.append style={fill=gray,no markers}\n\t\t\t\t\t]\n\t\t\t\t\t\\addplot coordinates {(Ravana,46.4) (Rama,28.3)};\n\t\t\t\t\\end{axis}\n\t\t\t\\end{tikzpicture}\n        }\n        \\caption{Fault-tolerant controllers throughput}\n        \\label{fig:controllers-throughput}\n    \\end{subfigure}\n    \\hspace{1cm}\n    \\begin{subfigure}[t]{0.25\\linewidth}\n        \\centering\n        \\resizebox{\\columnwidth}{!}{\n            \\begin{tikzpicture}\n\t\t\t\t\\begin{axis}[\n\t\t\t\t\t\tybar=0pt,\n\t\t\t\t\t\tbar width=12pt,\n\t\t\t\t\t\tx=1.7cm,\n\t\t\t\t\t\tymin=0,\n\t\t\t\t\t\taxis on top,\n\t\t\t\t\t\tymax=60,\n\t\t\t\t\t\tylabel=Throughput (K Responses\/s),\n\t\t\t\t\t\txtick=data,\n\t\t\t\t\t\tenlarge x limits=0.6,\n\t\t\t\t\t\tsymbolic x coords={Ravana,Rama},\n\t\t\t\t\t\taxis lines*=left,\n\t\t\t\t\t\tclip=false,\n\t\t\t\t\t\ttranspose legend,\n\t\t\t\t\t\tlegend style={draw=none,at={(0.5,1.3)},anchor=north, column sep=1.5mm},\n                        legend cell align=left,\n                        nodes near coords=\\pgfmathfloatifflags{\\pgfplotspointmeta}{0}{}{\\pgfmathprintnumber{\\pgfplotspointmeta}},\n                        every node near coord\/.append style={rotate=90, anchor=west},\n\t\t\t\t\t]\n\t\t\t\t\t\\addplot coordinates {(Ravana,52) (Rama,35.6)};\\label{legend-blue}\n\t\t\t\t\t\\addplot coordinates {(Ravana,0) (Rama,50.1)};\\label{legend-red}\n\t\t\t\t\t\\addplot coordinates {(Ravana,46) (Rama,28.3)};\\label{legend-brown}\n\t\t\t\t\t\\legend{Exactly-once events, Exactly-once commands, Both}\n\t\t\t\t\\end{axis}\n\t\t\t\\end{tikzpicture}\n        }\n        \\caption{Throughput with different consistency guarantees}\n        \\label{fig:ravana-vs-rama}\n    \\end{subfigure}\n    \\hspace{1cm}\n    \\begin{subfigure}[t]{0.35\\linewidth}\n        \\centering\n        \\resizebox{\\columnwidth}{!}{\n            \\begin{tikzpicture}\n    \\begin{axis}[\n    xlabel={Number of switches},\n    ylabel={Throughput (Responses\/s)},\n    xmin=1, xmax=7,\n    ymin=0, ymax=60,\n    xtick={1,2,3,4,5,6,7},\n    xticklabels={1,2,4,8,16,32,64},\n    ytick={0,10,20,30,40,50,60},\n    yticklabels={0K,10K,20K,30K,40K,50K,60K},\n   \n    legend style={draw=none,at={(0.5,1.3)},anchor=north},\n    ymajorgrids=true,\n    grid style=dashed,\n    ]\n    \\addplot[\n    color=blue,\n    mark=*,\n    ]\n    coordinates {\n    (1,22.0)(2,23.5)(3,28)(4,31.8)(5,35.6)(6,36.5)(7,36.1)\n    };\n    \\addplot[\n    color=red,\n    mark=*,\n    ]\n    coordinates {\n    (1,32.0)(2,36.8)(3,44.2)(4,45.3)(5,50.7)(6,51.9)(7,51.0)\n    };\n    \\addplot[\n    color=brown,\n    mark=*,\n    ]\n    coordinates {\n    (1,15.3)(2,16.4)(3,22.1)(4,24.3)(5,28.3)(6,28.7)(7,29.1)\n    };\n    \\legend{Exactly-once events, Exactly-once commands, Both}\n    \\end{axis}\n\t\t\t\\end{tikzpicture}\n        }\n        \\caption{Rama throughput with different number of switches}\n        \\label{fig:rama-throughput-switches}\n    \\end{subfigure}\n\\caption{Throughput} \n\\label{fig:rama-throughput}\n\\end{figure*}\n\n\\subsection{Rama performance}\n\nWe have compared the performance of Rama against Ravana~\\cite{katta2015}.\nFigure \\ref{fig:controllers-throughput} shows the throughput for each controller (for Ravana we use the results reported in \\cite{katta2015}, as its authors considered a similar setup).\nFor Rama measurements we run Cbench emulating 16 swit\\-ches.\n\nRama achieves a throughput close to 30K responses per second.\nThis figure is lower than Ravana's, as our solution incurs in higher costs compared to Ravana for the consistency guarantees provided.\nThe additional overhead is caused by two requirements of our protocol.\nFirst, current switches' lack of mechanisms to allow temporary storage of OpenFlow events and commands require Rama to instruct switches to send all events to all replicas, increasing network overhead.\nSecond, the lack of acknowledgement messages in OpenFlow leads Rama to a more expensive solution -- bundles -- to achieve similar purposes.\nThe overhead introduced by these mechanisms is translated into reduced throughput when compared with Ravana.\n\nIn figure \\ref{fig:ravana-vs-rama} we show, separately, throughput results considering the different levels of consistency provided by both Rama and Ravana.\nThe exactly-once events consistency level (\\ref{legend-blue}) ensures that no events are lost and that controllers do not process repeated events.\nAdditionally, controllers must agree on a total order of events to be delivered to applications.\nFor the latter, both Rama and Ravana rely on ZooKeeper to build a shared log across controllers.\nIn our case, the master controller batches events in multiple requests to ZooKeeper, waits for replies, and orders the events before adding them to the Pipeline Queue.\nNote that neither Rama nor Ravana wait for ZooKeeper to persistently store requests on disk (they both use ZooKeeper in-memory).\nIn our case, the multi-request is sent asynchronously (i.e., threads are freed to continue operation) and a callback function is registered. \nThis function will be activated when ZooKeeper replies to our multi request and enqueues the logged events (in order) in the Pipeline Queue to be processed by the modules.\nIn Ravana the processing is equivalent.\n\nThe Exactly-once commands semantics (\\ref{legend-red}) ensures that commands sent by controllers are not lost and that switches do not receive duplicate commands.\nRavana relies on switches to explicitly acknowledge each command and filter repeated ones.\nFor Rama, this includes maintaining state of all opened bundles for switches, and sending additional messages to the switches.\nInstead of replying only with a Packet Out as in Floodlight, Rama must send messages to open the bundle, add the Packet Out to it, close the bundle and commit it.\nTo evaluate this case, we modified Cbench to make switches increase their counters only when they receive a Commit Request message from the controller.\nThis allows a fair evaluation of the performance of Rama in a real system -- indeed, in Rama a packet will only be forwarded after committing the bundle on the switch to guarantee consistent processing.\n\nAs show in Figure \\ref{fig:ravana-vs-rama}, some guarantees are costlier to ensure than others\\footnote{Note that we do not include the results from Exactly-once commands in Ravana as these are not available in~\\cite{katta2015}. It is possible, however, to extrapolate that the results will be inline with the rest of the analysis.}.\nFor instance, the cost of providing Exactly-once events semantics is higher than Exactly-once commands semantics.\nThis result brings with it an important insight: the system bottleneck is the coordination service.\nIn other words, the additional mechanisms Rama uses to guarantee the desired consistency properties add overhead but, crucially, system performance is not limited by these mechanisms.\n\nFigure \\ref{fig:rama-throughput-switches} shows how maintaining multiple switch connections affects Rama throughput.\nAs switches send events at the highest possible rate, the throughput of the system saturates with around 16 switches.\nImportantly, the throughput does not decrease with a higher number of switches.\n\n\n\\subsection{Event batching}\n\nRama batches events to reduce the communication overhead of contacting ZooKeeper.\nIn practice, events are sent to ZooKeeper after reaching a configurable number of events in the batch (batching size) or after a configurable timeout (batching time).\n\nTo evaluate batching we conducted a series of tests with different configurations to understand how the batching size and time affects Rama performance (Figure \\ref{fig:rama-batch-size}).\nIntuitively, a larger batching size will increase throughput, but as downside will also increase latency.\nAs batching size increases, throughput increases due to the reduction of RPC calls required to replicate events.\n\n\n\\begin{figure}[t]\n  \\centering\n  \\resizebox{.75\\linewidth}{!}{\n  \\begin{tikzpicture}\n  \\begin{axis}[\n  xlabel={Batch size},\n  ylabel={Throughput (Responses\/s)},\n  xmin=1, xmax=7,\n  ymin=10, ymax=30,\n  xtick={1,2,3,4,5,6,7},\n  xticklabels={10,100,200,400,600,800,1000},\n  ytick={15,20,25,30},\n  yticklabels={15K,20K,25K,30K},\n  ymajorgrids=true,\n  grid style=dashed,\n  ]\n  \\addplot[\n  color=blue,\n  mark=*,\n  ]\n \n \n  coordinates {\n  (1,16.0)\n  (2,16.9)\n  (2.5,17.3)\n  (3,17.9)\n  (3.5,18.4)\n  (4,19.1)\n  (4.5,20.5)\n  (5,22.6)\n  (5.5,24)\n  (6,24.9)\n  (6.5,25.5)\n  (7,28.3)\n  };\n  \\end{axis}\n  \\end{tikzpicture}\n  }\n  \\caption{Variation of Rama throughput with batch size}\n  \\label{fig:rama-batch-size}\n  \n\\end{figure}\n\n\\subsection{Failover Time}\n\nTo measure the time for Rama to react to failures we use mininet~\\cite{mininet}, OpenvSwitch~\\cite{pfaff2015}, and iperf.\nWe setup a simple topology in Mininet with one switch and two hosts, one to act as iperf server and another as client.\nWe start the client and server in UDP mode, with the client generating 1 Mbit\/sec for 10 seconds.\nThe switch connects to two Rama instances and sends all events to both controllers.\nEach Rama instance is connected to the ZooKeeper server running on another machine (as before) with a negotiated session timeout of 500ms.\nTo make sure that no rules are installed on the switch -- so that events are sent to the controllers each time a packet arrives -- we run Rama with a module that only forwards packets (using Packet Out messages) without modifying the switch's tables.\n\nFigure \\ref{fig:rama-failover} shows the reported bandwidth from the iperf server and indicates the time taken by Rama to react to failures. \nNamely, the slave replica takes around 550ms to react to faults.\nThis includes the time for: (a) ZooKeeper to detect the failure and notify the slave replicas (500ms); (b) electing a new leader for the swit\\-ches; (c) the new leader to transition to master (finish processing logged events from the old master to reach the same internal state); (d) append buffered events to the log and start delivering unprocessed events in the log to applications so they start sending commands to the switches.\nAs is clear, the major factor affecting failover time is the time ZooKeeper needs to detect the failure of the master controller.\n\n\\begin{figure}[h!]\n  \\centering\n  \\resizebox{.75\\linewidth}{!}{\n    \\begin{tikzpicture}\n    \\begin{axis}[\n\t\txlabel={Time (s)},\n\t\tylabel={Bandwidth (Mbits\/sec)},\n\t\txmin=1, xmax=10,\n\t\tymin=0, ymax=1.5,\n\t\txtick={1,2,3,4,5,6,7,8,9,10},\n\t\n\t\n\t\tytick={0,1},\n\t\n\t\n\t\n\t\tlegend style={legend pos=south east},\n\t\tymajorgrids=true,\n\t\tgrid style=dashed,\n\t\t]\n\t\t\\addplot[\n\t\n\t\tcolor=blue,\n\t\tmark=none,\n\t\tline width=0.5mm\n\t\t]\n\t\tcoordinates {\n        (1,1)\n        (3,1)\n        (3.4,0.6)\n        (3.5,0)\n        (4,0)\n        (4,1)\n        (10,1)\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t};\n        \\draw[gray, dashed, thick] (3.4,0) -- (3.4,1.5);\n        \\draw[gray, dashed, thick] (4.0,0) -- (4.0,1.5);\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\legend{}\n    \\end{axis}\n  \\end{tikzpicture}\n  }\n\\caption{Rama failover time}\n\\label{fig:rama-failover}\n\\end{figure}\n\n\n\n\\subsection{Summary}\n\nRama comes close, but does not achieve the performance of Ravana.\nThis is due to the fact that our system incurs in higher costs. \nRama requires more messages to be sent over the network and introduces new mechanisms, such as bundles, which increase the overhead of the solution in order to achieve the same properties as Ravana.\nDespite the (relatively small) loss in performance, the value proposition of Rama of guaranteeing consistent command and event processing without requiring modifications to switches or to the OpenFlow protocol still makes it an effective enabler for immediate adoption of fault-tolerant SDN solutions.\n\n\\section{Related work}\n\\label{related}\n\n\\textbf{Consistent SDN.} Levin et al. \\cite{levin2012} have explored the trade-offs of state distribution in a distributed control plane, motivating the importance of strong consistency in applications' performance.\nOn the one hand, view staleness affects the correct operation of applications, which may lead to poor network performance.\nOn the other, applications need to be more complex in order to be aware of possible network inconsistencies.\n\nHaving a strongly consistent network view across the controllers may be critical to the operation of some applications (e.g., load balancing) in terms of correctness and performance \\cite{levin2012}.\nHowever, as noted in the CAP theorem, a system can not provide availability while also achieving strong consistency in the presence of network partitions.\nBecause of this, fault-tolerant and distributed SDN architectures must use techniques to explicitly handle partitions in order to optimize consistency and availability (and thus achieving a tradeoff between them)~\\cite{brewer2012}.\n\nPart of the strong consistency in the controllers comes from a consistent packet processing (i.e., packets received from switches). OF.CPP \\cite{perevsini2013} explores the consistency and performance problems associated with packet processing at the controller and proposes the use of transactional semantics to solve them.\nThese semantics are achieved by using multi-commit transactions, where each event is a sub transaction, which can commit or abort, of the related packet (the main transaction).\nHowever, this transactional semantics in packet processing is not enough: controllers should also coordinate to guarantee the same semantics in the switches' state. \nSpecifically, the commands sent by the controllers should be processed exactly once by the corresponding switches -- a problem our work addresses.\n\n\\textbf{Consistent network updates.} The concepts of per-packet and per-flow consistency in SDN were introduced in \\cite{reitblatt2011} to provide a useful abstraction for applications: consistent network updates.\nWith consistent updates, packets or flows in flight are processed exclusively by the old or by the new network policy (never a mix of both).\nFor example, with per-packet consistency,  every packet traversing the network is processed by exactly one consistent global network configuration.\nThe authors extend this work in \\cite{reitblatt2012} and implement Kinetic, which runs on top of NOX \\cite{gude2008} to offer these abstractions in a control plane to be used by applications.\nThe main mechanism used to guarantee consistent network updates is the use of a two-phase protocol to update the rules on the switches.\nFirst, the new configuration is installed in an unobservable way (no packets go through these rules yet).\nAfterwards, the switch's ingress ports are updated one-by-one to stamp packets with a new version number (using VLAN tags).\nOnly packets with the new version number are processed by the new rules.\n\nIn \\cite{canini2013}, Canini et al. extend Kinetic to a distributed control plane and formalize the notion of fault-tolerant policy composition.\nTheir algorithm also requires a bounded number of tags, regardless of the number of installed updates, as opposed to the unbounded number of tags in \\cite{reitblatt2012}.\n\nThis class of proposals addresses consistent network updates, which is an orthogonal problem to the one addressed here.\n\n\\textbf{Fault-tolerance in SDN.} Botelho et al.~\\cite{botelho2014} and Katta et al.~\\cite{katta2015} both address fault tolerance in the control plane while achieving strong consistency.\nIn~\\cite{botelho2014} the authors proposes SMaRtLight, a fault-tolerant controller architecture for SDN. \nTheir architecture uses a hybrid replication approach: passive replication in the controllers (one primary and multiple backups) and active replication in an external distributed data store, to achieve durability and strong consistency.\nThe controllers are coordinated through the data store and ca\\-ching mechanisms are employed to achieve acceptable performance.\nIn~\\cite{botelho2016} the authors extend their solution to a distributed deployment.\nIn contrast to our solution, SMaRtLight requires applications to be modified to use the data store directly.\nMore importantly, the solution does not consider the consistency if switch state in the system model.\nRavana~\\cite{katta2015} was the first fault-tolerant controller that integrates switches into the problem.\nThe techniques proposed by its authors guarantee correctness of event processing and command execution in SDN.\nThe main differentiating factor of our work to Ravana is that our solution does not require changes to the OpenFlow protocol nor to switches.\n\n\\textbf{Distributed SDN controllers}. The need for scalability and dependability has been a motivating factor for distribution and fault-tolerance in SDN control planes.\nOnix~\\cite{koponen2010}, the first distributed, dependable, production-level solution considered these problems from the outset.\nAs the choice of the ``right'' consistency model was perceived as fundamental by its authors, Onix offered two data stores to maintain the network state: an eventually consistent and a strong consistent option.\nONOS~\\cite{berde2014} is an open-source solution that shares with Onix the fact that controller state is stored in an external consistent data-store.\nBoth approaches ensure a consistent ordering of events between controller replicas, but they do not include switch state and hence can lead to the network anomalies of traditional replication solutions. \n\n\\textbf{Traditional fault-tolerance techniques.} Viewstamped Replication~\\cite{oki1988}, Paxos~\\cite{lamport1998}, and Raft~\\cite{ongaro2014} are well-known distributed consensus protocols used for replication of state machines in client-server models.\nNone of these widely-used protocols is directly applicable in the context of SDN, where to guarantee correctness it is necessary not only to have consistent controller state, but also switch state.\n\n\\section{Conclusions}\n\nIn a fault-tolerant SDN, maintaining consistent controller state is not enough to achieve correctness.\nUnlike traditional distributed systems, in SDN it is necessary to consistently handle switch state to avoid loss or repetition of commands and events under controller failures.\nTo address these challenges we propose Rama, a consistent and fault-tolerant SDN controller that handles the entire event processing cycle transactionally.\n\nRama differs from the existing alternative, Ravana ~\\cite{katta2015}, by not requiring modifications to the OpenFlow protocol nor to switches.\nThis comes at a cost, as the techniques introduced in Rama incur in a higher overhead when compared to Ravana.\nAs the overhead leads to a relatively modest decrease in performance, we expect, in practice, this to be compensated by the fact that our solution is immediately deployable.\nWe make our software available open source\\footnote{https:\/\/github.com\/fvramos\/rama} to further foster adoption of fault-tolerant SDN.\n\nAs for future work, besides devising a formal proof on the consistency guarantees Rama provides, we plan to address correctness in distributed SDN deployments and to consider richer fault models. \n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{s:intro}\n\\setcounter{equation}{0}\nGeneral Relativity is presently the most successful theory for describing the gravitational interaction\nat the classical level.\nIts own failure is marked by the prediction of the formation of geodesic singularities whenever a trapped surface arises from the gravitational collapse of a compact object.\\footnote{We\nalso recall that pointlike sources are mathematically incompatible with the Einstein field equations~\\cite{geroch}.}\nSuch considerations open up the possibility that significant departures from General Relativity\nmight occur where our experimental data do not yet place strong enough constraints,\nlike for example in regions of strong gravity near a very massive source.\nHowever, Einstein's field equations are not linear and this makes it difficult to modify the laws of gravity in the\nstrong-field regime without affecting also the weak-field behavior, since these regimes are likely to be related\nnontrivially in any nonlinear theories.\n\\par\nThe bootstrapped Newtonian gravity~\\cite{BootN,Casadio:2020mch} is an attempt at\ninvestigating these issues in a somewhat simplified context.\nThe approach, based on Deser's conjecture~\\cite{deser}, consists of retrieving the full Einstein's theory including gravitational self-coupling terms in the Fierz-Pauli action.\\footnote{This\nidea is indeed older, see {\\em e.g.}~Ref.~\\cite{Feynman}.} \nThese additional terms must be consistent with diffeomorphism invariance, in order to preserve the covariance of any (modified) metric theory.\nWe can obtain different modified gravitational theories depending on the choice of boundary conditions in the reconstruction procedure~\\cite{rubio}.\nA key observation is that a practically effective dynamics can be derived only starting with a ``small'' contribution of matter sources.\nFor large astrophysical sources, this implies that the matter source must also be included in\na nonperturbative way.\nIn the present approach this task is addressed starting from the Fierz-Pauli action corresponding to the potential generated by an arbitrarly large static source, and putting in extra terms representing gravitational self-coupling.\nFurthermore, the coupling constants for the additional terms are not fixed to their Einstein-Hilbert values in order to accommodate for diverse underlying dynamics.\nThis approach then results in a nonlinear equation including pressure effects and the gravitational self-interaction terms to next-to-leading order in the Newton constant, whose solution is the gravitational potential operating on test particles at rest.\nSuch equation was useful to investigate compact objects~\\cite{Casadio:2019cux,Casadio:2020kbc,Casadio:2019pli} and coherent quantum states~\\cite{Casadio:2016zpl,Casadio:2017cdv}.\\footnote{These quantum states show some of the\nproperties~\\cite{ciarfella} found in the corpuscular model of black holes~\\cite{DvaliGomez}.\nHowever, we shall not discuss quantum aspects in this work.}\n\\par\n The motion of (test) particles and photons in the surrondings of a compact object represents the most immediate tool to gather information on the gravitational potential in which they revolve.\nIn Ref.~\\cite{Casadio:2021gdf}, a full (effective) metric tensor was obtained from the bootstrapped\nNewtonian potential, which allows one to study these trajectories in general, and to compare them with\nresults from General Relativity.\nThe requirement that the resulting theory of gravity is covariant is satisfied by the use of an effective metric tensor, since the bootstrapped Newtonian dynamics is implicitly assumed to be invariant after coordinate transformations.\nNonetheless, the particular metric found in Ref.~\\cite{Casadio:2021gdf} differs from the Schwarzschild \ngeometry; hence, it is not a solution of the Einstein equations in the vacuum.\nAn effective fluid is therefore present, as was already noted in the cosmological context~\\cite{cosmo}.\n\\par\nThe bootstrapped effective metric is given as a function of parameterized\npost-Newtonian (PPN) parameters~\\cite{weinberg} in the weak-field expansion.\nThese parameters can be consistently chosen so as to minimize deviations from the Schwarzschild metric\nonly up to a point.\nIn fact, some of the PPN parameters are uniquely related, and at the PPN order determined in\nRef.~\\cite{Casadio:2021gdf}, they can be expressed in terms of one free parameter. \nIn this work, we report on a phenomenological investigation aiming at placing bounds on \nthis remaining free parameter from the measured precessions in the Solar System ~\\cite{DeMartino2018,Moyer200,Will2018}and from the study\nof S-star orbits around the black hole in the center of the Galaxy ~\\cite{Eckart1996,Eckart1997,Gillessen:2009,Gillessen2009L,Ghez1998,Ghez:2008}.\n\\par\nThe paper is organized as follows.\nIn Sec.~\\ref{s:boot}, we briefly review the equation for the bootstrapped Newtonian \npotential and its solution in the vacuum.\nWe then just recall the full effective metric reconstructed from this potential, which is \nthen used to analyze Solar System data and S-star motions in Sec.~\\ref{s:app}. We conclude with comments and an outlook in Sec.~\\ref{s:conc}.\n\\section{Bootstrapped Newtonian vacuum}\n\\label{s:boot}\n\\setcounter{equation}{0}\nWe shall only review briefly the derivation of the bootstrapped Newtonian equation, since all the\ndetails can be found in Refs.~\\cite{Casadio:2017cdv,BootN,Casadio:2019cux,Casadio:2019pli}. We shall use units with\nthe speed of light $c=1$ in this section.\nWe start from the Lagrangian for the Newtonian potential $V=V(r)$ generated by a static\nand spherically symmetric source of density $\\rho=\\rho(r)$, to wi\n\\begin{equation}\nL_{\\rm N}[V]\n=\n-4\\,\\pi\n\\int_0^\\infty\nr^2\\,\\d r\n\\left[\n\\frac{\\left(V'\\right)^2}{8\\,\\pi\\,G_N}\n+V \\rho\n\\right]\n\\ .\n\\label{LVn}\n\\end{equation}\nThe corresponding Euler-Lagrange field equation is given by Poisson's\n\\begin{equation}\n\\dfrac{1}{r^2}\\,\\dfrac{d}{d r}\n\\left(r^2\\,\\frac{dV}{dr}\\right)=\n4\\,\\pi\\,G_N\\,\\rho\n\\ ,\n\\label{poisson}\n\\end{equation}\nwhere we recall that the radial coordinate $r$\nis the one obtained from harmonic coordinates~\\cite{weinberg,Casadio:2021gdf}.\nWe next couple $V$ to a gravitational current proportional to its own energy density,\n\\begin{equation}\nJ_V\n\\simeq\n4\\,\\frac{d U_{ N}}{d \\mathcal{V}} \n=\n-\\dfrac{\\left[V'(r)\\right]^2}{2\\,\\pi\\,G_N}\n\\ ,\n\\end{equation}\nwhere $\\mathcal{V}$ is the spatial volume and $U_{\\rm N}$ is the Newtonian potential energy.\nWe also add the ``one loop term'' $J_{\\rho}\\simeq-2\\,V^2$, which couples to $\\rho$, and\nthe pressure term $p$~\\cite{Casadio:2019cux}.\nThe total Lagrangian then reads \n\\begin{align}\nL[V]\n=&\n-4\\,\\pi\n\\int_0^\\infty\nr^2\\,\\d r\n\\left[\n\\frac{\\left(V'\\right)^2}{8\\,\\pi\\,G_N}\n\\left(1-4\\,q_V\\,V\\right)\n\\right.\\nonumber\\\\&\\left.+\\left(\\rho+3\\,q_p\\,p\\right)V\n\\left(1-2\\,q_\\rho\\,V\\right)\\right]\\,\n\\label{LagrV}\n\\end{align}\nwhere the coupling constants $q_V$, $q_p$ and $q_\\rho$ can be used to track the effects of the different\ncontributions.\nFor instance, the case $q_V=q_p=q_\\rho=1$ reproduces the Einstein-Hilbert action at next-to-leading order\nin perturbations around Minkowski~\\cite{Casadio:2017cdv,Casadio:2019cux,Casadio:2019pli}.\nFinally, the bootstrapped Newtonian field equation reads\n\\begin{eqnarray}\n\\dfrac{1}{r^2}\\,\\dfrac{d}{d r}\n\\left(r^2\\,\\dfrac{d V}{d r}\\right)\n&&=\n4\\,\\pi\\,G_N\n\\dfrac{1-4\\,q_\\rho\\,V}{1-4\\,q_V\\,V}\n\\left(\\rho+3\\,q_p\\,p\\right)\n\\nonumber\\\\&&+\\dfrac{2\\,q_V\\left(V'\\right)^2}\n{1-4\\,q_V\\,V}\n\\ ,\n\\label{EOMV}\n\\end{eqnarray}\nwhich must be solved along with the conservation equation $p' = -V'\\left(\\rho+p\\right)$. \n\\subsection{Vacuum potential}\nIn vacuum, we have $\\rho=p=0$, and Eq.~\\eqref{EOMV} simplifies to\n\\begin{equation}\n\\frac{1}{r^2}\\,\\frac{d}{d r}\n\\left(r^2\\,\\frac{d V}{d r}\\right)\n=\n\\frac{2\\,q \\left(V'\\right)^2}{1-4\\,q\\,V}\n\\ ,\n\\label{EOMV0}\n\\end{equation}\nwhere we renamed $q\\equiv q_V$ for simplicity.\nThe exact solution was found in Ref.~\\cite{BootN} and reads\n\\begin{eqnarray}\n\\label{potential}\nV(r)\n=\n\\frac{1}{4\\,q}\n\\left[1-\\left(1+\\frac{6\\,q\\,G_N\\,M}{r}\\right)^{2\/3}\\right]\n\\ .\n\\end{eqnarray}\nThe asymptotic expansion away from the source yields\n\\begin{equation}\nV_{2}\n\\simeq\n-\\frac{G_N\\,M}{r}\n+q\\,\\frac{G_N^2\\,M^2}{r^2}\n-q^2\\,\\frac{8\\,G_N^3\\,M^3}{3\\,r^3}\n\\ ,\n\\label{Vasy}\n\\end{equation}\nso that the Newtonian behavior is always recovered (for $q=0$) and the post-Newtonian terms\nare seen to depend on the coupling $q$ (see Fig.~\\ref{f:V}).\n\\\n\\\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[scale=0.9]{FIG1.pdf}\n    \\caption{Bootstrapped Newtonian potential $V$ in Eq.~\\eqref{potential} compared to\n    its expansion $V_2$ from Eq.~\\eqref{Vasy} and to the Newtonian potential $V_{\\rm N}$\n    (for $q=1$).}\n    \\label{f:V}\n\\end{figure}\n\n\\subsection{Vacuum effective metric}\n\\label{ss:metric}\nA complete spacetime metric was reconstructed from the vacuum potential~\\eqref{potential}\nin Ref.~\\cite{Casadio:2021gdf}.\nThe procedure is rather cumbersome, and we shall only recall here a few main steps leading\nto the necessary expressions in the weak-field regime. We explicitly show the speed of light\n$c$ from here on.\nOne starts from the PPN form~\\cite{weinberg\n\\begin{widetext}\n\\begin{equation}\nds^2\\simeq\\left[1-\\alpha\\dfrac{2R_g}{\\bar{r}}+(\\beta-\\alpha\\gamma)\\dfrac{2R_g^2}{\\bar{r}^2}+(\\zeta-1)\\dfrac{8R_g^3}{\\bar{r}^3}\\right]c^2dt^2 +\\left[1+\\gamma\\dfrac{2R_g}{\\bar{r}}+\\xi\\dfrac{4R_g^2}{\\bar{r}^2}+\\sigma\\dfrac{8R_g^3}{\\bar{r}^3}\\right]d\\bar{r}^2+\\bar{r}^2d\\Omega^2\n\\end{equation}\n\\end{widetext}\nwhere $R_g=\\frac{G_N\\,M}{c^2}$ and $\\bar{r}$ is the areal radius, which differs from the radial coordinate $r$ in which\nthe potential~\\eqref{potential}\nis expressed.\nThe latter is obtained from harmonic coordinates and the two radial coordinates are\nrelated by~\\cite{Casadio:2021gdf}\n\\begin{eqnarray}\nr\n\\simeq&&\n\\bar{r}\n+\n(1-3\\,\\gamma)\\,\\frac{R_g}{2}\n+\\nonumber\\\\\n&&+(1-3\\,\\gamma+2\\,\\gamma^2-2\\,\\Xi)\\,\n\\frac{R_g^2}{\\bar{r}}\n\\ ,\n\\end{eqnarray}\nin which $\\Xi$ is a free parameter.\nFurthermore, we have\n\\begin{equation}\nq\n=\n\\beta+\\frac{\\gamma-1}{2}\n\\ .\n\\label{eq:q}\n\\end{equation}\n\\par\nWe can next set $\\alpha=1$ by simply absorbing this coefficient in the definition of the mass $M$~\\cite{adm},\nand  $\\beta=\\gamma=1$ in order to satisfy the experimental constraints $|\\gamma-1|\\simeq|\\beta-1|\\ll 1$.\nFrom Eq.~\\eqref{eq:q}, this is tantamount to setting $q=1$, as expected.\nThe higher order PPN parameters are then fully determined by $\\Xi$ according to \n\\begin{eqnarray}\n\\xi\n&\\!\\!=\\!\\!&\n1+\\Xi\n\\label{eq:xi}\n\\\\\n\\zeta\n&\\!\\!=\\!\\!&\n1-\\frac{5+6\\,\\Xi}{12}\n=\n\\frac{13-6\\,\\xi}\n{12}\n\\label{eq:zeta}\n\\\\\n\\sigma\n&\\!\\!=\\!\\!&\n\\frac{9+14\\, \\Xi}\n{4}\n\\ .\n\\label{eq:sigma}\n\\end{eqnarray}\nAs already noted in Ref.~\\cite{Casadio:2021gdf}, the General Relativistic PPN combination $\\xi=\\zeta=1$\ncannot be obtained for any value of $\\Xi$, and the bootstrapped metric for which we have the minimum deviation\nfrom the Schwarzschild form is thus given by \n\n\\begin{eqnarray}\nds^2\n\\simeq&&\n-\\left[\n1\n-\\frac{2\\,R_g}{r}\n-(5+6\\,\\Xi)\\,\\frac{2\\,R_g^3}{3\\,c^6\\,r^3}\n\\right]\nc^2\\,dt^2\n\\nonumber\n\\\\\n&&\n+\n\\left[\n1\n+\n\\frac{2\\,R_g}{r}\n+\n(1+\\Xi)\\,\\frac{4\\,R_g^2}{r^2}\n\\right.\\nonumber\\\\&&\\left.\n+\n(9+14\\, \\Xi)\\frac{2\\,R_g^3}{r^3}\n\\right]\ndr^2\n+\nr^2\\,d\\Omega^2\n\\ ,\n\\label{eq:g}\n\\end{eqnarray}\nin which we drop the bar from the areal coordinate for simplicity from now on.\nWe can see that there are contributions in the metric coefficients which cannot be reduced to the\nSchwarzschild expressions.\nThis deviation from the Schwarzschild solution is encoded by the free parameter $\\Xi$,\nwhose value is \\textit{a priori} unknown and must be constrained by observations.\nIn particular, we will test these corrections by analyzing the \nplanets in the Solar System and S-stars motion around Sgr~A*.\nThe geodesic equation\n\\begin{equation}\n\\Ddot{x}^{\\mu}+\\Gamma^{\\mu}_{\\alpha\\beta}\\,\\dot{x}^{\\alpha}\\,\\dot{x}^{\\beta}\n=\n0,\n\\end{equation}\nwhere a dot indicates the derivative with respect to the proper time, can be equivalently computed using the Euler-Lagrange equations\n\\begin{equation}\n\\dfrac{d}{d s}\n\\dfrac{\\partial\\mathcal{L}}{\\partial\\dot{x}^{\\mu}}\n-\\dfrac{\\partial\\mathcal{L}}{\\partial x^{\\mu}}\n=\n0\n\\ , \n\\end{equation}\nwith $\\mathcal{L}=g_{\\alpha\\beta}\\,\\dot x^\\alpha\\,\\dot x^\\beta=-1$ for a massive object.\nFrom the metric in Eq.~\\eqref{eq:g}, one then finds  \n\\par\n\\begin{widetext}\n\\begin{align}\n\\Ddot{r}\n&\n=\n\\frac{R_g\n\\left\\{\n4\\, (1+\\Xi)\\, R_g\\,r\\,\\dot{r}^2\n+R_g^2\n\\left[3\n\\left(\n9+14\\Xi\n\\right)\n\\dot{r}^2\n-c^2\n\\left(5+6\\Xi\\right)\n\\dot{t}^2\n\\right]\n+r^2\\left(\\dot{r}^2-c^2\\,\\dot{t}^2\\right)\n\\right\\}\n+r^5\\,(\\dot{\\theta}^2+\\dot{\\phi}^2\\,\\sin^2\\theta)}\n{r\n\\left[2\\, (9+14\\, \\Xi)\\, R_g^3+4\\,(1+\\Xi)\\, R_g^2\\, r+2\\,R_g\\,r^2+\\,r^3\n\\right]}\n\\label{eq:ddr}\n\\\\\n\\Ddot{\\theta}\n&=\n\\dot{\\phi}^2\\,\\sin\\theta\\,\\cos\\theta\n-\\frac{2\\,\\dot{r}\\,\\dot{\\theta}}{r}\n\\\\\n\\Ddot{\\phi}\n&=\n-\\frac{2\\,\\dot{\\phi}}{r}\\,(\\dot{r}+r\\,\\dot{\\theta}\\,\\cot\\theta)\n\\\\\n\\Ddot{t}\n&\n=\n\\frac{6\\,\\dot{r}\\,\\dot{t}\n\\left[\n(5+6\\,\\Xi)\\, R_g^3\n+R_g\\,r^2\n\\right]}\n{2\\,(5+6\\,\\Xi)\\, R_g^3\\,r\n+6\\,R_g\\,r^3\n-3\\,r^4}\n\\ .\n\\label{eq:ddt}\n\\end{align}\n\\end{widetext}\n The third and fourth equations are the usual conservation equations for the angular momentum\nand energy conjugated to $t$, respectively.\nSpherical symmetry as usual implies that the orbital motion occurs on a plane which we can\narbitrarily set at $\\theta=0=\\dot\\theta$.\n\n\nThe above parametric system of nonlinear differential equations can be integrated numerically\nin order to study the orbits.\n\n\\subsubsection{Precession}\n\\label{ss:precession}\n\\par\nIt is easy to express the perihelion precession in terms of the PPN parameters~\\cite{weinberg}.\nAt leading order, one has\n\\begin{equation}\n\\label{eq:eq1}\n\\Delta\\phi^{(1)}\n=\n2\\,\\pi\n\\left(2-\\beta+2\\,\\gamma\\right)\n\\dfrac{R_g}{\\ell}\n\\ ,\n\\end{equation}\nwhere $\\ell=a\\,(1-e^2)$ is the \\textit{semilatus rectum}, $a$ is the semimajor axis\nand $e$ is the eccentricity.\nFor $\\beta=\\gamma=1$, Eq.~(\\ref{eq:eq1}) reproduces the General Relativistic result\n\\begin{equation}\n\\label{eq:eq2}\n\\Delta\\phi_S^{(1)}\n=\n6\\,\\pi\\,\\frac{R_g}{\\ell}\n\\ .\n\\end{equation}\nThe second order correction depends on $\\xi$ and $\\zeta$, and for $\\beta=\\gamma=1$,\nit reads~\\cite{Casadio:2021gdf}\n\\begin{align}\n\\Delta\\phi^{(2)}\n&\n=\n\\pi\\left[(41+10\\xi-24\\,\\zeta)\n+\n(16\\,\\xi-13)\\,\\frac{e^2}{2}\\right]\n\\frac{R_g^2}{\\ell^2}\n\\nonumber\n\\\\\n&\n\\simeq\n\\pi\\left[(37+22\\,\\Xi)+(3+16\\,\\Xi)\\,\\frac{e^2}{2}\\right]\n\\frac{R_g^2}{\\ell^2}\n\\nonumber\n\\\\\n&\n\\simeq\n\\Delta\\phi_S^{(2)}\n+\n2\\,\\pi\\left[\n11\\,\\xi-7+4\\,(\\xi-1)\\,e^2\\right]\n\\frac{R_g^2}{\\ell^2}\n\\ ,\n\\end{align}\nwhere the General Relativistic result $\\Delta\\phi_S^{(2)}$ corresponds to $\\xi=\\zeta=1$.\nFrom Eqs.~\\eqref{eq:xi} and \\eqref{eq:zeta}, it follows that we cannot have $\\xi=\\zeta=1$ for any value of $\\Xi$,\nand a deviation from General Relativity remains. \n\\section{Astronomical tests}\n\\label{s:app}\n\\setcounter{equation}{0}\nIn order to constrain the free parameter of the bootstrapped Newtonian potential, $\\Xi$,\nwe confronted the theoretical results exposed in Sec.~\\ref{ss:metric} with astronomical \ndata.\n\nTo infer a range of validity for $\\Xi$, we compared the analytical expression of the precession with the observed values of the perihelion advance of Solar System's planets (Sec.~\\ref{ss:precession2}).\n\nThen, we turned our attention to the Galactic Center, and we studied the motion of S-stars\norbiting around Sgr A*.\nTo constrain $\\Xi$, we let it vary in a given range and fit the corresponding simulated\norbits to astrometric observations.\nIn particular, we adopted a fully relativistic approach which consists of integrating numerically\nEqs.~\\eqref{eq:ddr}-\\eqref{eq:ddt} in order to get the mock orbits, instead of solving Newton's\nlaw with the standard potential replaced by the modified one. \n\\subsection{Perihelion precession in the Solar System}\n\\label{ss:precession2}\nIn order to constrain $\\Xi$ we can start from the Solar System planets whose orbital precession\nhas been measured, namely Mercury, Venus, Earth, Mars, Jupiter and Saturn~\\cite{2015MNRAS.451.3034N}.\nThe confidence region for $\\Xi$ can be identified as the set of values such that the precession\n\\begin{equation}\n\\Delta\\phi=\\Delta\\phi^{(1)}+\\Delta\\phi^{(2)}\n\\label{eq:eq3}\n\\end{equation}\nis compatible with the observations.\nThe planetary parameters\\footnote{The reported values are taken from NASA fact sheet at\nhttps:\/\/nssdc.gsfc.nasa.gov\/planetary\/factsheet\/.},\nthe corresponding observed values of the precession~\\cite{2015MNRAS.451.3034N} and\nthe General Relativistic value obtained by Eq.~(\\ref{eq:eq2}) are reported in Table~\\ref{tab:tab1}\nfrom first to seventh columns.\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{cccccccc}\n    \\hline\\hline\n    Planet & $a(10^6km)$ & $P(years)$ & $i(^{\\circ})$ & $e$ & $\\Delta\\phi_{obs}(''\/cy)$ & $\\Delta\\phi_S(''\/cy)$ & $[\\Xi_{min};\\Xi_{max}]$ \\\\\n    \\hline\n    $\\textbf{Mercury}$&$57.909$&$0.24$&$7.005$&$0.2056$&$43.1000\\pm0.5000$&$42.9822$&$[-89708.7;144995]$\\\\\n    $\\textbf{Venus}$&$108.209$&$0.61$&$3.395$&$0.0067$&$8.6247\\pm0.0005$&$8.6247$&$[-1149.67;1167.47]$\\\\\n    $\\textbf{Earth}$&$149.596$&$1.00$&$0.000$&$0.0167$&$3.8387\\pm0.0004$&$3.83881$&$[-3660.86;2094.96]$\\\\\n    $\\textbf{Mars}$&$227.923$&$1.88$&$1.851$&$0.0935$&$1.3565\\pm0.0004$&$1.35106$&$[155248.;179879.]$\\\\\n    $\\textbf{Jupiter}$&$778.570$&$11.86$&$1.305$&$0.0489$&$0.6000\\pm0.3000$&$0.0623142$&$[5.46709\\times10^8;1.92679\\times10^9]$\\\\\n    $\\textbf{Saturn}$&$1433.529$&$29.45$&$2.485$&$0.0565$&$0.0105\\pm0.0050$&$0.0136394$&$[-1.57315\\times10^8;3.59618\\times10^7]$\\\\\n    \\hline\\hline\n    \\end{tabular}\n\\caption{Values of semimajor axis ($a$), orbital period ($P$), tilt angle ($i$), eccentricity ($e$), observed orbital precession\n($\\Delta\\phi_{obs}$), orbital precession as predicted by General Relativity ($\\Delta\\phi_S$) and constraints on $\\Xi$ for Solar System's planets.}\n    \\label{tab:tab1}\n\\end{table*}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.6]{Fig2.pdf}\n    \\caption{Bootstrapped orbital precession as a function of the parameter $\\Xi$.\n    Black lines give the theoretical prediction from Eq.~\\eqref{eq:eq3}, blue dashed lines represent\n    the measurements adapted from Ref.~\\cite{2015MNRAS.451.3034N} and red lines depict the General Relativistic\n    values as in Eq.~\\eqref{eq:eq2}.\n    Confidence regions for $\\Xi$ are shaded in gray.}\n    \\label{fig:fig1}\n\\end{figure*}\nThe allowed region of $\\Xi$ for each planet is defined as the range of values compatible with data, having as extremes the\nvalues of $\\Xi$ solving the equation\n\\begin{equation}\n\\Delta\\phi=\\Delta\\phi_{obs}\n\\ .\n\\end{equation}\nThe inferred lower and upper limits on $\\Xi$ are reported in the last column of Table~\\ref{tab:tab1}, and the included area\nis depicted in Fig.~\\ref{fig:fig1} for each planet (gray shades).\nIt is worth noticing the discrepancy between the General Relativistic value (the red line) and the observed precession\n(blue dashed lines) for Mars and Jupiter; it could be attributed to the incomplete subtraction of nonrelativistic effects\nfrom the observed value, complicated by the presence of the asteroid belt between Mars and Jupiter, and the presence\nof an anomalous residual precession \\cite{2015MNRAS.451.3034N,2013MNRAS.432.3431P}.\n\\par\nThe tightest interval on the parameter $\\Xi$ is obtained with Venus, for which it can vary between $-1149.67$ and $1167.47$.\nWe can use the values defining such an interval to predict the precession for Uranus, Neptune and Pluto,\nfor which no observation is available.\nThe results, summarized in Table~\\ref{tab:tab2}, show that the bootstrapped theory predictions are in perfect agreement\nwith General Relativity. \n\\begin{table*}\n    \\centering\n    \\begin{tabular}{ccccccc}\n    \\hline\\hline\n    Planet & $a(10^6km)$ & $P(years)$ & $i(^{\\circ})$ & $e$ &  $\\Delta\\phi_{S}(''\/cy)$ & $[\\Delta\\phi_{min};\\Delta\\phi_{max}]$ \\\\\n    \\hline\n    $\\textbf{Uranus}$&$2872.463$&$84.01$&$0.772$&$0.0457$&$0.00238404$&$[0.00238404;0.00238405]$\\\\\n    $\\textbf{Neptune}$&$4495.060$&$164.786$&$1.769$&$0.0113$&$0.000775374$&$[0.000775373;0.000775375]$\\\\\n    $\\textbf{Pluto}$&$5869.656$&$247.936$&$17.16$&$0.2444$&$0.000419669$&$[0.000419669;0.00041967]$\\\\\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{Orbital parameters from Nasa Fact Sheet, the General Relativistic prediction for the precession in the sixth column\n    and the values predicted by the bounds on the parameter $\\Xi$ of the bootstrapped theory deduced for Venus (see Table~\\ref{tab:tab1}).}\n    \\label{tab:tab2}\n\\end{table*}\n\\par\nNow it is useful to move to a different scale and analyze $S2$ (see Table \\ref{tab4}), the only one among the S-stars whose precession was observed \\cite{GRAVITY:2020gka}.\nWe can next calculate the precession for Mars, Jupiter, and $S2$ with the values of $\\Xi$ as obtained by Mercury, Venus, Earth\nand Saturn to check agreement with the corresponding Schwarzschild value and with the observations (Table \\ref{tab3}). The results confirm the compatibility of our predictions with General Relativity.\nThe mean value of the parameter $\\Xi$ such that\n\\begin{equation}\n    \\Delta\\phi\n    =\n    \\Delta\\phi_S\n\\end{equation}\nis given by\n\\begin{equation}\n\\Xi\n=\n-1.64236\\pm 0.10305\n\\ .\n\\label{eq:c2}\n\\end{equation}\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{cccccccc}\n    \\hline\\hline\n    Star & $a(AU)$ & $P(years)$ & $i(^{\\circ})$ & $e$ & $\\Delta\\phi_{obs}(''\/\\text{orbit})$ & $\\Delta\\phi_S(''\/\\text{orbit})$ & $[\\Xi_{min};\\Xi_{max}]$ \\\\\n    \\hline\n    $\\textbf{S2}$&$1031.32$&$16.0455$&$134.567$&$0.884649$&$730.382\\times(1.10\\pm0.19)$&$730.382$&$[-103.066;326.398]$\\\\\n\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{For the star $S2$, orbital parameters~\\cite{GRAVITY:2020gka}, observed orbital precession ($\\Delta\\phi_{obs}$),\n    orbital precession as predicted by General Relativity ($\\Delta\\phi_S$), and constraints on $\\Xi$.}\n    \\label{tab4}\n\\end{table*}\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{cccccc}\n    \\hline\\hline\n   Object  & $\\Delta\\phi_S$ & $\\Delta\\phi(\\Xi_{Mercury})$ &$\\Delta\\phi(\\Xi_{Venus})$&$\\Delta\\phi(\\Xi_{Earth})$&$\\Delta\\phi(\\Xi_{Saturn})$\\\\\n    \\hline\n    $\\textbf{Mars}$&$1.35106$&$[1.34814;1.35577]$&$[1.35102;1.3511]$&$[1.35094;1.35113]$&$[-3.75855;2.5191]$\\\\\n    $\\textbf{Jupiter}$&$0.0623142$&$[0.0622752;0.0623773]$&$[0.0623137;0.0623147]$&$[0.0623126;0.0623151]$&$[-0.00607962;0.0779489]$\\\\\n    $\\textbf{S2}$&$730.382$&$[-57243.9;94435.7]$&$[-11.7295;1485.75]$&$[-1634.61;2085.15]$&$[-1.01666*10^8;2.32414*10^7]$\\\\\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{Precession for Mars, Jupiter, and $S2$ as predicted by confidence regions for $\\Xi$ inferred from Mercury, Venus, Earth and Saturn.}\n    \\label{tab3}\n\\end{table*}\n\n\\subsection{S-star dynamics}\n\\label{ss:dynamic}\nWe can confirm the bounds on $\\Xi$ deduced from orbital precessions by comparing them with results deduced from the analysis\nof stellar orbits at the Galactic Center.\nThis further analysis consists in comparing simulated orbits in bootstrapped Newtonian gravity, obtained by integrating numerically\nEqs.~\\eqref{eq:ddr}-\\eqref{eq:ddt}, with observed orbits of three S-stars constructed by astrometric observations\n(see Sec.~\\ref{sec:data}).\nIn particular, we focused on stars $S2$, $S38$ and $S55$ for two main reasons:\namong the brightest stars they are those with the shortest period.\nThese properties are desired because highly bright stars are less prone to being confused with other sources,\nand a short period allows us to observe a larger part of the orbit in a given observation session.\nFor simplicity, we neglected perturbations from other members of the cluster and any extended matter structures. \n\\subsubsection{Astrometric data}\n\\label{sec:data}\nAstrometric data are taken from Ref.~\\cite{Gillessen:2017}~\\footnote{Data are publicly available on the electronic version of\nRef.~\\cite{Gillessen:2017} at the link https:\/\/iopscience.iop.org\/article\/10.3847\/1538-4357\/aa5c41\/meta.}\nand cover $25$ years of observations performed in the near-infrared (NIR), where the interstellar extinction amounts to\nabout three magnitudes.\nDifferent instruments have been used, which we briefly describe below.\n\\begin{enumerate}\n\\item\nSHARP.- First high-resolution data of the Galactic Center were obtained in $1992$ with the SHARP camera at the European Southern Observatory's\n(ESO) $3.5\\,$m New Technology Telescope (NTT) in Chile, operating in Speckle mode with exposure times of $0.3\\,$s, $0.5\\,$s and $1.0\\,$s.\nThe data, described in detail in Ref.~\\cite{Schodel:2003gy}, led to the detection of high proper motion near the central massive object.\n\\item\nNACO.- The first Adaptive Optics (AO) imaging data were produced by Naos-Conica (NACO) system,\nmounted at the telescope Yepun ($8.0\\,$m) of the VLT and starting to operate in $2002$.\nIt followed a great improvement due to larger telescope aperture, and the higher Strehl ratio (about $40\\%$).\n\\item\nGEMINI.- The dataset includes observations obtained by the $8\\,$m telescope Gemini North in Mauna Kea, Hawaii.\nThese images, obtained using the AO system in combination with the NIR camera Quirc, were processed by the Gemini team.\n\\end{enumerate}\nThe astrometric calibration of these data, treated in Ref.~\\cite{Gillessen:2009ht}, consists in the following steps:\nobtaining high-quality maps of the S-stars, extracting pixel positions, and transforming them to a common astrometric\ncoordinate system.\nIn particular, the astrometric reference frame is implemented relating the S-stars positions to a set of selected reference\nstars, which are in turn related to a set of Silicon Monoxide (SiO) maser stars whose positions relative to Sgr~A* is known.\n\\subsubsection{Fitting procedure}\nThe first step of the fitting procedure is the numerical integration of the system of parametric nonlinear\ndifferential equations~\\eqref{eq:ddr}-\\eqref{eq:ddt} to produce stellar simulated orbits in bootstrapped\nNewtonian gravity.\n\\begin{table*}\n\\begin{centering}\n    \\begin{tabular}{cccc}\n    \\hline\\hline\nParameter           & S2                     & S38               & S55               \\\\ \\hline\n$a$ (mas)           & $125.058\\pm0.041$      & $141.6\\pm0.2$     & $107.8\\pm1.0$     \\\\\n$\\Omega$ ($^\\circ$) & $228.171\\pm0.031$      & $101.06\\pm0.24$   & $325.5\\pm4.0$     \\\\\n$e$                 & $0.884649\\pm0.000066$  & $0.8201\\pm0.0007$ & $0.7209\\pm0.0077$ \\\\\n$i$ ($^\\circ$)      & $134.567\\pm0.033$     & $171.1\\pm2.1$     & $150.1\\pm2.2$     \\\\\n$\\omega$ ($^\\circ$) & $66.263\\pm0.031$       & $17.99\\pm0.25$    & $331.5\\pm3.9$     \\\\\n$t_p$ (yr)          & $2018.37900\\pm0.00016$ & $2003.19\\pm0.01$  & $2009.34\\pm0.04$  \\\\\n$T$ (yr)            & $16.0455\\pm0.0013$     & $19.2\\pm0.02$     & $12.80\\pm0.11$  \\\\ \n$m_K$     &\n13.95          &\n17.       &\n17.5     \\\\\nRef.          &\n\\cite{GRAVITY:2020gka} &\n\\cite{Gillessen:2017}              &\n\\cite{Gillessen:2017} \\\\\n\\hline\\hline\n    \\end{tabular}\n        \\caption{Orbital parameters of $S2$, $S38$, and $S55$:\n        semimajor axis  $a$, eccentricity $e$, inclination $i$, angle of the line of node $\\Omega$,\n        angle from ascending node to pericenter $\\omega$, orbital period $T$, and the time of the\n        pericenter passage $t_p$. }\n        \\label{tab1}\n        \\end{centering}\n\\end{table*}\n\\begin{table*}\n\\begin{centering}\n    \\begin{tabular}{cccc}\n    \\hline\\hline\n     Star&$M(M_{\\odot})$&$R(kpc)$&Ref.\\\\\n    \\hline\n    S2&$(4.261\\pm0.012)\\times10^6$&$8.2467\\pm0.0093$&GRAVITY Collaboration \\cite{GRAVITY:2020gka} \\\\\n    S38&$(4.35\\pm0.13)\\times10^6$&$8.33\\pm0.12$&Gillessen et al. \\cite{Gillessen:2017}\\\\\n    S55&$(4.35\\pm0.13)\\times10^6$&$8.33\\pm0.12$&Gillessen et al. \\cite{Gillessen:2017}\\\\\n    \\hline\\hline\n    \\end{tabular}\n        \\caption{Parameters of the central BH: the mass $M$ and the distance $R$.} \\label{tab2}\n        \\end{centering}\n\\end{table*}\n\\par\nPreliminarily, we fix the Keplerian elements and the parameters of the central mass to the values reported\nin Tables~\\ref{tab1} and \\ref{tab2}.\nIn particular, for the study of $S2$, we used the values obtained by the GRAVITY Collaboration~\\cite{GRAVITY:2020gka},\nand for $S38$ and $S55$, we used those obtained in Ref.~\\cite{Gillessen:2017}.\nIn order to have a well-defined Cauchy problem, we must provide initial conditions for the four-dimensional coordinates\nand their derivatives with respect to the proper time: \\{$r(0)$, $\\dot{r}(0)$, $\\theta(0)$, $\\dot{\\theta}(0)$, $\\phi(0)$,\n$\\dot{\\phi}(0)$, $t(0)$, $\\dot{t}(0)$\\}. \nWe assume that the star initially lies on the equatorial plane of the reference system, for which $\\theta(0)=\\pi\/2$,\nand that its initial velocity is parallel to the equatorial plane, that is $\\dot{\\theta}(0)=0$.\nIt then follows that $\\ddot{\\theta}(0)=0$ identically.\nIn particular, we set the initial conditions for $r$ and $\\phi$ at the time of passage of the apocenter,\nwhen the Cartesian coordinates of the star expressed in the orbital plane are given by\n\\begin{equation}\n(x_{orb},y_{orb})\n=\n\\left(-a\\,(1+e),0\\right)\n\\end{equation}\nand the Cartesian components of its velocity read\n\\begin{equation}\n(v_{x,orb},v_{y,orb})\n=\n\\left(0,\\dfrac{2\\,\\pi\\,a^2}{T\\,r}\\,\\sqrt{1-e^2}\\right)\n\\ .\n\\end{equation}\nThe initial condition for $\\dot{t}$ can be retrieved from the normalization of four-velocities requiring\nthat the geodesic is timelike.\n\\par\nStarting from the initial conditions of each star, we proceed with an explicit Runge-Kutta numerical integration\nof the relativistic equations of motion.\nThe results are the stars mock orbit in the orbital plane, described by a four-dimensional array\n$\\{t(\\tau),r(\\tau),\\theta(\\tau),\\phi(\\tau)\\}$.\nTo compare the theoretical orbits with those observed from the Earth, we must project any point\n$(x_{\\rm orb}, y_{\\rm orb})$ on the orbital plane into the point $(x, y)$ on the observer's sky plane.\nSuch a transformation is realized by applying the Thiele-Innes formulas~\\cite{1930MNRAS,aitken}:\n\\begin{align}\n    x&=l_1\\,x_{\\rm orb}+l_2\\,y_{\\rm orb}\n    \\\\\n    y&=m_1\\,x_{\\rm orb}+m_2\\,y_{\\rm orb}\n    \\ .\n\\end{align}\nThe Thiele-Innes elements $l_1$, $l_2$, $m_1$ and $m_2$ depend on the Keplerian elements by according to\n\\begin{align}\n    l_1\n    &=\n    \\cos\\Omega\\,\\cos\\omega-\\sin\\Omega\\, \\sin\\omega\\, \\cos i\n    \\\\\n    l_2\n    &=\n    -\\cos \\Omega\\, \\sin\\omega-\\sin\\Omega\\, \\cos\\omega \\cos i\n    \\\\\n    m_1\n    &=\n    \\sin\\Omega\\, \\cos\\omega+\\cos\\Omega\\, \\sin\\omega \\cos i\n    \\\\\n    m_2\n    &=\n    -\\sin\\Omega\\,\\sin\\omega+\\cos\\Omega\\, \\cos\\omega\\, \\cos i\n     .\n     \\end{align}\n\\par     \nThe second step consists in the fitting procedure itself, and has the aim to constrain the parameter $\\Xi$.\nGuided by the results obtained from the precession in Sec.~\\ref{ss:precession}, we let it vary freely in an\nappropriate range including the value~\\eqref{eq:c2}.\nFor each value of $\\Xi$ we repeated the aforementioned procedure to get the true positions $(x_i,y_i)$\nand velocities $(\\dot{x}_i,\\dot{y}_i)$ of the stars at all the observed epochs.\nAfter transforming the true positions into the apparent positions $(x_i^{th},y_i^{th})$, we computed the\nreduced-$\\chi^2$ distribution to quantify the discrepancy between theory and observations as\n\\begin{equation}\n  \\chi^2_{\\rm red}\n  =\n  \\frac{1}{2\\,N-1}\\,\n  \\sum_i^N\\left[\\left(\\frac{x_i^{\\rm obs}-x_i^{\\rm th}}{\\sigma_{x_i^{\\rm obs}}}\\right)^2\n  +\\left(\\frac{y_i^{\\rm obs}-y_i^{\\rm th}}{\\sigma_{y_i^{\\rm obs}}}\\right)^2\n  \\right]\n  \\ ,\n\\end{equation}\nwhere $(x_i^{\\rm obs},y_i^{\\rm obs})$ and $(x_i^{\\rm th},y_i^{\\rm th})$ are respectively the observed and\nthe predicted positions, $N$ is the number of observations and $(\\sigma_{x_i^{\\rm obs}},\\sigma_{y_i^{\\rm obs}})$\nare the observative uncertainties.\nFinally, we calculated the likelihood probability distribution, $2\\,\\log\\mathcal{L}=-\\chi^2_{\\rm red}(\\Xi)$.\nThe best-fit value for $\\Xi$ was derived as the point that maximizes the likelihood distribution.\n\\subsubsection{Results}\n\\begin{table}\n    \\centering\n    \\begin{tabular}{cc}\n    \\hline\\hline\n     Star&$\\Xi$\\\\\n    \\hline\n    $S2$&$-5900_{-44964.9}^{+39358.8}$\\\\\n    $S38$&$25500_{-23312.88}^{+22607.1}$\\\\\n    $S55$&$60400_{-87446.9}^{+81386}$\\\\\n    Multi-star&$17400_{-32244.3}^{+30555.6}$\\\\\n    \\hline\\hline\n    \\end{tabular}\n    \\caption{Best-fit values for $\\Xi$.}\n    \\label{tab5}\n\\end{table}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.45]{Fig3.pdf}\n    \\caption{Comparisons between the NTT\/VLT astrometric observations with their errors (black circles)\n    and the theoretical best-fit orbits using parameters reported in the first three rows of Table~\\ref{tab5}.\n    The results for $S2$, $S55$ and $S38$ are illustrated respectively in the top left, top right, and bottom panels.}\n    \\label{fig:fig3}\n\\end{figure*}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.6]{Fig4.pdf}\n    \\caption{Top panels show the comparison between the observed and fitted coordinates, and\n    bottom panels show the corresponding (O-C) residuals for $S2$, $S38$ and $S55$.}\n    \\label{fig:fig4}\n\\end{figure*}\n\\begin{figure*}[!ht]\n    \\centering\n    \\includegraphics[scale=0.7]{Fig5.pdf}\n    \\caption{Best relativistic multistar orbit fit of $S2$, $S38$ and $S55$.}\n    \\label{fig:fig5}\n\\end{figure*}\nOur results are summarized in Table~\\ref{tab5} and represented in Figs.~\\ref{fig:fig3}, \\ref{fig:fig4} and \\ref{fig:fig5}.\n\\par\nIn Fig.~\\ref{fig:fig3} we show the comparison between best fit and observed orbits of the selected stars:\nthe top left panel, the top right panel, and the bottom panel illustrate the results respectively for $S2$, $S55$ and $S38$.\nAstrometric data are reported with their own error bars to note the effectiveness of our fitting procedure.\n\\par\nFigure~\\ref{fig:fig4} depicts the comparisons between the observed and simulated coordinates with the corresponding residuals.\nThe left column contains the right ascension (RA), while the right column reports the declination (Dec).\nIt is worth noticing that in all stars and for both coordinates, residuals are larger at the beginning observing epochs,\nand decrease as astrometric accuracy improves.\n\\par\nFinally, we show in Fig.~\\ref{fig:fig5} the orbits of the studied S-stars corresponding to the best multistar fit for\n$\\Xi=17400_{-32244.3}^{+30555.6}$ (last row of Table~\\ref{tab5}).\nAs expected, the parameter $\\Xi$ is compatible with the the mean value~\\eqref{eq:c2} such that the bootstrapped\nNewtonian precession recovers General Relativity.\n\\section{Conclusions}\n\\label{s:conc}\n\\setcounter{equation}{0}\nIn this paper we tested astronomically the bootstrapped Newtonian gravity.\nThe starting point is the complete spacetime metric~\\eqref{eq:g} derived in Ref.~\\cite{Casadio:2021gdf}.\nThe leading order deviation from the Schwarzschild solution cannot be eliminated and is encoded in the free parameter\n$\\Xi$, which is not \\textit{a priori} known and must be constrained by observations.\n\\par\nFirst, we show that bounds on $\\Xi$ can be deduced from the comparison between the measurements\nof the orbital precession of Solar System bodies and the theoretical predictions arising from bootstrapped\nNewtonian metric computed in Ref.~\\cite{Casadio:2021gdf}.\nThe inferred confidence region for $\\Xi$ for each planet is reported in Table~\\ref{tab:tab1} and graphically\ndepicted in Fig.~\\ref{fig:fig1}.\nBased on the tightest interval obtained with Venus, we found that $\\Xi$ lies in the range $[-1149.67\\,;\\,+1167.47]$.\nWith these values of the parameter $\\Xi$ we predicted the orbital precession for Uranus, Neptune and Pluto,\nand we found a theoretical precession in great agreement with the General Relativistic value.\nSuch a compatibility was confirmed by turning our attention to the Galactic Center and repeating the same\nanalysis for the star $S2$~\\cite{GRAVITY:2020gka}.\nThe mean value of the parameter $\\Xi$ such that the bootstrapped Newtonian precession equals the Schwarzschild\nvalue is \n\\begin{equation}\n\\Xi=-1.64236\\pm0.10305\n\\ .\n\\end{equation}\n\\par\nWe next focused on the Galactic Center scale to constrain $\\Xi$ by investigating the orbital motion of S-stars.\nWe used a fully relativistic approach based on an agnostic method:\nfor each value of $\\Xi$, we solved the geodesic equations numerically starting from initial conditions at the\napocenter.\nAfter applying the Thiele-Innes formulas to the mock positions, we were able to compare the resulting solution\nwith the observed stellar orbits.\nFinally, we quantified the discrepancy between the simulated and observed orbits performing a $\\chi^2$-statistics.\nThe inferred confidence region for $\\Xi$ is compatible with the bounds obtained by the precession analysis,\nand thus with General Relativity.\nIndeed we found $17400_{-32244.3}^{+30555.6}$.\nSince S-stars are at a distance of about  $r>1000\\, R_g$ from the source, strong-field effects are not relevant,\nand such a result was expected. \n\\par\nThe proposed approach is completely general and represents a useful tool in the classification of extended\ntheories of gravity.\nMoreover, this approach has already been used to test a Yukawa-like gravitational potential by means of dynamical tests at the\nGalactic Center~\\cite{yuk1,yuk2,yuk3,DellaMonica2021}, where no significant deviations from General Relativity were found.\nNevertheless, the definitive confirmation\/exclusion of a given extended theory of gravity requires the improvement\nof the constraints on its free parameters based on the observation of various strong-field effects.\nThis task can be accomplished taking advantage of the increasing high accuracy observations of second\ngeneration instruments like GRAVITY~\\cite{Gillessen:2010ei}.\n\\par\nIn particular, we focus on finding stars with short semimajor axis and highly eccentric orbits within the\npericenter of $S2$.\nThe existence of such a population of stars can be inferred from the recent discovery of the sources $S62$,\n$S4711$ and $S4714$~\\cite{peissker2020,peissker20202}.\nObserving stars at smaller radii is essential to detect strong-field effects, which become no longer negligible\nfor distances of the pericenter $r\\simeq 10\\,R_g$, and therefore any deviations from General Relativity\nto find out the underlying gravitational theory.\n\n\n\n\n\\begin{acknowledgments}\nR.C.~is partially supported by the INFN grant FLAG. M.D.L. and A.D. acknowledges INFN Sez. di Napoli (Iniziativa Specifica TEONGRAV). \nA.G.~is supported by the European Union's Horizon 2020 research and innovation programme under the \nMarie Sk\\l{}odowska-Curie Actions (grant agreement No.~895648--CosmoDEC).\nThe work of R.C.~and A.G~has also been carried out in the framework\nof activities of the National Group of Mathematical Physics (GNFM, INdAM). \n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA Borel set $E$ in an Abelian Polish group $X$ is said to be \\emph{Haar Null} if there is a Borel probability measure $\\mu$ on $X$ such that $\\mu(x+E)=0$ for every $x\\in X$. Haar null sets were introduced for the first time by J.P.R.\\ Christensen in \\cite{chris_1972} in order to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups, where the Haar measure is not defined. Indeed, Haar null sets and sets with zero Haar measure agree on locally compact Abelian Polish groups and, as in the locally compact case, Haar null sets form a $\\sigma$-ideal in the Borel $\\sigma$-algebra of $X$. We say that a Borel set $E\\subseteq X$ is \\emph{Haar positive} if $E$ is not Haar null. The reader is invited to have a look at the survey papers \\cite{bog_2018} and \\cite{en_2020}, as well as at \\cite{benlin}, Chapter 6, for a detailed exposition of this topic.\n\nAlthough the definition of Haar positive sets is measure-theoretical in nature, in \\cite{daverava2} a measure-free characterisation of Haar positive closed, convex sets in separable Banach spaces is provided.\n\n\\begin{theo}\n\\label{theo:intro}\nLet $C$ be a closed and convex set in a separable Banach space $X$ with unit ball $B_X$. The following assertions are equivalent.\n\\begin{enumerate}\n\\item $C$ is Haar positive.\n\\item There is $r>0$ with the property that, for every compact set $K\\subseteq rB_X$, there is $x\\in X$ such that $x+K\\subseteq C$.\n\\end{enumerate}\n\\end{theo} \n\\noindent Equivalently, a closed, convex set $C$ in a separable Banach space $X$ is Haar null if and only if $C$ is \\emph{Haar meagre}. That is, there is a compact metric space $M$ and a continuous function $\\func{f}{M}{X}$ such that $f^{-1}(x+C)$ is meagre in $M$ for every $x\\in X$. For a more general treatment of Haar meagre sets, we refer the reader to \\cite{dar_2013}, where they were first introduced.\n\nTheorem \\ref{theo:intro} motivates the introduction of several geometric radii associated to such sets. These are defined in Section \\ref{sec:radii} and multiple properties about these quantities are shown. In Section \\ref{sec:weaks} we exploit these radii to prove a new characterisation of Haar positive, closed, convex sets. Namely, a closed, convex set $C$ in a separable Banach space $X$ is Haar positive if and only if its weak$^*$ closure in the second dual $X^{**}$ has nonempty interior with respect to the norm topology. This improves the well-known fact that closed, convex subsets of a Euclidean space $\\mathbb{R}^d$ have positive Lebesgue measure if and only if their interior is nonempty and a theorem of Eva Matou\\v{s}kov\\'{a} (\\cite{eva3}) which states that, in separable, reflexive Banach spaces, closed convex sets are Haar positive if and only if they have nonempty interior. As a corollary, it is shown in Section \\ref{sec:top} that the family of Haar positive, closed, convex and bounded sets is open in the space of all nonempty, closed, convex and bounded subsets of $X$, endowed with the Hausdorff distance.\n\nThe standard notation of Banach space theory is used throughout the paper. Given a Banach space $X$, $B_X$ and $S_X$ stand for the closed unit ball and the unit sphere of $X$ respectively. $X^*$ denotes the dual of $X$, whereas $X^{**}$ is the second dual. The closure of a set $E\\subseteq X$ is denoted by $\\textup{cl}(E)$ and in a dual space we denote by $\\textup{w}^*\\textup{-cl}(E)$ the closure of $E$ in the weak$^*$ topology. We use the notation $\\mathcal{C}(X)$ for the space of all nonempty, closed, convex and bounded subsets of $X$. This turns into a complete metric space if endowed with the Hausdorff distance $d_\\textup{H}$ given by\n\\[ d_\\textup{H}(C,D)=\\inf\\{\\epsilon>0\\,:\\,C\\subseteq\\textup{cl}(D+\\epsilon B_X) \\text{ and }D\\subseteq\\textup{cl}(C+\\epsilon B_X)\\}. \\]\nAll Banach spaces are assumed to be real.\n\n\\section{Geometric radii of closed, convex and bounded sets} \n\\label{sec:radii}\nGiven $\\rho>0$ and a nonempty, closed, convex set $C$ in a Banach space $X$, we denote by $\\textup{ir}(C,\\rho)$ the $\\rho\\,$-\\emph{inner radius} of $C$. That is, the supremum of all $r\\geq 0$ such that $C$ contains a closed ball of the form $x+rB_X$, where $x\\in\\rho B_X$. Clearly, $C$ has nonempty interior if and only if $\\textup{ir}(C,\\rho)>0$ for some $\\rho>0$. The $\\rho\\,$-\\emph{compact radius} $\\textup{kr}(C,\\rho)$ is defined as follows: it is the supremum of all $r\\geq 0$ with the property that, for every compact set $K\\subseteq rB_X$, there is $x\\in\\rho B_X$ such that $x+K\\subseteq C$. The $\\rho\\,$-\\emph{finite radius} $\\textup{fr}(C,\\rho)$ is defined similarly. Namely, it is the supremum of all $r\\geq 0$ with the property that, for every finite set $F\\subseteq rB_X$, there is $x\\in\\rho B_X$ such that $x+F\\subseteq C$. Finally, we introduce the $\\rho\\,$-\\emph{loose radius} $\\textup{lr}(C,\\rho)$ as follows: it is the supremum of all $r\\geq 0$ with the property that, for every finite set $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ and every $\\epsilon>0$, there are a finite set $G=\\{y_1,\\dots,y_n\\}$ and $z\\in\\rho B_X$ such that $\\norm{x_j-y_j}{X}<\\epsilon$ for every $j\\in\\{1,\\dots,n\\}$ and $z+G\\subseteq C$.\n\n\\begin{theo}\n\\label{theo:radii}\nLet $C$ be a nonempty, closed, convex set in a Banach space $X$. The following inequalities hold for every $\\rho>0$.\n\\[ \\textup{ir}(C,\\rho)\\leq\\textup{kr}(C,\\rho)\\leq\\textup{fr}(C,\\rho)\\leq\\textup{lr}(C,\\rho)\\leq 2\\textup{kr}(C,\\rho). \\]\n\\end{theo}\n\\begin{proof}\nThe only nontrivial inequality is the last one and we will prove it using a variation of the Banach-Dieudonn\\'{e} Theorem, similar to the one which appears in \\cite{eva1}. If $\\textup{lr}(C,\\rho)=0$, the claim is obvious. Assume that $\\textup{lr}(C,\\rho)>0$. We aim to show that, for every $r\\geq 0$ such that $2r<\\textup{lr}(C,\\rho)$, every compact set $K\\subseteq rB_X$ can be translated into $C$ via some $z\\in\\rho B_X$. Let $K$ be a compact subset of $rB_X$ and find a finite set $F_1=\\{x_{1,1},\\dots,x_{1,k(1)}\\}\\subseteq 2rB_X$ such that $2^{-1}F_1$ is an $(r\/4)$-net for $K$. Find a further finite set $G_1=\\{y_{1,1},\\dots,y_{1,k(1)}\\}$ and $z_1\\in\\rho B_X$ such that $\\norm{y_{1,j}-x_{1,j}}{X}<r\/2$ for every $j\\in\\{1,\\dots,k(1)\\}$ and such that $z_1+G_1\\subseteq C$. Notice that $2^{-1}G_1$ is an $(r\/2)$-net of $K$. Define \n\\[ K_1=2\\Bigl((K-2^{-1}G_1)\\cap\\frac{r}{2}B_X\\Bigr), \\]\nand observe that $K_1$ is a compact subset of $rB_X$ such that $K\\subseteq 2^{-1}G_1+2^{-1}K_1$. Working by induction, let $n$ be a positive integer and assume that we have already found finite sets $G_1,\\dots,G_n\\subseteq X$, $z_1,\\dots,z_n\\in\\rho B_X$ and a compact set $K_n\\subset rB_X$ such that $z_j+G_j\\subseteq C$ for every $j\\in\\{1,\\dots,n\\}$ and \n\\begin{equation}\n\\label{eq:loose1}\nK\\subseteq \\sum_{j=1}^n2^{-j}G_j+2^{-n}K_n.\n\\end{equation}\nLet $F_{n+1}=\\{x_{n+1,1},\\dots,x_{n+1,k(n+1)}\\}\\subseteq 2rB_X$ be a finite set such that $2^{-1}F_{n+1}$ is an $(r\/4)$-net for $K_n$. Find a further finite set $G_{n+1}=\\{y_{n+1,1},\\dots,y_{n+1,k(n+1)}\\}$ and $z_{n+1}\\in\\rho B_X$ such that $\\norm{y_{n+1,j}-x_{n+1,j}}{X}<r\/2$ for every $j\\in\\{1,\\dots,k(n+1)\\}$ and $z_{n+1}+G_{n+1}\\subseteq C$. Notice again that $2^{-1}G_{n+1}$ is an $(r\/2)$-net of $K_n$. Define \n\\[ K_{n+1}=2\\Bigl((K_n-2^{-1}G_{n+1})\\cap\\frac{r}{2}B_X\\Bigr), \\]\nand observe that $K_{n+1}\\subseteq rB_X$ is compact and that $K_n\\subseteq 2^{-1}G_{n+1}+2^{-1}K_{n+1}$ holds. Now we have \n\\[ K\\subseteq \\sum_{j=1}^{n+1}2^{-j}G_j+2^{-(n+1)}K_{n+1}, \\]\ntherefore we can claim by induction that we can find a sequence ${(z_n)}_{n=1}^\\infty$ in $\\rho B_X$, a sequence ${(G_n)}_{n=1}^\\infty$ of finite subsets of $X$ and a sequence ${(K_n)}_{n=1}^\\infty$ of compact subsets of $rB_X$ such that $z_n+G_n\\subseteq C$ and (\\ref{eq:loose1}) hold for every $n$. In particular, this implies \n\\[ K\\subseteq\\textup{cl}\\biggl(\\,\\sum_{n=1}^\\infty 2^{-n}G_n\\biggr). \\]\nPut $z=\\sum_{n=1}^\\infty 2^{-n}z_n$ and observe that $z\\in\\rho B_X$. Then\n\\[ z+K\\subseteq\\sum_{n=1}^\\infty 2^{-n}z_n+\\textup{cl}\\biggl(\\,\\sum_{n=1}^\\infty 2^{-n}G_n\\biggr)=\\textup{cl}\\biggl(\\,\\sum_{n=1}^\\infty 2^{-n}z_n+\\sum_{n=1}^\\infty 2^{-n}G_n\\biggr)\\subseteq C, \\]\nwhere the last inclusion holds because $C$ is closed and convex. Since $K\\subseteq rB_X$ is arbitrary, we are done.\n\\end{proof}\n\nThe following argument also appears with some minor differences in \\cite{eva1}. We report it here for completeness and formulate it in the language we have just introduced.\n\\begin{lemma}\n\\label{lemma:kr}\nLet $C$ be a closed, convex set in a separable Banach space $X$. $C$ is Haar positive if and only if there is $\\rho>0$ such that $\\textup{kr}(C,\\rho)>0$. It follows in particular that $C$ is Haar positive if and only if $\\textup{lr}(C,\\rho)>0$ for some $\\rho>0$.\n\\end{lemma}\n\\begin{proof}\nIt is a direct consequence of Theorem \\ref{theo:intro} that $\\textup{kr}(C,\\rho)=0$ for every $\\rho>0$ if $C$ is Haar null. To show the converse statement, assume by way of contradiction that, for every positive integer $n$, there is a compact set $K_n\\subseteq n^{-1}B_X$ such that, whenever $x+K_n\\subseteq C$ for some $x\\in X$, we have $\\norm{x}{X}>n$. By Theorem \\ref{theo:intro}, there is $r>0$ such that every compact subset of $rB_X$ can be translated into $C$. In particular, this holds for\n\\[ K=\\{0\\}\\cup\\bigcup_{n>r^{-1}}K_n. \\]\nIf $x$ is such that $x+K\\subseteq C$, then $x+K_n\\subseteq C$ for every $n>r^{-1}$. It then follows by the choice of $K_n$ that $\\norm{x}{X}>n$ for every positive integer $n$, which does not make any sense. The last assertion is a consequence of Theorem \\ref{theo:radii}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:lr-fr}\nGiven $\\rho>0$ and a nonempty, closed, convex set $C$ in a Banach space $X$, we have\n\\[ \\textup{lr}(C,\\rho)=\\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr). \\]\n\\end{lemma}\n\\begin{proof}\nSet $r=\\textup{lr}(C,\\rho)$. Pick $\\epsilon>0$ and find $F=\\{x_1,\\dots,x_n\\}\\subset(r+\\epsilon)B_X$ and $\\delta_0>0$ such that every finite set $G=\\{y_1,\\dots,y_n\\}$ which fulfills the condition $\\norm{y_j-x_j}{X}<2\\delta_0$ for every $j\\in\\{1,\\dots,n\\}$ cannot be translated into $C$ via some $z\\in\\rho B_X$. Suppose that there is $z\\in\\rho B_X$ such that $z+F\\subset\\textup{cl}(C+\\delta_0B_X)$. This would imply that, for every $j\\in\\{1,\\dots,n\\}$, we can find $w_j\\in C$ such that $\\norm{z+x_j-w_j}{X}<2\\delta_0$. Put $y_j=w_j-z$ for every $j$. Then the set $G=\\{y_1,\\dots,y_n\\}$ satisfies $z+G\\subseteq C$ and $\\norm{x_j-y_j}{X}<2\\delta_0$ for every $j$, in contradiction with the choice of $F$ and $\\delta_0$. Thus, $F$ cannot be translated into $\\textup{cl}(C+\\delta_0B_X)$ via some $z\\in\\rho B_X$, which yields\n\\[ \\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr)\\leq\\textup{fr}\\bigl(\\textup{cl}(C+\\delta_0B_X),\\rho\\bigr)\\leq r+\\epsilon. \\]\nSince $\\epsilon$ is arbitrary, we get\n\\[ \\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr)\\leq r. \\]\nTo show the opposite inequality, observe that it is obvious if $r=0$. Assume that $r>0$, choose $\\delta\\in(0,r)$ and pick a finite set $F=\\{x_1,\\dots,x_n\\}\\subset(r-\\delta)B_X$. Find $G=\\{y_1,\\dots,y_n\\}$ and $z\\in\\rho B_X$ such that $z+G\\subseteq C$ and $\\norm{x_j-y_j}{X}<\\delta$ for each $j$. Notice that\n\\[ z+x_j=z+y_j+x_j-y_j\\in C+\\delta B_X. \\]\nfor each $j$, hence $z+F\\subseteq\\textup{cl}(C+\\delta B_X)$. Since $\\delta>0$ and $F\\subset (r-\\delta)B_X$ are arbitrary, we conclude that\n\\[ \\lim_{\\delta\\to 0^+}\\textup{fr}\\bigl(\\textup{cl}(C+\\delta B_X),\\rho\\bigr)\\geq\\lim_{\\delta\\to 0^+}(r-\\delta)=r, \\]\nas wished.\n\\end{proof}\n\n\\section{Weak$^*$ closures of Haar positive closed, convex sets}\n\\label{sec:weaks}\nGiven a Banach space $X$ and a positive integer $n$, recall that we can endow the Banach space $X^n$, the product of $n$ copies of $X$, with the $\\infty$-product norm:\n\\[ \\norm{x}{X^n}=\\max_{1\\leq j\\leq n}\\norm{x_j}{X} \\]\nfor every $x=(x_1,\\dots,x_n)\\in X^n$. In this way we have $B_{X^n}={(B_X)}^n$. We denote by $\\func{\\Delta}{X}{X^n}$ the diagonal embedding $x\\mapsto(x,\\dots,x)$. The second dual space of $X^n$ is simply ${(X^{**})}^n$, endowed with the same $\\infty$-product norm. Notice that, in ${(X^{**})}^n$, we have $\\textup{w}^*\\textup{-cl}(\\Delta(\\rho B_X))=\\Delta(\\rho B_{X^{**}})$. We are now ready to state our main theorem.\n\\begin{theo}\n\\label{theo:hpws}\nLet $\\rho>0$, let $C$ be a nonempty, closed, convex set in a Banach space $X$ and let $r\\geq 0$. The following assertions are equivalent.\n\\begin{enumerate}\n\\item $\\textup{lr}(C,\\rho)\\geq r$.\n\\item For every finite $F\\subseteq rB_X$ there is $z\\in\\rho B_{X^{**}}$ such that $z+F\\subseteq\\textup{w}^*\\textup{-cl}(C)$.\n\\item There is $z_0\\in\\rho B_{X^{**}}$ such that $z_0+rB_{X^{**}}\\subseteq\\textup{w}^*\\textup{-cl}(C)$.\n\\end{enumerate}\nIn particular, $\\textup{lr}(C,\\rho)=\\textup{ir}(\\textup{w}^*\\textup{-cl}(C),\\rho)$ and, in case $X$ is separable, $C$ is Haar positive if and only if $\\textup{w}^*\\textup{-cl}(C)$ has nonempty interior in the norm topology of $X^{**}$.\n\\end{theo}\n\\begin{proof}\n(1)$\\Rightarrow$(2). Let $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ be a finite set and, for each positive integer $k$, find $G_k=\\{y_{k,1},\\dots,y_{k,n}\\}$ and $z_k\\in\\rho B_X$ such that $\\norm{x_{k,j}-y_{k,j}}{X}<k^{-1}$ for each $j\\in\\{1,\\dots,n\\}$ and $z_k+G_k\\subseteq C$. Since ${(z_k)}_{k=1}^\\infty$ is a bounded sequence, it admits a subsequence with a weak$^*$ limit $z\\in\\rho B_{X^{**}}$. Now, it is not hard to see that $z+F\\subseteq\\textup{w}^*\\textup{-cl}(C)$.\n\n(2)$\\Rightarrow$(3). Let $\\mathcal{U}$ be the directed set of all weak$^*$ neighbourhoods of $0$ in $X^{**}$, ordered by reverse inclusion. Since $rB_X$ is weakly$^*$ dense in $rB_{X^{**}}$ and $rB_{X^{**}}$ is weakly$^*$ compact, for every $U\\in\\mathcal{U}$ we can find a finite set $F_U\\subseteq rB_X$ and $z_U\\in \\rho B_{X^{**}}$ such that $rB_{X^{**}}\\subseteq F_U+U$ and $z_U+F_U\\subseteq\\textup{w}^*\\textup{-cl}(C)$. Since ${(z_U)}_{U\\in\\mathcal{U}}$ is a bounded net, there exist another directed set $I$ and a map $\\func{\\psi}{I}{\\mathcal{U}}$ such that ${(z_{\\psi(\\alpha)})}_{\\alpha\\in I}$ is a subnet of ${(z_U)}_{U\\in\\mathcal{U}}$ which converges in the weak$^*$ topology to some $z_0\\in\\rho B_{X^{**}}$. Assume by contradiction that there is $x_0\\in rB_{X^{**}}\\setminus(\\textup{w}^*\\textup{-cl}(C)-z_0)$. The Hahn-Banach Theorem delivers $f\\in X^*$ and $t\\in\\mathbb{R}$ such that $f(x_0)>t>f(x)$ for every $x\\in\\textup{w}^*\\textup{-cl}(C)-z_0$. Define\n\\[ s=\\frac{f(x_0)-t}{2} \\]\nand $V=\\{x\\in X^{**}\\,:\\,|f(x)|<s\\}$. Find $\\alpha_0\\in I$ such that $\\psi(\\alpha)\\subseteq V$ for every $\\alpha\\geq\\alpha_0$. Further, find $\\beta_0\\in I$ such that $z_{\\psi(\\beta)}-z_0\\in V$ for every $\\beta\\geq\\beta_0$. Take $\\gamma\\in I$ such that $\\gamma\\geq\\alpha_0$ and $\\gamma\\geq\\beta_0$ and find $y_0\\in F_{\\psi(\\gamma)}$ such that $x_0\\in y_0+\\psi(\\gamma)\\subseteq y_0+V$. Then we have $|f(y_0)-f(x_0)|<s$, which implies\n\\[ f(y_0)>f(x_0)-s=\\frac{f(x_0)+t}{2}. \\]\nAt the same time though, we have $z_{\\psi(\\gamma)}+y_0-z_0\\in\\textup{w}^*\\textup{-cl}(C)-z_0$, therefore $|f(z_{\\psi(\\gamma)})-f(z_0)+f(y_0)|<t$, which implies\n\\[ f(y_0)<t+|f(z_{\\psi(\\gamma)})-f(z_0)|<t+s=\\frac{f(x_0)+t}{2}, \\]\na contradiction.\n\n(3)$\\Rightarrow$(1). Assume by contradiction that $\\textup{lr}(C,\\rho)<r$. By Lemma \\ref{lemma:lr-fr}, there is $\\delta>0$ such that $\\textup{fr}(\\textup{cl}(C+\\delta B_X),\\rho)<r$, which in turn means that there is a finite set $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ such that $(z+F)\\setminus\\textup{cl}(C+\\delta B_X)\\ne\\varnothing$ for every $z\\in\\rho B_X$. In $X^n$ define $D=(C-x_1)\\times\\cdots\\times(C-x_n)$ and observe that\n\\[ \\textup{cl}(D+\\delta B_{X^n})=\\bigl(\\textup{cl}(C+\\delta B_X)-x_1\\bigr)\\times\\cdots\\times\\bigl(\\textup{cl}(C+\\delta B_X)-x_n\\bigr). \\]\nNotice moreover that $\\Delta(\\rho B_X)\\cap\\textup{cl}(D+\\delta B_{X^n})=\\varnothing$, otherwise it would hold that $z+F\\subseteq\\textup{cl}(C+\\delta B_X)$ for every $(z,\\dots,z)\\in \\Delta(\\rho B_X)\\cap\\textup{cl}(D+\\delta B_{X^n})$, against how $F$ has been chosen. Hence, by the Hahn-Banach theorem, there are $f\\in{(X^n)}^*$ and $t_1,t_2\\in\\mathbb{R}$ such that $t_1<t_2$, $f(x)<t_1$ for every $x\\in\\Delta(\\rho B_X)$ and $f(x)>t_2$ for every $x\\in D$. By taking the weak$^*$ closures of both sets in ${(X^n)}^{**}$, we have $f(x)\\leq t_1$ for every $x\\in\\Delta(\\rho B_{X^{**}})$ and $f(x)\\geq t_2$ for every $x\\in\\textup{w}^*\\textup{-cl}(D)$, thus $\\Delta(\\rho B_{X^{**}})\\cap\\textup{w}^*\\textup{-cl}(D)=\\varnothing$. This is a contradiction, because\n\\[ z_0\\in\\bigcap_{j=1}^n\\bigl(\\textup{w}^*\\textup{-cl}(C)-x_j\\bigr), \\]\ni.e.\\ $(z_0,\\dots,z_0)\\in(\\textup{w}^*\\textup{-cl}(C)-x_1)\\times\\cdots\\times(\\textup{w}^*\\textup{-cl}(C)-x_n)=\\textup{w}^*\\textup{-cl}(D)$.\n\nThe last statement is a consequence of Lemma \\ref{lemma:kr}.\n\\end{proof}\n\nTheorem \\ref{theo:hpws} offers an interesting connection with the theory of weak$^*$ derived sets. If $X$ is a Banach space and $E$ is a subset of $X^*$, the first weak$^*$ derived set of $E$ is given by\n\\[ E^{(1)}=\\bigcup_{n=1}^\\infty\\textup{w}^*\\textup{-cl}(E\\cap nB_{X^*}) \\]\nand corresponds to the set of all possible limits of bounded, weak$^*$ convergent nets with elements in $E$. Clearly $E^{(1)}\\subseteq\\textup{w}^*\\textup{-cl}(E)$. We refer the reader to \\cite{ostr_2023} and the references therein for a detailed account on weak$^*$ derived sets. In particular, it has to be remarked that, in general, $E^{(1)}$ and $\\textup{w}^*\\textup{-cl}(E)$ can be different. However, we have the following result.\n\n\\begin{cor}\nLet $C$ be a closed, convex set in a Banach space $X$. In the second dual $X^{**}$, $C^{(1)}$ has empty interior in the norm topology if and only if $\\textup{w}^*\\textup{-cl}(C)$ does.\n\\end{cor}\n\\begin{proof}\nOne of the implications follows from $C^{(1)}\\subseteq\\textup{w}^*\\textup{-cl}(C)$. Conversely, assume that $\\textup{w}^*\\textup{-cl}(C)$ has nonempty interior in the norm topology. Then, by Theorem \\ref{theo:hpws} and Theorem \\ref{theo:radii}, there are $\\rho>0$ and $r>0$ such that $\\textup{kr}(C,\\rho)\\geq r$. Now, it is not hard to see that $\\textup{kr}(C\\cap nB_X,\\rho)\\geq r$ for every positive integer $n>\\rho+r$, hence $\\textup{w}^*\\textup{-cl}(C\\cap n B_X)$ has nonempty interior in the norm topology for every $n>\\rho+r$ and so does $C^{(1)}$.\n\\end{proof}\n\n\\section{Haar positive sets and the Hausdorff metric}\n\\label{sec:top}\n\nLet $C$ be a closed and convex set in a Banach space $X$. Under the additional assumption that $C$ is bounded we define \n\\begin{align*} \n\\textup{bir}(C)=\\sup_{\\rho>0}\\textup{ir}(C,\\rho),&\\quad\\textup{bkr}(C)=\\sup_{\\rho>0}\\textup{kr}(C,\\rho),\\\\\n\\textup{bfr}(C)=\\sup_{\\rho>0}\\textup{fr}(C,\\rho),&\\quad\\textup{blr}(C)=\\sup_{\\rho>0}\\textup{lr}(C,\\rho). \n\\end{align*}\nAll these values are finite, as $\\textup{diam}(C)$ is an upper estimate for all of them. Moreover, the chain of inequalities \n\\begin{equation} \n\\label{eq:radii}\n\\textup{bir}(C)\\leq\\textup{bkr}(C)\\leq\\textup{bfr}(C)\\leq\\textup{blr}(C)\\leq 2\\textup{bkr}(C)\n\\end{equation}\nis a direct consequence of Theorem \\ref{theo:radii}. We call $\\textup{bir}(C)$ the \\emph{bounded inner radius} of $C$. It is the supremum of all $r\\geq 0$ such that $C$ contains a closed ball of radius $r$. $\\textup{bkr}(C)$ is the \\emph{bounded compact radius} of $C$ and corresponds to the supremum of all $r\\geq 0$ with the property that, for every compact set $K\\subseteq rB_X$, there is $x\\in X$ such that $x+K\\subseteq C$. The \\emph{bounded finite radius} $\\textup{bfr}(C)$ is, similarly, the supremum of all $r\\geq 0$ with the property that, for every finite set $F\\subseteq rB_X$, there is $x\\in X$ such that $x+F\\subseteq C$. Finally, the \\emph{bounded loose radius} $\\textup{blr}(C)$ is the supremum of all $r\\geq 0$ with the property that, for every finite set $F=\\{x_1,\\dots,x_n\\}\\subseteq rB_X$ and every $\\epsilon>0$, there are a finite set $G=\\{y_1,\\dots,y_n\\}$ and $z\\in X$ such that $\\norm{x_j-y_j}{X}<\\epsilon$ for every $j$ and $z+G\\subseteq C$. In case $X$ is separable, $C$ is Haar positive if and only if $\\textup{blr}(C)>0$. This follows from ($\\ref{eq:radii}$) and Theorem \\ref{theo:intro}.\n\nTheorem \\ref{theo:hpws} allows to prove that the family $\\mathcal{H}^+(X)$ of Haar positive, closed, convex and bounded subsets of a separable Banach space $X$ is an open subset of $\\mathcal{C}(X)$. This result follows from a few lemmas we are going to show.\n\n\\begin{lemma}\n\\label{lemma:hpws}\nGiven a closed, convex and bounded set $C$ in a Banach space $X$, we have $\\textup{blr}(C)=\\textup{bir}(\\textup{w}^*\\textup{-cl}(C))$.\n\\end{lemma}\n\\begin{proof}\nUsing Theorem \\ref{theo:hpws}, we have\n\\[ \\textup{blr}(C)=\\sup_{\\rho>0}\\textup{lr}(C,\\rho)=\\sup_{\\rho>0}\\textup{ir}\\bigl(\\textup{w}^*\\textup{-cl}(C),\\rho\\bigl)=\\textup{bir}\\bigl(\\textup{w}^*\\textup{-cl}(C)\\bigr). \\qedhere \\]\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:wscliso}\nGiven a Banach space $X$, the function $\\func{\\textup{w}^*\\textup{-cl}}{\\mathcal{C}(X)}{\\mathcal{C}(X^{**})}$ is an isometry.\n\\end{lemma}\n\\begin{proof}\nTake $\\epsilon>0$ and let $C,D\\in\\mathcal{C}(X)$ be such that $d_\\textup{H}(C,D)\\leq\\epsilon$. This means that $C\\subseteq\\textup{cl}(D+\\epsilon B_X)$ and $D\\subseteq\\textup{cl}(C+\\epsilon B_X)$. Take $x\\in\\textup{w}^*\\textup{-cl}(C)$ and let ${(x_\\alpha)}_{\\alpha\\in I}$ be a net in $C$ whose weak$^*$ limit is $x$. Choose $\\delta>0$ and, for each $\\alpha\\in I$, find $y_\\alpha\\in D$ and $z_\\alpha\\in\\epsilon B_X$ such that $\\norm{x_\\alpha-(y_\\alpha+z_\\alpha)}{X}\\leq\\delta$. By considering a subnet if necessary, we can assume that the nets ${(y_\\alpha)}_{\\alpha\\in I}$ and ${(z_\\alpha)}_{\\alpha\\in I}$ have weak$^*$ limits $y\\in\\textup{w}^*\\textup{-cl}(D)$ and $z\\in\\epsilon B_{X^{**}}$ respectively. Moreover, \n\\[ \\norm{x-(y+z)}{X^{**}}\\leq\\liminf_{\\alpha\\in I}\\norm{x_\\alpha-(y_\\alpha+z_\\alpha)}{X}\\leq\\delta, \\]\ni.e.\\ $x\\in\\textup{w}^*\\textup{-cl}(D)+(\\epsilon+\\delta)B_{X^{**}}$. Since $\\delta$ is arbitrary, we have\n\\[ x\\in\\bigcap_{\\delta>0}\\bigl(\\textup{w}^*\\textup{-cl}(D)+(\\epsilon+\\delta)B_{X^{**}}\\bigr)=\\textup{cl}\\bigl(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}}\\bigr). \\]\nAs $x$ is also arbitrary, we conclude that $\\textup{w}^*\\textup{-cl}(C)\\subseteq\\textup{cl}(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}})$. The inclusion $\\textup{w}^*\\textup{-cl}(D)\\subseteq\\textup{cl}(\\textup{w}^*\\textup{-cl}(C)+\\epsilon B_{X^{**}})$ is shown similarly, hence we get that $d_\\textup{H}(\\textup{w}^*\\textup{-cl}(C),\\textup{w}^*\\textup{-cl}(D))\\leq\\epsilon$. \n\nConversely, suppose that $d_\\textup{H}(C,D)>\\epsilon$ and, by swapping $C$ and $D$ if necessary, assume that there is $x_0\\in C\\setminus\\textup{cl}(D+\\epsilon B_X)$. Using the Hahn-Banach Theorem, find $f\\in X^*$ and $t\\in\\mathbb{R}$ such that $f(x_0)>t>f(x)$ for every $x\\in\\textup{cl}(D+\\epsilon B_X)$. Then $f(x)\\leq t$ for every $x\\in\\textup{w}^*\\textup{-cl}(D+\\epsilon B_X)$. Now observe that \n\\begin{align*} \n\\textup{cl}(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}}) &= \\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}}= \\\\\n&=\\textup{w}^*\\textup{-cl}(D)+\\textup{w}^*\\textup{-cl}(\\epsilon B_{X})\\subseteq\\textup{w}^*\\textup{-cl}(D+\\epsilon B_X). \n\\end{align*}\nThus we get $x_0\\notin\\textup{cl}(\\textup{w}^*\\textup{-cl}(D)+\\epsilon B_{X^{**}})$, from which $d_\\textup{H}(\\textup{w}^*\\textup{-cl}(C),\\textup{w}^*\\textup{-cl}(D))>\\epsilon$ follows. Since $C$, $D$ and $\\epsilon$ are arbitrary, the proof is complete.\n\\end{proof}\n\nFinally, we want to show the continuity of the bounded inner radius in the metric space of all nonempty, closed, convex and bounded subsets of a Banach space, endowed with the Hausdorff metric. Although it seems that this result cannot be found in the literature, it might be well-know and belong to the folklore. We prove it here for the sake of completeness.\n\\begin{lemma}\n\\label{lemma:ir}\nIn a Banach space $X$, the function $\\func{\\textup{bir}}{\\mathcal{C}(X)}{[0,+\\infty)}$ is Lipschitz continuous with Lipschitz constant $1$.\n\\end{lemma}\n\\begin{proof}\nThe proof is based on the following claim: if $C$ is a nonempty, closed, convex and bounded set in $X$ and $\\epsilon>0$, then\n\\begin{equation}\n\\label{eq:ir+eps}\n\\textup{bir}\\bigl(\\textup{cl}(C+\\epsilon B_X)\\bigr)=\\textup{bir}(C)+\\epsilon.\n\\end{equation}\n\nLet us see first that $\\textup{bir}(\\textup{cl}(C+\\epsilon))\\leq\\textup{bir}(C)+\\epsilon$. Set $r=\\textup{bir}(C)$ and, looking for a contradiction, assume that $\\textup{bir}(\\textup{cl}(C+\\epsilon))>r+\\epsilon$. Then there is $\\delta>0$ such that $r+\\epsilon+\\delta<\\textup{bir}(\\textup{cl}(C+\\epsilon))$ and, without loss of generality, we may assume that $(r+\\epsilon+\\delta)B_X\\subseteq\\textup{cl}(C+\\epsilon B_X)$. Since $(r+\\delta)B_X\\setminus C\\ne\\varnothing$, the Hahn-Banach Theorem provides $x_0\\in(r+\\delta)B_X$, $t\\in\\mathbb{R}$ and $f\\in S_{X^*}$ such that $f(x_0)>t>f(x)$ for every $x\\in C$. Further, we have $t<\\norm{f}{X^*}\\norm{x_0}{X}\\leq r+\\delta$. Find $\\delta'$ such that $r+\\delta-\\delta'-t>0$ and $x_1\\in(r+\\epsilon+\\delta)B_X$ such that $f(x_1)>r+\\epsilon+\\delta-\\delta'$. Since $(r+\\epsilon+\\delta)B_X\\subseteq\\textup{cl}(C+\\epsilon B_X)$, there exist $x\\in C$ and $y\\in\\epsilon B_X$ such that\n\\[ \\norm{x_1-(x+y)}{X}<r+\\delta-\\delta'-t \\]\nOn the other hand, we have\n\\[ \\norm{x_1-(x+y)}{X}\\geq f(x_1)-f(x)-f(y)>r+\\epsilon+\\delta-\\delta'-t-\\epsilon=r+\\delta-\\delta'-t, \\]\na contradiction. \n\nTo see that $\\textup{bir}(C)+\\epsilon\\leq\\textup{bir}(\\textup{cl}(C+\\epsilon B_X))$, set again $r=\\textup{bir}(C)$, choose $\\delta\\in(0,r)$ and find $x_0\\in X$ such that $x_0+(r-\\delta)B_X\\subseteq C$. Then $x_0+(r+\\epsilon-\\delta)B_X\\subseteq\\textup{cl}(C+\\epsilon B_X)$. Since $\\delta$ is arbitrary, it follows that $\\textup{bir}(\\textup{cl}(C+\\epsilon B_X))\\geq r+\\epsilon$, as wished.\n\nTo prove the statement, pick $C,D\\in\\mathcal{C}(X)$ and $\\epsilon>0$. If $d_\\textup{H}(C,D)\\leq\\epsilon$, then $D\\subseteq\\textup{cl}(C+\\epsilon B_X)$, which implies by (\\ref{eq:ir+eps}) that $\\textup{bir}(D)\\leq\\textup{bir}(C)+\\epsilon$. The inequality $\\textup{bir}(C)\\leq\\textup{bir}(D)+\\epsilon$ follows similarly. Thus, $|\\textup{bir}(C)-\\textup{bir}(D)|\\leq\\epsilon$. Since $C,D$ and $\\epsilon$ are arbitrary, this lets us conclude that $\\textup{bir}$ is $1$-Lipschitz.\n\\end{proof}\n\n\\begin{theo}\nLet $X$ be a Banach space. The function $\\func{\\textup{blr}}{\\mathcal{C}(X)}{\\mathbb{R}}$ is $1$-Lipschitz. In particular, in case $X$ is separable, the family $\\mathcal{H}^+(X)$ of all Haar positive closed, convex and bounded subsets of $X$ is open. Equivalently, a convergent sequence ${(C_n)}_{n=1}^\\infty\\subset\\mathcal{C}(X)$ of Haar null sets has a Haar null limit.\n\\end{theo}\n\\begin{proof}\nWe have $\\textup{blr}=\\textup{bir}\\circ\\textup{w}^*\\textup{-cl}$ by Lemma \\ref{lemma:hpws}, Lemma \\ref{lemma:wscliso} and Lemma \\ref{lemma:ir}, hence $\\textup{blr}$ is a composition of $1$-Lipschitz maps and therefore it is $1$-Lipschitz. Since\n\\[ \\mathcal{H}^+(X)=\\textup{blr}^{-1}\\bigl((0,+\\infty)\\bigr), \\]\nit follows immediately that $\\mathcal{H}^+(X)$ is open.\n\\end{proof}\n\n\\section*{Acknowledgements}\nThe author would like to thank Professors Eva Kopeck\\'{a} and Christian Bargetz for the many helpful remarks, and Professor Mikhail Ostrovskii for pointing out a mistake in an earlier version of the preprint. The author's research is supported by the Austrian Science Fund (FWF): P 32523-N.\n\n\\printbibliography\n\n\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe area of high-dimensional statistics deals with estimation in the\n``large \\ensuremath{p}, small \\ensuremath{n}'' setting, where $\\ensuremath{p}$ and $\\ensuremath{n}$\ncorrespond, respectively, to the dimensionality of the data and the\nsample size. Such high-dimensional problems arise in a variety of\napplications, among them remote sensing, computational biology and\nnatural language processing, where the model dimension may be\ncomparable or substantially larger than the sample size.  It is\nwell-known that such high-dimensional scaling can lead to dramatic\nbreakdowns in many classical procedures.  In the absence of additional\nmodel assumptions, it is frequently impossible to obtain consistent\nprocedures when $\\ensuremath{p} \\gg \\ensuremath{n}$. Accordingly, an active line of\nstatistical research is based on imposing various restrictions on the\nmodel----for instance, sparsity, manifold structure, or graphical\nmodel structure----and then studying the scaling behavior of different\nestimators as a function of sample size $\\ensuremath{n}$, ambient dimension\n$\\ensuremath{p}$ and additional parameters related to these structural\nassumptions.\n\nIn this paper, we study the following problem: given $\\ensuremath{n}$ i.i.d.\nobservations $\\{X^{(\\obsind)}\\}_{\\ensuremath{k}=1}^{\\ensuremath{n}}$ of a zero mean\nrandom vector $X \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p}}$, estimate both its covariance\nmatrix \\mbox{$\\ensuremath{\\ensuremath{\\Sigma}^*}$,} and its inverse covariance or\nconcentration matrix $\\ensuremath{\\Theta^*} \\ensuremath{:  =} \\inv{\\ensuremath{\\ensuremath{\\Sigma}^*}}$.  Perhaps\nthe most natural candidate for estimating $\\ensuremath{\\ensuremath{\\Sigma}^*}$ is the\nempirical sample covariance matrix, but this is known to behave poorly\nin high-dimensional settings. For instance, when $\\ensuremath{p}\/\\ensuremath{n}\n\\rightarrow c > 0$, and the samples are drawn i.i.d. from a\nmultivariate Gaussian distribution, neither the eigenvalues nor the\neigenvectors of the sample covariance matrix are consistent estimators\nof the population\nversions~\\cite{Johnstone2001,JohnstoneLu2004}. Accordingly, many\nregularized estimators have been proposed to estimate the covariance\nor concentration matrix under various model assumptions.  One natural\nmodel assumption is that reflected in shrinkage estimators, such as in\nthe work of \\citet{LedoitWolf2003}, who proposed to shrink the sample\ncovariance to the identity matrix. An alternative model assumption,\nrelevant in particular for time series data, is that the covariance or\nconcentration matrix is banded, meaning that the entries decay based\non their distance from the diagonal. \\citet{Furrer2007} proposed to\nshrink the covariance entries based on this distance from the\ndiagonal. \\citet{Wu2003} and \\citet{Huang2006} estimate these banded\nconcentration matrices by using thresholding and $\\ell_1$-penalties\nrespectively, as applied to a Cholesky factor of the inverse\ncovariance matrix. \\citet{BickelLevina2008} prove the consistency of\nthese banded estimators so long as $\\frac{(\\log\\,\\ensuremath{p})^2}{\\ensuremath{n}}\n\\rightarrow 0$ and the model covariance matrix is banded as well, but\nas they note, these estimators depend on the presented order of the\nvariables.\n\nA related class of models are based on positing some kind of sparsity,\neither in the covariance matrix, or in the inverse covariance.\n\\citet{BickelLevina2007} study thresholding estimators of covariance\nmatrices, assuming that each row satisfies an $\\ell_q$-ball sparsity\nassumption.  In independent work, \\citet{Karoui2007} also studied\nthresholding estimators of the covariance, but based on an alternative\nnotion of sparsity, one which captures the number of closed paths of\nany length in the associated graph.  Other work has studied models in\nwhich the inverse covariance or concentration matrix has a sparse\nstructure.  As will be clarified in the next section, when the random\nvector is multivariate Gaussian, the set of non-zero entries in the\nconcentration matrix correspond to the set of edges in an associated\nGaussian Markov random field (GMRF).  In this setting, imposing\nsparsity on the concentration matrix can be interpreted as requiring\nthat the graph underlying the GMRF have relatively few edges.  A line\nof recent papers~\\citep{AspreBanG2008,FriedHasTib2007,YuanLin2007}\nhave proposed an estimator that minimizes the Gaussian negative\nlog-likelihood regularized by the $\\ell_1$ norm of the entries (or the\noff-diagonal entries) of the concentration matrix. The resulting\noptimization problem is a log-determinant program, which can be solved\nin polynomial time with interior point methods~\\citep{Boyd02}, or by\nfaster co-ordinate descent\nalgorithms~\\citep{AspreBanG2008,FriedHasTib2007}.  In recent work,\n\\citet{Rothman2007} have analyzed some aspects of high-dimensional\nbehavior of this estimator; assuming that the minimum and maximum\neigenvalues of $\\ensuremath{\\ensuremath{\\Sigma}^*}$ are bounded, they show that consistent\nestimates can be achieved in Frobenius and operator norm, in\nparticular at the rate ${\\mathcal{O}}(\\sqrt{\\frac{(\\ensuremath{s} + \\ensuremath{p}) \\log\n\\ensuremath{p}}{\\ensuremath{n}}})$.\n\n\nThe focus of this paper is the problem of estimating the concentration\nmatrix $\\Theta^*$ under sparsity conditions.  We do not impose\nspecific distributional assumptions on $X$ itself, but rather analyze the estimator in terms of \nthe tail behavior of the maximum deviation $\\max_{i,j}\n|\\estim{\\Sigma}^n_{ij} - \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}|$ of the sample and\npopulation covariance matrices.  To estimate $\\Theta^*$, we\nconsider minimization of an $\\ell_1$-penalized log-determinant Bregman\ndivergence, which is equivalent to the usual $\\ell_1$-penalized\nmaximum likelihood when $X$ is multivariate Gaussian.  We analyze the\nbehavior of this estimator under high-dimensional scaling, in which\nthe number of nodes $\\ensuremath{p}$ in the graph, and the maximum node degree\n$\\ensuremath{\\ensuremath{d}}$ are all allowed to grow as a function of the sample size\n$\\ensuremath{n}$.\n\nIn addition to the triple $(\\ensuremath{n}, \\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$, we also\nexplicitly keep track of certain other measures of model complexity,\nthat could potentially scale as well.  The first of these measures is\nthe $\\ell_\\infty$-operator norm of the covariance matrix\n$\\ensuremath{\\ensuremath{\\Sigma}^*}$, which we denote by $\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{:  =}\n\\matnorm{\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty}$.  The next quantity involves the\nHessian of the log-determinant objective function, $\\ensuremath{\\Gamma^*} \\ensuremath{:  =}\n(\\ensuremath{\\Theta^*})^{-1} \\otimes (\\ensuremath{\\Theta^*})^{-1}$.  When the distribution\nof $X$ is multivariate Gaussian, this Hessian has the more explicit\nrepresentation $\\ensuremath{\\Gamma^*}_{(j,k), (\\ell, m)} = \\operatorname{cov}\\{X_j\nX_k, \\; X_\\ell X_m \\}$, showing that it measures the covariances of\nthe random variables associated with each edge of the graph.  For this\nreason, the matrix $\\ensuremath{\\Gamma^*}$ can be viewed as an edge-based\ncounterpart to the usual node-based covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.\nUsing $\\ensuremath{S}$ to index the variable pairs $(i,j)$ associated with\nnon-zero entries in the inverse covariance.  our analysis involves the\nquantity $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} = \\matnorm{(\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}})^{-1}}{\\infty}$. Finally, we also impose a mutual\nincoherence or irrepresentability condition on the Hessian $\\ensuremath{\\Gamma^*}$;\nthis condition is similar to assumptions imposed on $\\ensuremath{\\ensuremath{\\Sigma}^*}$ in\nprevious work~\\cite{Tropp2006,Zhao06,MeinsBuhl2006,Wainwright2006_new} on\nthe Lasso.  We provide some examples where the Lasso\nirrepresentability condition holds, but our corresponding condition on\n$\\ensuremath{\\Gamma^*}$ fails; however, we do not know currently whether one\ncondition strictly dominates the other. \n\n\nOur first result establishes consistency of our estimator\n$\\estim{\\Theta}$ in the elementwise maximum-norm, providing a rate\nthat depends on the tail behavior of the entries in the random matrix\n$\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n} - \\ensuremath{\\ensuremath{\\Sigma}^*}$. For the special case of sub-Gaussian\nrandom vectors with concentration matrices having at most $d$\nnon-zeros per row, a corollary of our analysis is consistency in\nspectral norm at rate \\mbox{$\\matnorm{\\estim{\\Theta} - \\Theta^*}{2} =\n{\\mathcal{O}}(\\sqrt{(\\ensuremath{\\ensuremath{d}}^2 \\,\\log \\ensuremath{p})\/\\ensuremath{n}})$,} with high\nprobability, thereby strengthening previous\nresults~\\cite{Rothman2007}.  Under the milder restriction of each\nelement of $X$ having bounded $4m$-th moment, the rate in\nspectral norm is substantially slower---namely,\n\\mbox{$\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{2} = {\\mathcal{O}}(\\ensuremath{\\ensuremath{d}}\\,\n\\ensuremath{p}^{1\/2m}\/\\sqrt{\\ensuremath{n}})$}---highlighting that the familiar\nlogarithmic dependence on the model size $\\ensuremath{p}$ is linked to\nparticular tail behavior of the distribution of $X$.  Finally, we show\nthat under the same scalings as above, with probability converging to\none, the estimate $\\estim{\\Theta}$ correctly specifies the zero\npattern of the concentration matrix $\\Theta^*$.\n\n\nThe remainder of this paper is organized as follows.  In\nSection~\\ref{SecBackground}, we set up the problem and give some\nbackground.  Section~\\ref{SecResult} is devoted to statements of our\nmain results, as well as discussion of their consequences.\nSection~\\ref{SecProof} provides an outline of the proofs, with the\nmore technical details deferred to appendices.  In\nSection~\\ref{SecExperiments}, we report the results of some simulation\nstudies that illustrate our theoretical predictions.\n\n\n\\myparagraph{Notation} For the convenience of the reader, we summarize\nhere notation to be used throughout the paper. Given a vector $\\ensuremath{u}\n\\in \\ensuremath{{\\mathbb{R}}}^\\ensuremath{d}$ and parameter $a \\in [1, \\infty]$, we use\n$\\|\\ensuremath{u}\\|_a$ to denote the usual $\\ell_a$ norm.  Given a matrix\n$\\ensuremath{U} \\in \\ensuremath{{\\mathbb{R}}}^{p \\times p}$ and parameters $a,b \\in [1, \\infty]$,\nwe use $\\matnorm{\\ensuremath{U}}{a,b}$ to denote the induced matrix-operator\nnorm $\\max_{\\|y\\|_a = 1} \\|\\ensuremath{U} y\\|_b$; see \\citet{Horn1985} for\nbackground.  Three cases of particular importance in this paper are\nthe \\emph{spectral norm} $\\matnorm{\\ensuremath{U}}{2}$, corresponding to the\nmaximal singular value of $\\ensuremath{U}$; the\n\\emph{$\\ell_\\infty\/\\ell_\\infty$-operator norm}, given by\n\\begin{eqnarray}\n\\label{EqnLinfOp}\n\\matnorm{\\ensuremath{U}}{\\infty} & \\ensuremath{:  =} & \\max \\limits_{j=1, \\ldots, p}\n\\sum_{k=1}^p |\\ensuremath{U}_{jk}|,\n\\end{eqnarray}\nand the \\emph{$\\ell_1\/\\ell_1$-operator norm}, given by\n$\\matnorm{\\ensuremath{U}}{1} = \\matnorm{\\ensuremath{U}^T}{\\infty}$.  Finally, we use\n$\\|\\ensuremath{U}\\|_\\infty$ to denote the element-wise maximum $\\max_{i,j}\n|\\ensuremath{U}_{ij}|$; note that this is not a matrix norm, but rather a norm\non the vectorized form of the matrix. For any matrix $\\ensuremath{U} \\in\n\\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times \\ensuremath{p}}$, we use $\\ensuremath{\\operatorname{vec}}(\\ensuremath{U})$ or equivalently\n$\\widebar{\\ensuremath{U}} \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p}^2}$ to denote its \\emph{vectorized\nform}, obtained by stacking up the rows of $\\ensuremath{U}$.  We use\n$\\tracer{\\ensuremath{U}}{\\ensuremath{V}} \\ensuremath{:  =} \\sum_{i,j} \\ensuremath{U}_{ij} \\ensuremath{V}_{ij}$ to\ndenote the \\emph{trace inner product} on the space of symmetric\nmatrices. Note that this inner product induces the \\emph{Frobenius\nnorm} $\\matnorm{\\ensuremath{U}}{F} \\ensuremath{:  =} \\sqrt{\\sum_{i,j}\n\\ensuremath{U}_{ij}^2}$. Finally, for asymptotics, we use the following\nstandard notation: we write $f(n) = {\\mathcal{O}}(g(n))$ if $f(n) \\leq c\ng(n)$ for some constant $c < \\infty$, and $f(n) = \\Omega(g(n))$ if\n$f(n) \\geq c' g(n)$ for some constant $c' > 0$.  The notation\n\\mbox{$f(n) \\asymp g(n)$} means that \\mbox{$f(n) = {\\mathcal{O}}(g(n))$} and\n\\mbox{$f(n) = \\Omega(g(n))$.}\n\n\\section{Background and problem set-up}\n\\label{SecBackground}\n\nLet $X = (X_1, \\ldots, X_\\ensuremath{p})$ be a zero mean $\\ensuremath{p}$-dimensional\nrandom vector. The focus of this paper is the problem of estimating\nthe covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*} \\ensuremath{:  =} \\ensuremath{{\\mathbb{E}}}[ X X^T]$ and\nconcentration matrix $\\ensuremath{\\Theta^*} \\ensuremath{:  =} \\invn{\\ensuremath{\\ensuremath{\\Sigma}^*}}$ of the\nrandom vector $X$ given $\\ensuremath{n}$ i.i.d. observations\n$\\{X^{(\\ensuremath{k})}\\}_{\\ensuremath{k}=1}^{\\ensuremath{n}}$.  In this section, we\nprovide background, and set up this problem more precisely.  We begin\nwith background on Gaussian graphical models, which provide one\nmotivation for the estimation of concentration matrices.  We then\ndescribe an estimator based based on minimizing an $\\ell_1$\nregularized log-determinant divergence; when the data are drawn from a\nGaussian graphical model, this estimator corresponds to\n$\\ell_1$-regularized maximum likelihood.  We then discuss the\ndistributional assumptions that we make in this paper.\n\n\\subsection{Gaussian graphical models}\n\nOne motivation for this paper is the problem of Gaussian graphical\nmodel selection.  A graphical model or a Markov random field is a\nfamily of probability distributions for which the conditional\nindependence and factorization properties are captured by a graph. Let\n$X = (X_1, X_2, \\ldots, X_\\ensuremath{p})$ denote a zero-mean Gaussian random\nvector; its density can be parameterized by the inverse covariance or\n\\emph{concentration matrix} $\\ensuremath{\\Theta^*} = (\\ensuremath{\\ensuremath{\\Sigma}^*})^{-1} \\in\n\\Symconepl{\\ensuremath{p}}$, and can be written as\n\\begin{eqnarray}\n\\label{EqnDefnGaussMRF}\nf(x_1, \\ldots, x_\\ensuremath{p}; \\ensuremath{\\Theta^*}) & = & \\frac{1}{\\sqrt{(2 \\pi)^\\ensuremath{p}\n \\det((\\ensuremath{\\Theta^*})^{-1})}} \\; \\exp \\big\\{ -\\frac{1}{2} x^T \\ensuremath{\\Theta^*}\n x \\big \\}.\n\\end{eqnarray}\n\\begin{figure}[ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\psfrag{#1#}{$1$} \\psfrag{#2#}{$2$} \\psfrag{#3#}{$3$}\n\\psfrag{#4#}{$4$} \\psfrag{#5#}{$5$}\n\\raisebox{.2in}{\\widgraph{0.3\\textwidth}{simple_gauss.eps}} & &\n\\widgraph{.25\\textwidth}{fig_simple_gauss_invcov.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{(a) Simple undirected graph.  A Gauss Markov random field has\na Gaussian variable $X_i$ associated with each vertex $i \\in \\ensuremath{V}$.\nThis graph has $\\ensuremath{p} = 5$ vertices, maximum degree $d = 3$ and $s=6$\nedges.  (b) Zero pattern of the inverse covariance $\\ensuremath{\\Theta^*}$\nassociated with the GMRF in (a).  The set $\\ensuremath{E}(\\ensuremath{\\Theta^*})$\ncorresponds to the off-diagonal non-zeros (white blocks); the diagonal\nis also non-zero (grey squares), but these entries do not correspond\nto edges.  The black squares correspond to non-edges, or zeros in\n$\\ensuremath{\\Theta^*}$.}\n\\label{FigMarkov}\n\\end{figure}\n\nWe can relate this Gaussian distribution of the random vector $X$ to a\ngraphical model as follows.  Suppose we are given an undirected graph\n$\\ensuremath{G} = (\\ensuremath{V}, \\ensuremath{E})$ with vertex set $\\ensuremath{V} = \\{1, 2, \\ldots,\n\\ensuremath{p} \\}$ and edge\\footnote{As a remark on notation, we would like to\ncontrast the notation for the edge-set $\\ensuremath{E}$ from the notation for\nan expectation of a random variable, $\\mathbb{E}(\\cdot)$.} set\n$\\ensuremath{E}$, so that each variable $X_i$ is associated with a\ncorresponding vertex $i \\in \\ensuremath{V}$. The Gaussian Markov random field\n(GMRF) associated with the graph $\\ensuremath{G}$ over the random vector $X$\nis then the family of Gaussian distributions with concentration\nmatrices $\\ensuremath{\\Theta^*}$ that respect the edge structure of the graph, in\nthe sense that $\\ensuremath{\\Theta^*}_{ij} = 0$ if $(i,j) \\notin\n\\ensuremath{E}$. Figure~\\ref{FigMarkov} illustrates this correspondence between\nthe graph structure (panel (a)), and the sparsity pattern of the\nconcentration matrix $\\ensuremath{\\Theta^*}$ (panel (b)).  The problem of\nestimating the entries of the concentration matrix $\\ensuremath{\\Theta^*}$\ncorresponds to estimating the Gaussian graphical model instance, while\nthe problem of estimating the off-diagonal zero-pattern of the\nconcentration matrix----that is, the set\n\\begin{eqnarray}\n\\label{EqnDefnEdgeSet}\n\\ensuremath{E}(\\ensuremath{\\Theta^*}) & \\ensuremath{:  =} & \\{i, j \\in \\ensuremath{V} \\mid \\, i \\neq j,\n\\ensuremath{\\Theta^*}_{ij} \\neq 0 \\}\n\\end{eqnarray}\ncorresponds to the problem of Gaussian graphical \\emph{model\nselection}. \n\nWith a slight abuse of notation, we define the \\emph{sparsity index}\n$\\ensuremath{s} \\ensuremath{:  =} |\\ensuremath{E}(\\ensuremath{\\Theta^*})|$ as the total number of non-zero\nelements in off-diagonal positions of $\\ensuremath{\\Theta^*}$; equivalently, this\ncorresponds to twice the number of edges in the case of a Gaussian\ngraphical model. We also define the \\emph{maximum degree or row\ncardinality}\n\\begin{eqnarray}\n\\label{EqnDefnDegmax}\n\\ensuremath{\\ensuremath{d}} & \\ensuremath{:  =} & \\max_{i = 1, \\ldots, \\ensuremath{p} } \\biggr|\\big \\{ j \\in\n\\ensuremath{V} \\, \\mid \\, \\ensuremath{\\Theta^*}_{ij} \\neq 0 \\big\\} \\biggr|,\n\\end{eqnarray}\ncorresponding to the maximum number of non-zeros in any row of\n$\\ensuremath{\\Theta^*}$; this corresponds to the maximum degree in the graph of\nthe underlying Gaussian graphical model.  Note that we have included\nthe diagonal entry $\\ensuremath{\\Theta^*}_{ii}$ in the degree count,\ncorresponding to a self-loop at each vertex.\n\nIt is convenient throughout the paper to use graphical terminology,\nsuch as degrees and edges, even though the distributional assumptions\nthat we impose, as described in Section~\\ref{SecDistAssum}, are milder\nand hence apply even to distributions that are not Gaussian MRFs.\n\n\n\n\\subsection{$\\ell_1$-penalized log-determinant divergence}\n\nAn important set in this paper is the cone\n\\begin{eqnarray}\n\\Symconepl{\\ensuremath{p}} & \\ensuremath{:  =} & \\big \\{ A \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times \\ensuremath{p}}\n\\mid A = A^T, \\; A \\succeq 0 \\big \\},\n\\end{eqnarray}\nformed by all symmetric positive semi-definite matrices in $\\ensuremath{p}$\ndimensions. We assume that the covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ and\nconcentration matrix $\\ensuremath{\\Theta^*}$ of the random vector $X$ are\nstrictly positive definite, and so lie in the interior of this cone\n$\\Symconepl{\\ensuremath{p}}$. \n\nThe focus of this paper is a particular type of $M$-estimator for the\nconcentration matrix $\\ensuremath{\\Theta^*}$, based on minimizing a Bregman\ndivergence between symmetric matrices.  A function is of Bregman type\nif it is strictly convex, continuously differentiable and has bounded\nlevel sets~\\cite{Bregman67a,Censor}.  Any such function induces a\n\\emph{Bregman divergence} of the form $\\Breg{A}{B} = g(A) - g(B) -\n\\trs{\\nabla g(B)}{A-B}$.  From the strict convexity of $g$, it follows\nthat $\\Breg{A}{B} \\geq 0$ for all $A$ and $B$, with equality if and\nonly if $A = B$.\n\n\nAs a candidate Bregman function, consider the log-determinant barrier\nfunction, defined for any matrix $A \\in \\Symconepl{\\ensuremath{p}}$ by\n\\begin{eqnarray}\n\\label{EqnDefnLogDet}\ng(A) & \\ensuremath{:  =} &  \\begin{cases} - \\log \\det(A)  & \\mbox{if $A \\succ 0$} \\\\\n                              + \\infty & \\mbox{otherwise.}\n                \\end{cases}\n\\end{eqnarray}\nAs is standard in convex analysis, we view this function as taking\nvalues in the extended reals $\\ensuremath{{\\mathbb{R}}}_* = \\ensuremath{{\\mathbb{R}}} \\cup \\{+\\infty \\}$.\nWith this definition, the function $g$ is strictly convex, and its\ndomain is the set of strictly positive definite matrices.  Moreover,\nit is continuously differentiable over its domain, with $\\nabla g(A) =\n- A^{-1}$; see Boyd and Vandenberghe~\\cite{Boyd02} for further\ndiscussion.  The Bregman divergence corresponding to this\nlog-determinant Bregman function $g$ is given by\n\\begin{eqnarray}\n\\label{EqnDefnBreg}\n\\Breg{A}{B} & \\ensuremath{:  =} & - \\log \\det A + \\log \\det B +\n\\tracer{B^{-1}}{A-B},\n\\end{eqnarray}\nvalid for any $A, B \\in \\Symconepl{\\ensuremath{p}}$ that are strictly positive\ndefinite.  This divergence suggests a natural way to estimate\nconcentration matrices---namely, by minimizing the divergence\n$\\Breg{\\ensuremath{\\Theta^*}}{\\Theta}$---or equivalently, by minimizing the\nfunction\n\\begin{equation}\n\\label{EqnPop}\n\\min_{\\Theta \\succ 0 } \\big \\{ \\tracer{\\Theta}{\\ensuremath{\\ensuremath{\\Sigma}^*}} - \\log\n\\det \\Theta \\big \\},\n\\end{equation}\nwhere we have discarded terms independent of $\\Theta$, and used the\nfact that the inverse of the concentration matrix is the covariance matrix\n(i.e., $(\\ensuremath{\\Theta^*})^{-1} = \\ensuremath{\\ensuremath{\\Sigma}^*}= \\ensuremath{{\\mathbb{E}}}[X X^T]$).  Of course,\nthe convex program~\\eqref{EqnPop} cannot be solved without knowledge\nof the true covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$, but one can take the\nstandard approach of replacing $\\ensuremath{\\ensuremath{\\Sigma}^*}$ with an empirical\nversion, with the possible addition of a regularization term.\n\n\nIn this paper, we analyze a particular instantiation of this strategy.\nGiven $\\ensuremath{n}$ samples, we define the \\emph{sample covariance matrix}\n\\begin{eqnarray}\n\\label{EqnDefnSamCov}\n\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n} & \\ensuremath{:  =} & \\frac{1}{\\ensuremath{n}} \\sum_{\\ensuremath{k}=1}^\\ensuremath{n}\n\\sp{X}{\\ensuremath{k}} (\\sp{X}{\\ensuremath{k}})^T.\n\\end{eqnarray}\nTo lighten notation, we occasionally drop the superscript $\\ensuremath{n}$,\nand simply write $\\ensuremath{\\widehat{\\Sigma}}$ for the sample covariance.  We also define\nthe \\emph{off-diagonal $\\ell_1$ regularizer}\n\\begin{eqnarray}\n\\ellreg{\\Theta} & \\ensuremath{:  =} & \\sum_{i \\neq j} |\\Theta_{ij}|,\n\\end{eqnarray}\nwhere the sum ranges over all $i, j = 1, \\ldots, \\ensuremath{p}$ with $i \\neq\nj$.  Given some regularization constant $\\ensuremath{\\lambda_\\ensuremath{n}} > 0$, we consider\nestimating $\\ensuremath{\\Theta^*}$ by solving the following\n\\emph{$\\ell_1$-regularized log-determinant program}:\n\\begin{eqnarray}\n\\label{EqnGaussMLE}\n\\ensuremath{\\widehat{\\Theta}} & \\ensuremath{:  =} & \\arg\\min_{\\Theta \\succ 0} \\big \\{\n\\tracer{\\Theta}{\\ensuremath{\\widehat{\\Sigma}}^\\ensuremath{n}} - \\log \\det(\\Theta) + \\ensuremath{\\lambda_\\ensuremath{n}}\n\\ellreg{\\Theta} \\big \\}.\n\\end{eqnarray}\nAs shown in Appendix~\\ref{AppLemMLECharac}, for any $\\ensuremath{\\lambda_\\ensuremath{n}} > 0$ and\nsample covariance matrix $\\ensuremath{\\widehat{\\Sigma}}^\\ensuremath{n}$ with strictly positive\ndiagonal, this convex optimization problem has a unique optimum, so\nthere is no ambiguity in equation~\\eqref{EqnGaussMLE}.  When the data\nis actually drawn from a multivariate Gaussian distribution, then the\nproblem~\\eqref{EqnGaussMLE} is simply $\\ell_1$-regularized maximum\nlikelihood.\n\n\n\\defc{c}\n\\defc{c}\n\n\\subsection{Tail conditions}\n\\label{SecDistAssum}\n\nIn this section, we describe the tail conditions that underlie our\nanalysis.  Since the estimator ~\\eqref{EqnGaussMLE} is based on using\nthe sample covariance $\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n}$ as a surrogate for the\n(unknown) covariance $\\ensuremath{\\ensuremath{\\Sigma}^*}$, any type of consistency requires\nbounds on the difference $\\ensuremath{\\estim{\\ensuremath{\\Sigma}}}^\\ensuremath{n} - \\ensuremath{\\ensuremath{\\Sigma}^*}$.  In\nparticular, we define the following tail condition:\n\\begin{defns}[Tail conditions]\n\\label{DefnTail}\nThe random vector $X$ satisfies tail condition $\\ensuremath{\\mathcal{T}}(f,\nv_*)$ if there exists a constant $v_* \\in (0, \\infty]$ and\na function $f: \\mathbb{N} \\times (0,\\infty) \\rightarrow (0,\n\\infty)$ such that for any $(i,j) \\in \\ensuremath{V} \\times \\ensuremath{V}$:\n\\begin{eqnarray}\n\\label{EqnSamTail}\n\\ensuremath{\\mathbb{P}}[|\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}_{ij} - \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}| \\geq \\delta] & \\leq\n& 1\/f(\\ensuremath{n},\\delta) \\qquad \\mbox{for all $\\delta \\in (0,\n1\/v_*]$.}\n\\end{eqnarray}\nWe adopt the convention $1\/0 \\ensuremath{:  =} + \\infty$, so that the value\n$v_* = 0$ indicates the inequality holds for any $\\delta \\in\n(0,\\infty)$.\n\\end{defns}\n\n\n\\newcommand{\\ensuremath{a}}{\\ensuremath{a}}\n\nTwo important examples of the tail function $f$ are the\nfollowing:\n\\begin{enumerate} \n\\item[(a)] an \\emph{exponential-type tail function}, meaning that\n$f(\\ensuremath{n},\\delta) = \\exp(c \\, \\ensuremath{n}\\,\n\\delta^{\\ensuremath{a}})$, for some scalar $c > 0$, and exponent\n$\\ensuremath{a} > 0$; and\n\\item[(b)] a \\emph{polynomial-type tail function}, meaning that\n$f(\\ensuremath{n},\\delta) = c \\, \\ensuremath{n}^{m} \\,\n\\delta^{2m}$, for some positive integer $m \\in\n\\mathbb{N}$ and scalar $c > 0$.\n\\end{enumerate}\nAs might be expected, if $X$ is multivariate Gaussian, then the\ndeviations of sample covariance matrix have an exponential-type tail\nfunction with $\\ensuremath{a} = 2$.  A bit more generally, in the following\nsubsections, we provide broader classes of distributions whose sample\ncovariance entries satisfy exponential and a polynomial tail bounds\n(see Lemmata~\\ref{LEM_SAM_COV_BOUND_SUBG}\nand~\\ref{LEM_SAM_COV_BOUND_MOMENT} respectively).\n\nGiven a larger number of samples $\\ensuremath{n}$, we expect the tail\nprobability bound $1\/f(\\ensuremath{n},\\delta)$ to be smaller, or\nequivalently, for the tail function $f(\\ensuremath{n},\\delta)$ to\nlarger.  Accordingly, we require that $f$ is monotonically\nincreasing in $\\ensuremath{n}$, so that for each fixed $\\delta >0$, we can\ndefine the inverse function\n\\begin{eqnarray}\n\\label{EqnSamTailN}\n\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(r; \\delta) & \\ensuremath{:  =} & \\arg \\max \\big \\{ n \\; \\mid \\;\nf(\\ensuremath{n}, \\delta) \\leq r \\big \\}.\n\\end{eqnarray}\nSimilarly, we expect that $f$ is monotonically increasing in\n$\\delta$, so that for each fixed $\\ensuremath{n}$, we can define the\ninverse in the second argument\n\\begin{eqnarray}\n\\label{EqnSamTailT}\n\\ensuremath{\\widebar{\\delta}_f}(r; \\ensuremath{n}) & \\ensuremath{:  =} & \\arg \\max \\big \\{ \\delta \\; \\mid\n\\; f(\\ensuremath{n}, \\delta) \\leq r \\big \\}.\n\\end{eqnarray}\nFor future reference, we note a simple consequence of the monotonicity\nof the tail function $f$---namely\n\\begin{eqnarray}\n\\label{EqnMonot}\n\\ensuremath{n} > \\ensuremath{{\\widebar{\\ensuremath{n}}_f}}( \\delta, r) \\quad \\mbox{for some $\\delta > 0$} &\n\\Longrightarrow & \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},r) \\leq \\delta.\n\\end{eqnarray}\nThe inverse functions $\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ and $\\ensuremath{\\widebar{\\delta}_f}$ play an\nimportant role in describing the behavior of our estimator.  We\nprovide concrete examples in the following two subsections.\n\n\n\\subsubsection{Sub-Gaussian distributions}\n\\label{SecSamCovBoundSubG}\nIn this subsection, we study the case of i.i.d. observations of\nsub-Gaussian random variables.\n\\begin{defns}\nA zero-mean random variable $Z$ is \\emph{sub-Gaussian} if there exists\na constant $\\ensuremath{\\sigma}\\in (0, \\infty)$ such that\n\\begin{eqnarray}\n\\label{EqnDefnSubgauss}\n\\ensuremath{\\mathbb{E}}[\\exp(t Z)] & \\leq & \\exp(\\ensuremath{\\sigma}^2 \\, t^2\/2) \\qquad \\mbox{for all\n$t \\in \\ensuremath{{\\mathbb{R}}}$.}\n\\end{eqnarray}\n\\end{defns}\nBy the Chernoff bound, this upper bound~\\eqref{EqnDefnSubgauss} on the\nmoment-generating function implies a two-sided tail bound of the form\n\\begin{eqnarray}\n\\label{EqnSubgaussChern}\n\\ensuremath{\\mathbb{P}}[|Z| > z] & \\leq & 2 \\exp \\big(- \\frac{z^2}{2 \\ensuremath{\\sigma}^2}\\big).\n\\end{eqnarray}\nNaturally, any zero-mean Gaussian variable with variance $\\sigma^2$\nsatisfies the bounds~\\eqref{EqnDefnSubgauss}\nand~\\eqref{EqnSubgaussChern}.  In addition to the Gaussian case, the\nclass of sub-Gaussian variates includes any bounded random variable\n(e.g., Bernoulli, multinomial, uniform), any random variable with\nstrictly log-concave density~\\cite{BulKoz,Ledoux01}, and any finite\nmixture of sub-Gaussian variables.\n\nThe following lemma, proved in\nAppendix~\\ref{APP_LEM_SAM_COV_BOUND_SUBG}, shows that the entries of\nthe sample covariance based on i.i.d. samples of sub-Gaussian random\nvector satisfy an exponential-type tail bound with exponent $\\ensuremath{a}\n= 2$.  The argument is along the lines of a result due to Bickel and\nLevina~\\cite{BickelLevina2007}, but with more explicit control of the\nconstants in the error exponent:\n\\begin{lems}\n\\label{LEM_SAM_COV_BOUND_SUBG}\nConsider a zero-mean random vector $(X_1, \\ldots, X_\\ensuremath{p})$ with\ncovariance $\\ensuremath{\\ensuremath{\\Sigma}^*}$ such that each $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$\nis sub-Gaussian with parameter $\\ensuremath{\\sigma}$.  Given $\\ensuremath{n}$\ni.i.d. samples, the associated sample covariance $\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}$\nsatisfies the tail bound\n\\begin{eqnarray*}\n\\ensuremath{\\mathbb{P}} \\big[ |\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}_{ij }- \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}| > \\delta\n \\big] & \\leq & 4 \\exp \\big \\{- \\frac{\\ensuremath{n} \\delta^2}{ 128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2 }\n \\big \\},\n\\end{eqnarray*}\nfor all $\\delta \\in \\big(0, \\max_{i}(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\,8(1 + 4\n\\ensuremath{\\sigma}^2)\\big)$.\n\n\\end{lems}\nThus, the sample covariance entries the tail condition $\\ensuremath{\\mathcal{T}}(f,\nv_*)$ with $v_* = \\big[\\max_{i}(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\,8(1 + 4\n\\ensuremath{\\sigma}^2)\\big]^{-1}$, and an exponential-type tail function with\n$\\ensuremath{a} = 2$---namely\n\\begin{eqnarray}\n\\label{EqnSubgaussF}\n\\qquad f(\\ensuremath{n}, \\delta) = \\frac{1}{4} \\exp( c_{*}\n\\ensuremath{n} \\delta^2), & \\mbox{with}  & c_{*} = \\big[ 128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2 \\big]^{-1}\n\\end{eqnarray}\nA little calculation shows that the associated inverse functions take\nthe form\n\\begin{equation}\n\\label{EqnExpInverse}\n\\ensuremath{\\widebar{\\delta}_f}(r; \\ensuremath{n} ) \\, = \\, \\sqrt{\\frac{\\log(4 \\,r)}{c_{*} \\,\n\\ensuremath{n}}}, \\quad \\mbox{and} \\quad \\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(r; \\delta) \\, = \\,\n\\frac{\\log(4 \\, r)}{c_{*} \\delta^2}.\n\\end{equation}\n\n\n\n\\subsubsection{Tail bounds with moment bounds}\n\\label{SecSamCovBoundMoment}\n\nIn the following lemma, proved in\nAppendix~\\ref{APP_LEM_SAM_COV_BOUND_MOMENT}, we show that given\ni.i.d. observations from random variables with bounded moments, the\nsample covariance entries satisfy a polynomial-type tail bound. See\nthe papers~\\cite{Zhao06,Karoui2007} for related results on tail bounds\nfor variables with bounded moments.\n\\begin{lems}\n\\label{LEM_SAM_COV_BOUND_MOMENT}\nSuppose there exists a positive integer $m$ and scalar $K_{\\momentpow} \\in\n\\ensuremath{{\\mathbb{R}}}$ such that for $i = 1,\\hdots,\\ensuremath{p}$,\n\\begin{eqnarray}\n\\label{EqnBoundedMoments}\n\\ensuremath{{\\mathbb{E}}} \\biggr[ \\big(\\frac{X_i}{\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}} \\big)^{4 m}\n\\biggr] & \\leq &   K_{\\momentpow}.\n\\end{eqnarray}\nFor i.i.d. samples $\\{\\Xsam{\\ensuremath{k}}_i \\}_{\\ensuremath{k}=1}^\\ensuremath{n}$, the\nsample covariance matrix $\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}$ satisfies the bound\n\\begin{eqnarray}\n\\ensuremath{\\mathbb{P}} \\big [ \\Big| \\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}^\\ensuremath{n}_{ij} - \\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} \\Big)\\Big|\n  > \\delta \\big] & \\leq & \\frac{\\big\\{m^{2m+1}\n  2^{2m} (\\max_i \\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}) ^{2m}\\, (K_{\\momentpow}\n  + 1 ) \\big\\}}{\\ensuremath{n}^{m}\\, \\delta^{2m}}.\n\\end{eqnarray}\n\\end{lems}\nThus, in this case, the sample covariance satisfies the tail condition\n$\\ensuremath{\\mathcal{T}}(f, v_*)$ with $v_* = 0$, so that the bound\nholds for all $\\delta \\in (0,\\infty)$, and with the polynomial-type\ntail function\n\\begin{equation}\n\\label{EqnPolyF}\nf(\\ensuremath{n},\\delta) = c_{*} \\ensuremath{n}^{m}\n\\delta^{2m} \\quad \\mbox{where $c_{*} =\n1\/\\big\\{m^{2m+1} 2^{2m} (\\max_i\n\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}) ^{2m}\\, (K_{\\momentpow} + 1 ) \\big\\}$.}\n\\end{equation}\nFinally, a little calculation shows that in this case, the inverse\ntail functions take the form\n\\begin{equation}\n\\label{EqnPolyInverse}\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},r) \\, = \\,\n\\frac{(r\/c_{*})^{1\/2m}}{\\sqrt{\\ensuremath{n}}}, \\quad\n\\mbox{and} \\quad \\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,r) \\, = \\,\n\\frac{(r\/c_{*})^{1\/m}}{\\delta^{2}}.\n\\end{equation}\n\n\n\n\\section{Main results and some consequences}\n\\label{SecResult}\n\nIn this section, we state our main results, and discuss some of their\nconsequences.  We begin in Section~\\ref{SecAssumptions} by stating\nsome conditions on the true concentration matrix $\\ensuremath{\\Theta^*}$ required in\nour analysis, including a particular type of incoherence or\nirrepresentability condition.  In Section~\\ref{SecEllinf}, we state\nour first main result---namely, Theorem~\\ref{ThmMain} on consistency\nof the estimator $\\ensuremath{\\widehat{\\Theta}}$, and the rate of decay of its error in\nelementwise $\\ell_\\infty$ norm.  Section~\\ref{SecModelCons} is devoted\nto Theorem~\\ref{ThmModel} on the model selection consistency of the\nestimator.  Section~\\ref{SecInco} is devoted the relation between the\nlog-determinant estimator and the ordinary Lasso (neighborhood-based\napproach) as methods for graphical model selection; in addition, we\nillustrate our irrepresentability assumption for some simple graphs.\nFinally, in Section~\\ref{SecFrob}, we state and prove some corollaries\nof Theorem~\\ref{ThmMain}, regarding rates in Frobenius and operator\nnorms.\n\n\\subsection{Conditions on covariance and Hessian}\n\\label{SecAssumptions}\n\nOur results involve some quantities involving the Hessian of the\nlog-determinant barrier~\\eqref{EqnDefnLogDet}, evaluated at the true\nconcentration matrix $\\ensuremath{\\Theta^*}$.  Using standard results on matrix\nderivatives~\\citep{Boyd02}, it can be shown that this Hessian takes\nthe form\n\\begin{eqnarray}\n\\label{EqnDefnHess}\n\\ensuremath{\\Gamma^*} & \\ensuremath{:  =} & \\nabla^2_{\\Theta} g(\\Theta) \\Big |_{\\Theta =\n  \\ensuremath{\\Theta^*}} \\; = \\; \\invn{\\ensuremath{\\Theta^*}} \\otimes \\invn{\\ensuremath{\\Theta^*}},\n\\end{eqnarray}\nwhere $\\otimes$ denotes the Kronecker matrix product.  By definition,\n$\\ensuremath{\\Gamma^*}$ is a $\\ensuremath{p}^{2} \\times \\ensuremath{p}^{2}$ matrix indexed by vertex\npairs, so that entry $\\ensuremath{\\Gamma^*}_{(j,k), (\\ell, m)}$ corresponds to the\nsecond partial derivative $ \\frac{\\partial^2 g}{\\partial \\Theta_{jk}\n\\partial \\Theta_{\\ell m}}$, evaluated at $\\Theta = \\ensuremath{\\Theta^*}$.  When\n$X$ has multivariate Gaussian distribution, then $\\ensuremath{\\Gamma^*}$ is the\nFisher information of the model, and by standard results on cumulant\nfunctions in exponential families~\\cite{Brown86}, we have the more\nspecific expression $\\ensuremath{\\Gamma^*}_{(j,k), (\\ell, m)} =\n\\operatorname{cov}\\{X_j X_k, \\; X_\\ell X_m \\}$.  For this reason,\n$\\ensuremath{\\Gamma^*}$ can be viewed as an edge-based counterpart to the usual\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.\n\nWe define the set of non-zero off-diagonal entries in the model\nconcentration matrix $\\ensuremath{\\Theta^*}$:\n\\begin{eqnarray}\n\\ensuremath{E}(\\ensuremath{\\Theta^*}) & \\ensuremath{:  =} & \\{ (i,j) \\in \\ensuremath{V} \\times \\ensuremath{V} \\,\n\t\\mid \\, i \\neq j, \\ensuremath{\\Theta^*}_{ij} \\neq 0 \\},\n\\end{eqnarray}\nand let $\\ensuremath{S}(\\ensuremath{\\Theta^*}) = \\{ \\ensuremath{E}(\\ensuremath{\\Theta^*}) \\cup \\{(1,1),\n\\ldots, (\\ensuremath{p}, \\ensuremath{p}) \\}$ be the augmented set including the\ndiagonal.  We let $\\ensuremath{\\EsetPlus^c}(\\ensuremath{\\Theta^*})$ denote the complement of\n$\\ensuremath{S}(\\ensuremath{\\Theta^*})$ in the set $\\{1, \\ldots, \\ensuremath{p} \\} \\times \\{1,\n\\ldots, \\ensuremath{p}\\}$, corresponding to all pairs $(\\ell, m)$ for which\n$\\ensuremath{\\Theta^*}_{\\ell m} = 0$.  When it is clear from context, we shorten\nour notation for these sets to $\\ensuremath{S}$ and $\\ensuremath{\\EsetPlus^c}$,\nrespectively.  Finally, for any two subsets $T$ and $T'$ of $\\ensuremath{V}\n\\times \\ensuremath{V}$, we use $\\ensuremath{\\Gamma^*}_{T T'}$ to denote the $|T| \\times\n|T'|$ matrix with rows and columns of $\\ensuremath{\\Gamma^*}$ indexed by $T$ and\n$T'$ respectively.\n\n\nOur main results involve the $\\ell_\\infty\/\\ell_\\infty$ norm applied to\nthe covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$, and to the inverse of a sub-block\nof the Hessian $\\ensuremath{\\Gamma^*}$.  In particular, we define\n\\begin{eqnarray}\n\\label{EqnCovConst}\n\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} & \\ensuremath{:  =} & \\matnorm{\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} \\; = \\; \\Big(\n \\max_{i=1, \\ldots ,\\ensuremath{p}} \\sum_{j=1}^\\ensuremath{p} |\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}| \\Big),\n\\end{eqnarray}\ncorresponding to the $\\ell_\\infty$-operator norm of the true\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$, and\n\\begin{eqnarray}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} & \\ensuremath{:  =} & \\matnorm{(\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}})^{-1}}{\\infty} \\; = \\; \\matnorm{([ {\\ensuremath{\\Theta^*}}^{-1}\n\\otimes {\\ensuremath{\\Theta^*}}^{-1}]_{\\ensuremath{S} \\ensuremath{S}})^{-1}}{\\infty}.\n\\end{eqnarray}\nOur analysis keeps explicit track of these quantities, so that they\ncan scale in a non-trivial manner with the problem dimension $\\ensuremath{p}$.\n\n\nWe assume the Hessian satisfies the following type of \\emph{mutual\nincoherence or irrepresentable condition}:\n\\begin{asss}\n\\label{AssInco}\nThere exists some $\\mutinco \\in (0,1]$ such that\n\\begin{eqnarray}\n\\label{EqnInco}\n\\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c} \\ensuremath{S}} (\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}})^{-1}}{\\infty} & \\leq & (1 - \\mutinco).\n\\end{eqnarray}\n\\end{asss}\n\nThe underlying intuition is that this assumption imposes control on\nthe influence that the non-edge terms, indexed by $\\ensuremath{\\EsetPlus^c}$, can\nhave on the edge-based terms, indexed by $\\ensuremath{S}$.  It is worth\nnoting that a similar condition for the Lasso, with the covariance\nmatrix $\\Sigma^*$ taking the place of the matrix $\\ensuremath{\\Gamma^*}$ above, is\nnecessary and sufficient for support recovery using the ordinary\nLasso~\\cite{MeinsBuhl2006,Tropp2006,Wainwright2006_new,Zhao06}.  See\nSection~\\ref{SecInco} for illustration of the form taken by\nAssumption~\\ref{AssInco} for specific graphical models.\n\nA remark on notation: although our analysis allows the quantities\n$\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ as well as the model size $\\ensuremath{p}$ and maximum\nnode-degree $\\ensuremath{\\ensuremath{d}}$ to grow with the sample size $\\ensuremath{n}$, we\nsuppress this dependence on $\\ensuremath{n}$ in their notation.\n\n\n\\subsection{Rates in elementwise $\\ell_\\infty$-norm}\n\\label{SecEllinf}\n\nWe begin with a result that provides sufficient conditions on the\nsample size $\\ensuremath{n}$ for bounds in the elementwise\n$\\ell_\\infty$-norm.  This result is stated in terms of the tail\nfunction $f$, and its inverses $\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ and $\\ensuremath{\\widebar{\\delta}_f}$ (equations~\\eqref{EqnSamTailN} and~\\eqref{EqnSamTailT}), and so covers a general range of possible\ntail behaviors.  So as to make it more concrete, we follow the general\nstatement with corollaries for the special cases of exponential-type\nand polynomial-type tail functions, corresponding to sub-Gaussian and\nmoment-bounded variables respectively.  \n\nIn the theorem statement, the choice of regularization constant\n$\\ensuremath{\\lambda_\\ensuremath{n}}$ is specified in terms of a user-defined parameter $\\tau >\n2$.  Larger choices of $\\tau$ yield faster rates of convergence in\nthe probability with which the claims hold, but also lead to more\nstringent requirements on the sample size.\n\\begin{theos}\n\\label{ThmMain}\nConsider a distribution satisfying the incoherence\nassumption~\\eqref{EqnInco} with parameter \\mbox{$\\mutinco \\in (0,1]$,}\nand the tail condition~\\eqref{EqnSamTail} with parameters\n$\\ensuremath{\\mathcal{T}}(f, v_*)$.  Let $\\ensuremath{\\widehat{\\Theta}}$ be the unique optimum of\nthe log-determinant program~\\eqref{EqnGaussMLE} with regularization\nparameter \\mbox{$\\ensuremath{\\lambda_\\ensuremath{n}} = (8\/\\mutinco) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$} for some $\\tau > 2$.  Then,\nif the sample size is lower bounded as\n\\begin{eqnarray}\n\\label{EqnSampleBound}\n\\ensuremath{n} & > & \\ensuremath{{\\widebar{\\ensuremath{n}}_f}} \\Biggr( 1 \\Big\/\\max \\Big \\{ v_*,\\; 6\n \\big(1 + 8\\mutinco^{-1} \\big) \\: \\ensuremath{\\ensuremath{d}}\\, \\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}\n \\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\Big \\} , \\; \\;\\ensuremath{p}^{\\tau}\n \\Biggr),\n\\end{eqnarray}\nthen with probability greater than $1-1\/\\ensuremath{p}^{\\tau - 2}\n\\rightarrow 1$, we have:\n\\begin{enumerate}\n\\item[(a)] The estimate $\\ensuremath{\\widehat{\\Theta}}$ satisfies the elementwise\n$\\ell_\\infty$-bound:\n\\begin{eqnarray}\n\\label{EqnEllinfBound}\n\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty & \\leq & \\big\\{2 \\big(1 + 8 \\mutinco^{-1}\\big)\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\big\\}\\;\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray}\n\\item[(b)] It specifies an edge set $\\ensuremath{E}(\\ensuremath{\\widehat{\\Theta}})$ that is a\nsubset of the true edge set $\\ensuremath{E}(\\ensuremath{\\Theta^*})$, and includes all\nedges $(i,j)$ with $|\\ensuremath{\\Theta^*}_{ij}| >  \\big\\{2 \\big(1 + 8 \\mutinco^{-1}\\big)\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\big\\}\\; \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$.\n\\end{enumerate}\n\\end{theos}\n\nIf we assume that the various quantities $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}},\n\\mutinco$ remain constant as a function of $(\\ensuremath{n}, \\ensuremath{p},\n\\ensuremath{\\ensuremath{d}})$, we have the elementwise $\\ell_\\infty$ bound \\mbox{$\\|\n\\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty =\n{\\mathcal{O}}(\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}))$}, so that the inverse\ntail function $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$ (see\nequation~\\eqref{EqnSamTailT}) specifies rate of convergence in the\nelement-wise $\\ell_\\infty$-norm.  In the following section, we derive\nthe consequences of this $\\ell_\\infty$-bound for two specific tail\nfunctions, namely those of exponential-type with $\\ensuremath{a} = 2$, and\npolynomial-type tails (see Section~\\ref{SecDistAssum}).  Turning to\nthe other factors involved in the theorem statement, the quantities\n$\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$ and $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ measure the sizes of the entries in the\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ and inverse Hessian $(\\ensuremath{\\Gamma^*})^{-1}$\nrespectively.  Finally, the factor $(1 + \\frac{8}{\\mutinco})$ depends\non the irrepresentability assumption~\\ref{AssInco}, growing in\nparticular as the incoherence parameter $\\mutinco$ approaches $0$.\n\n\\subsubsection{Exponential-type tails}\n\nWe now discuss the consequences of Theorem~\\ref{ThmMain} for\ndistributions in which the sample covariance satisfies an\nexponential-type tail bound with exponent $\\ensuremath{a} = 2$.  In\nparticular, recall from Lemma~\\ref{LEM_SAM_COV_BOUND_SUBG} that\nsuch a tail bound holds when the variables are sub-Gaussian.\n\\defc_1{c_1}\n\\defc_2{c_2}\n\n\n\\begin{cors} \n\\label{CorEllinfSubg}\nUnder the same conditions as Theorem~\\ref{ThmMain}, suppose moreover\nthat the variables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are sub-Gaussian with\nparameter $\\ensuremath{\\sigma}$, and the samples are drawn independently.  Then if\nthe sample size $\\ensuremath{n}$ satisfies the bound\n\\begin{eqnarray}\n\\ensuremath{n} & > & \\ensuremath{C_1} \\; \\ensuremath{\\ensuremath{d}}^2 \\, (1 + \\frac{8}{\\mutinco})^2\n\\;\\big (\\tau \\log \\ensuremath{p} + \\log 4 \\big)\n\\end{eqnarray}\nwhere $\\ensuremath{C_1} \\ensuremath{:  =} \\big\\{48\\sqrt{2} \\,(1 + 4 \\csubg^2) \\,\\max_{i} (\\CovMatStar_{ii})  \\, \\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\big\\}^{2}$, then with\nprobability greater than $1-1\/\\ensuremath{p}^{\\tau -2}$, the estimate\n$\\ensuremath{\\widehat{\\Theta}}$ satisfies the bound,\n\\begin{eqnarray*}\n\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty & \\leq & \n\\big\\{16\\sqrt{2} \\,(1 + 4 \\csubg^2) \\,\\max_{i} (\\CovMatStar_{ii})  \\, (1 + 8\\mutinco^{-1})\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\big\\}\\; \\sqrt{\\frac{\\tau \\log \\ensuremath{p} + \\log 4}{\\ensuremath{n}}}.\n\\end{eqnarray*}\n\\end{cors}\n\\begin{proof}\nFrom Lemma~\\ref{LEM_SAM_COV_BOUND_SUBG}, when the rescaled variables\n$X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are sub-Gaussian with parameter\n$\\ensuremath{\\sigma}$, the sample covariance entries satisfies a tail bound\n$\\ensuremath{\\mathcal{T}}(f, v_*)$ with with $v_* = \\big[\\max_{i}(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\,8(1 + 4 \\ensuremath{\\sigma}^2)\\big]^{-1}$ \nand $f(\\ensuremath{n},\\delta) = (1\/4) \\exp(c_{*} \\ensuremath{n} \\delta^2)$, where \\mbox{$c_{*} =\n\\big[128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2\\big]^{-1}$.}  As a consequence, for this particular model, the\ninverse functions $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$ and\n$\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,\\ensuremath{p}^{\\tau})$ take the form\n\\begin{subequations}\n\\label{EqnExpTailInv}\n\\begin{eqnarray}\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) & = &\n\\sqrt{\\frac{\\log(4\\,\\ensuremath{p}^{\\tau})}{c_{*} \\, \\ensuremath{n}}} \\; = \\;\n\\sqrt{ 128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2} \\; \\sqrt{\\frac{\\tau \\log \\ensuremath{p} + \\log 4}{\\ensuremath{n}}},\\\\\n\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,\\ensuremath{p}^{\\tau}) &= &\n\\frac{\\log(4\\,\\ensuremath{p}^{\\tau})} {c_{*} \\delta^2} \\; = \\; \n128(1 + 4\\csubg^{2})^{2}\\max_{i} (\\CovMatStar_{ii})^2  \\;\n\\biggr(\\frac{\\tau \\log \\ensuremath{p} + \\log 4}{\\delta^{2}}\\biggr).\n\\end{eqnarray}\n\\end{subequations}\nSubstituting these forms into the claim of Theorem~\\ref{ThmMain} and\ndoing some simple algebra yields the stated corollary.\n\\end{proof}\n\nWhen $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco$ remain constant as a function of\n$(\\ensuremath{n}, \\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$, the corollary can be summarized\nsuccinctly as a sample size of \\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\log\n\\ensuremath{p})$} samples ensures that an elementwise\n$\\ell_\\infty$ bound \\mbox{$\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty =\n{\\mathcal{O}}\\big( \\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}}\\big)$} holds with high probability. \nIn practice, one frequently considers graphs with maximum node degrees\n$\\ensuremath{\\ensuremath{d}}$ that either remain bounded, or that grow sub-linearly with\nthe graph size (i.e., $\\ensuremath{\\ensuremath{d}} = o(\\ensuremath{p})$). In such cases, the sample\nsize allowed by the corollary can be substantially smaller than the\ngraph size, so that for sub-Gaussian random variables, the method can\nsucceed in the $\\ensuremath{p} \\gg \\ensuremath{n}$ regime.\n\n\\subsubsection{Polynomial-type tails}\n\nWe now state a corollary for the case of a polynomial-type tail\nfunction, such as those ensured by the case of random variables with\nappropriately bounded moments.\n\\begin{cors}\n\\label{CorEllinfPoly}\nUnder the assumptions of Theorem~\\ref{ThmMain}, suppose the rescaled\nvariables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have $4m^{th}$\nmoments upper bounded by $K_{\\momentpow}$, and the sampling is i.i.d.\nThen if the sample size $\\ensuremath{n}$ satisfies the bound\n\\begin{eqnarray}\n\\label{EqnPolyTailSampSize}\n\\ensuremath{n} & > & \\ensuremath{C_2} \\, \\ensuremath{\\ensuremath{d}}^{2}\\, \\big(1 + \\frac{8}{\\mutinco}\n\\big)^2\\, \\ensuremath{p}^{\\tau\/m},\n\\end{eqnarray}\nwhere $\\ensuremath{C_2} \\ensuremath{:  =} \\big\\{12 m \\,[m (K_{\\momentpow} +\n  1)]^{\\frac{1}{2m}}\\, \\max_i(\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\\max\n\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{4} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\big\\}^{2}$,\nthen with probability greater than $1-1\/\\ensuremath{p}^{\\tau -2}$, the\nestimate $\\ensuremath{\\widehat{\\Theta}}$ satisfies the bound,\n\\begin{eqnarray*}\n\\| \\estim{\\Theta} - \\ensuremath{\\Theta^*}\\|_\\infty & \\leq & \\{4m\n [m (K_{\\momentpow} + 1)]^{\\frac{1}{2m}}\\, \\big(1 +\n \\frac{8}{\\mutinco} \\big) \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\}\\;\n \\sqrt{\\frac{\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}.\n\\end{eqnarray*}\n\\end{cors}\n\\begin{proof}\nRecall from Lemma~\\ref{LEM_SAM_COV_BOUND_MOMENT} that when the\nrescaled variables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have bounded\n$4m^{th}$ moments, then the sample covariance $\\ensuremath{\\widehat{\\ensuremath{\\Sigma}}}$\nsatisfies the tail condition $\\ensuremath{\\mathcal{T}}(f, v_*)$ with\n$v_* = 0$, and with $f(\\ensuremath{n},\\delta) = c_{*}\n\\ensuremath{n}^{m} \\delta^{2m}$ with $c_{*}$ defined as\n$c_{*} = 1\/\\big\\{m^{2m+1} 2^{2m} (\\max_i\n\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}) ^{2m}\\, (K_{\\momentpow} + 1 ) \\big\\}$. As a\nconsequence, for this particular model, the inverse functions take the\nform\n\\begin{subequations}\n\\label{EqnPolyTailInv}\n\\begin{eqnarray}\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) & = &\n\\frac{(\\ensuremath{p}^{\\tau}\/c_{*})^{1\/2m}}{\\sqrt{\\ensuremath{n}}}\n\\,=\\, \\{2m [m (K_{\\momentpow} +\n1)]^{\\frac{1}{2m}} \\max_i \\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} \\}\\;\n\\sqrt{\\frac{\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}},\\\\\n\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}(\\delta,\\ensuremath{p}^{\\tau}) & = &\n\\frac{(\\ensuremath{p}^{\\tau}\/c_{*})^{1\/m}}{\\delta^{2}} \\,=\\,\n\\{2m [m (K_{\\momentpow} + 1)]^{\\frac{1}{2m}}\n\\max_i \\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} \\}^{2}\\;\n\\big(\\frac{\\ensuremath{p}^{\\tau\/m}}{\\delta^{2}}\\big).\n\\end{eqnarray}\n\\end{subequations}\nThe claim then follows by substituting these expressions into Theorem~\\ref{ThmMain} and performing some algebra.\n\\end{proof}\n\nWhen the quantities $(\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco)$ remain constant\nas a function of $(\\ensuremath{n}, \\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$,\nCorollary~\\ref{CorEllinfPoly} can be summarized succinctly as\n\\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\, \\ensuremath{p}^{\\tau\/m})$}\nsamples are sufficient to achieve a convergence rate in elementwise\n$\\ell_\\infty$-norm of the order \\mbox{$\\| \\estim{\\Theta} -\n\\ensuremath{\\Theta^*}\\|_\\infty = {\\mathcal{O}}\\big(\n\\sqrt{\\frac{\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}\\big)$,} with high\nprobability.  Consequently, both the required sample size and the rate\nof convergence of the estimator are polynomial in the number of\nvariables $\\ensuremath{p}$.  It is worth contrasting these rates with the case\nof sub-Gaussian random variables, where the rates have only\nlogarithmic dependence on the problem size $\\ensuremath{p}$.\n\n\n \n\n\n\n\\subsection{Model selection consistency}\n\\label{SecModelCons}\n\nPart (b) of Theorem~\\ref{ThmMain} asserts that the edge set\n$\\ensuremath{E}(\\ensuremath{\\widehat{\\Theta}})$ returned by the estimator is contained within the\ntrue edge set $\\ensuremath{E}(\\ensuremath{\\Theta^*})$---meaning that it correctly\n\\emph{excludes} all non-edges---and that it includes all edges that\nare ``large'', relative to the $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$\ndecay of the error.  The following result, essentially a minor\nrefinement of Theorem~\\ref{ThmMain}, provides sufficient conditions\nlinking the sample size $\\ensuremath{n}$ and the minimum value\n\\begin{eqnarray}\n\\label{EqnDefnThetaMin}\n\\ensuremath{\\theta_{\\operatorname{min}}} & \\ensuremath{:  =} & \\min_{(i,j) \\in \\ensuremath{E}(\\ensuremath{\\Theta^*})} |\\ensuremath{\\Theta^*}_{ij}|\n\\end{eqnarray}\nfor model selection consistency.  More precisely, define the event\n\\begin{eqnarray}\n\\ensuremath{\\mathcal{M}}(\\ensuremath{\\widehat{\\Theta}}; \\ensuremath{\\Theta^*}) & \\ensuremath{:  =} & \\big \\{ \\textrm{sign}(\\ensuremath{\\widehat{\\Theta}}_{ij})\n= \\textrm{sign}(\\ensuremath{\\Theta^*}_{ij}) \\quad \\forall (i,j) \\in \\ensuremath{E}(\\ensuremath{\\Theta^*})\n\\big \\}\n\\end{eqnarray}\nthat the estimator $\\ensuremath{\\widehat{\\Theta}}$ has the same edge set as $\\ensuremath{\\Theta^*}$,\nand moreover recovers the correct signs on these edges.  With this\nnotation, we have:\n\\begin{theos}\n\\label{ThmModel}\n\nUnder the same conditions as Theorem~\\ref{ThmMain}, suppose that\nthe sample size satisfies the lower bound\n\\begin{eqnarray}\n\\label{EqnNumobsModel}\n\\ensuremath{n} & > & \\ensuremath{{\\widebar{\\ensuremath{n}}_f}} \\Biggr( 1 \\big\/\\max \\Big \\{ 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} (1 +\n8\\mutinco^{-1})\\, \\ensuremath{\\theta_{\\operatorname{min}}}^{-1}, \\; v_*, \\; 6 \\big (1 +\n8\\mutinco^{-1} \\big) \\: \\ensuremath{\\ensuremath{d}}\\, \\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}},\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2} \\} \\Big\\} , \\; \\;\\ensuremath{p}^{\\tau}\n\\Biggr).\n\\end{eqnarray}\nThen the estimator is model selection consistent with high probability\nas $\\ensuremath{p} \\rightarrow \\infty$,\n\\begin{eqnarray}\n\\ensuremath{\\mathbb{P}} \\big[ \\ensuremath{\\mathcal{M}}(\\ensuremath{\\widehat{\\Theta}}; \\ensuremath{\\Theta^*}) \\big] & \\geq & 1 -\n1\/\\ensuremath{p}^{\\tau - 2} \\; \\rightarrow \\; 1.\n\\end{eqnarray}\n\\end{theos}\n\nIn comparison to Theorem~\\ref{ThmMain}, the sample size\nrequirement~\\eqref{EqnNumobsModel} differs only in the additional term\n$\\frac{2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} (1 + \\frac{8}{\\mutinco})}{\\ensuremath{\\theta_{\\operatorname{min}}}}$ involving the\nminimum value.  This term can be viewed as constraining how quickly\nthe minimum can decay as a function of $(\\ensuremath{n}, \\ensuremath{p})$, as we\nillustrate with some concrete tail functions.\n\n\n\\subsubsection{Exponential-type tails} \n\nRecall the setting of Section~\\ref{SecSamCovBoundSubG}, where the\nrandom variables $\\{X^{(\\obsind)}_{i}\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}\\}$ are\nsub-Gaussian with parameter $\\ensuremath{\\sigma}$.  Let us suppose that the\nparameters $(\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco)$ are viewed as constants\n(not scaling with $(\\ensuremath{p}, \\ensuremath{\\ensuremath{d}})$.  Then, using the\nexpression~\\eqref{EqnExpTailInv} for the inverse function\n$\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ in this setting, a corollary of Theorem~\\ref{ThmModel}\nis that a sample size \n\\begin{eqnarray}\n\\label{EqnModelSampSub}\n\\ensuremath{n} & = & \\Omega \\big( (\\ensuremath{\\ensuremath{d}}^2 + \\ensuremath{\\theta_{\\operatorname{min}}}^{-2}) \\, \\tau \\log\n\\ensuremath{p} \\big)\n\\end{eqnarray}\nis sufficient for model selection consistency with probability greater\nthan $1-1\/\\ensuremath{p}^{\\tau-2}$.  Alternatively, we can state that $\\ensuremath{n}\n= \\Omega(\\tau \\ensuremath{\\ensuremath{d}}^2 \\log \\ensuremath{p})$ samples are sufficient, as\nalong as the minimum value scales as \\mbox{$\\ensuremath{\\theta_{\\operatorname{min}}} =\n\\Omega(\\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}})$.}\n\n\\subsubsection{Polynomial-type tails} \n\nRecall the setting of Section~\\ref{SecSamCovBoundMoment}, where the\nrescaled random variables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have bounded\n$4m^{th}$ moments. Using the expression~\\eqref{EqnPolyTailInv}\nfor the inverse function $\\ensuremath{{\\widebar{\\ensuremath{n}}_f}}$ in this setting, a corollary of\nTheorem~\\ref{ThmModel} is that a sample size\n\\begin{eqnarray}\n\\label{EqnModelSampPoly}\n\\ensuremath{n} & = & \\Omega\\big( (\\ensuremath{\\ensuremath{d}}^2 + \\ensuremath{\\theta_{\\operatorname{min}}}^{-2})\\,\n\\ensuremath{p}^{\\tau\/m} \\big)\n\\end{eqnarray}\nis sufficient for model selection consistency with probability greater\nthan $1-1\/\\ensuremath{p}^{\\tau-2}$.  Alternatively, we can state than $\\ensuremath{n}\n= \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\ensuremath{p}^{\\tau\/m})$ samples are\nsufficient, as long as the minimum value scales as \\mbox{$\\ensuremath{\\theta_{\\operatorname{min}}} =\n\\Omega(\\ensuremath{p}^{\\tau\/(2m)}\/{\\sqrt{\\ensuremath{n}}})$.}\n\n\n\n\\subsection{Comparison to neighbor-based graphical model selection}\n\\label{SecInco}\n\nSuppose that $X$ follows a multivariate Gaussian distribution, so that\nthe structure of the concentration matrix $\\ensuremath{\\Theta^*}$ specifies the\nstructure of a Gaussian graphical model.  In this case, it is\ninteresting to compare our sufficient conditions for graphical model\nconsistency of the log-determinant approach, as specified in\nTheorem~\\ref{ThmModel}, to those of the neighborhood-based method,\nfirst proposed by \\citet{MeinsBuhl2006}. The latter method estimates\nthe full graph structure by performing an $\\ell_1$-regularized linear\nregression (Lasso)---of the form $X_i = \\sum_{j \\neq i} \\theta_{ij}\nX_j + W$--- of each node on its neighbors and using the support of the\nestimated regression vector $\\theta$ to predict the neighborhood set.\nThese neighborhoods are then combined, by either an OR rule or an AND\nrule, to estimate the full graph.  Various aspects of the\nhigh-dimensional model selection consistency of the Lasso are now\nunderstood~\\cite{MeinsBuhl2006,Wainwright2006_new,Zhao06}; for\ninstance, it is known that mutual incoherence or irrepresentability\nconditions are necessary and sufficient for its\nsuccess~\\cite{Tropp2006,Zhao06}. In terms of scaling,\nWainwright~\\cite{Wainwright2006_new} shows that the Lasso succeeds\nwith high probability if and only if the sample size scales as\n\\mbox{$\\ensuremath{n} \\asymp c (\\{\\ensuremath{\\ensuremath{d}} + \\theta_{\\operatorname{min}}^{-2}\n\\} \\log \\ensuremath{p})$,} where $c$ is a constant determined by the covariance\nmatrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.  By a union bound over the $\\ensuremath{p}$ nodes in the\ngraph, it then follows that the neighbor-based graph selection method\nin turn succeeds with high probability if $\\ensuremath{n} = \\Omega(\\{\\ensuremath{\\ensuremath{d}}\n+ \\ensuremath{\\theta_{\\operatorname{min}}}^{-2} \\} \\log \\ensuremath{p})$.\n\nFor comparison, consider the application of Theorem~\\ref{ThmModel} to\nthe case where the variables are sub-Gaussian (which includes the\nGaussian case). For this setting, we have seen that the scaling\nrequired by Theorem~\\ref{ThmModel} is $\\ensuremath{n} = \\Omega( \\{ \\ensuremath{\\ensuremath{d}}^2\n+ \\ensuremath{\\theta_{\\operatorname{min}}}^{-2} \\} \\log \\ensuremath{p})$, so that the dependence of the\nlog-determinant approach in $\\ensuremath{\\theta_{\\operatorname{min}}}$ is identical, but it depends\nquadratically on the maximum degree $\\ensuremath{\\ensuremath{d}}$.  We suspect that that\nthe quadratic dependence $\\ensuremath{\\ensuremath{d}}^2$ might be an artifact of our\nanalysis, but have not yet been able to reduce it to $\\ensuremath{\\ensuremath{d}}$.\nOtherwise, the primary difference between the two methods is in the\nnature of the irrepresentability assumptions that are imposed: our\nmethod requires Assumption~\\ref{AssInco} on the Hessian $\\ensuremath{\\Gamma^*}$,\nwhereas the neighborhood-based method imposes this same type of\ncondition on a set of $\\ensuremath{p}$ covariance matrices, each of size\n$(\\ensuremath{p} -1) \\times (\\ensuremath{p}-1)$, one for each node of the graph.  Below\nwe show two cases where the Lasso irrepresentability condition holds,\nwhile the log-determinant requirement fails.  However, in general, we\ndo not know whether the log-determinant irrepresentability strictly\ndominates its analog for the Lasso.\n\n\n\\subsubsection{Illustration of irrepresentability: Diamond graph} \nConsider the following Gaussian graphical model example from\n\\citet{Meins2008}.  Figure~\\ref{FigSimpGraph}(a) shows a\ndiamond-shaped graph $G = (V,E)$, with vertex set $V = \\{1,2,3,4\\}$\nand edge-set as the fully connected graph over $V$ with the edge\n$(1,4)$ removed.\n\\begin{figure}[htb]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\psfrag{#1#}{$1$} \\psfrag{#2#}{$2$} \\psfrag{#3#}{$3$}\n\\psfrag{#4#}{$4$} \\widgraph{.3\\textwidth}{meins.eps} & \\hspace*{.2in}\n& \\psfrag{#1#}{$1$} \\psfrag{#2#}{$2$} \\psfrag{#3#}{$3$}\n\\psfrag{#4#}{$4$} \\widgraph{.3\\textwidth}{small_star.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{(a) Graph of the example discussed by~\\citet{Meins2008}.  (b)\nA simple $4$-node star graph.}\n\\label{FigSimpGraph}\n\\end{figure}\nThe covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ is parameterized by the\ncorrelation parameter $\\rho \\in [0,1\/\\sqrt{2}]$: the diagonal entries\nare set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{i} = 1$, for all $i \\in \\ensuremath{V}$; the entries\ncorresponding to edges are set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = \\rho$ for $(i,j)\n\\in \\ensuremath{E} \\backslash \\{(2,3)\\}$, $\\ensuremath{\\ensuremath{\\Sigma}^*}_{23} = 0$; and finally\nthe entry corresponding to the non-edge is set as $\\ensuremath{\\ensuremath{\\Sigma}^*}_{14} =\n2 \\rho^2$. \\citet{Meins2008} showed that the $\\ell_1$-penalized\nlog-determinant estimator $\\ensuremath{\\widehat{\\Theta}}$ fails to recover the graph\nstructure, for any sample size, if $\\rho > -1 + (3\/2)^{1\/2} \\approx\n0.23$.  It is instructive to compare this necessary condition to the\nsufficient condition provided in our analysis, namely the incoherence\nAssumption~\\ref{AssInco} as applied to the Hessian $\\ensuremath{\\Gamma^*}$.  For\nthis particular example, a little calculation shows that\nAssumption~\\ref{AssInco} is equivalent to the constraint\n\\begin{eqnarray*}\n4 |\\rho| (|\\rho| + 1) & < & 1,\n\\end{eqnarray*}\nan inequality which holds for all $\\rho \\in (-0.2017, 0.2017)$.  Note\nthat the upper value $0.2017$ is just below the necessary threshold\ndiscussed by \\citet{Meins2008}. On the other hand, the\nirrepresentability condition for the Lasso requires only that $2\n|\\rho| < 1$, i.e., $\\rho \\in (-0.5,0.5)$.  Thus, in the regime $|\\rho|\n\\in [0.2017,0.5)$, the Lasso irrepresentability condition holds while\nthe log-determinant counterpart fails.\n\n\n\\subsubsection{Illustration of irrepresentability: Star graphs} \nA second interesting example is the star-shaped graphical model,\nillustrated in Figure~\\ref{FigSimpGraph}(b), which consists of a\nsingle hub node connected to the rest of the spoke nodes. We consider\na four node graph, with vertex set $V = \\{1,2,3,4\\}$ and edge-set $E =\n\\{(1,s) \\mid s \\in \\{2,3,4\\}\\}$. The covariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$\nis parameterized the correlation parameter $\\rho \\in [-1,1]$: the\ndiagonal entries are set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} = 1$, for all $i \\in V$;\nthe entries corresponding to edges are set to $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} =\n\\rho$ for $(i,j) \\in E$; while the non-edge entries are set as\n$\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = \\rho^2$ for $(i,j) \\notin E$.  Consequently, for\nthis particular example, Assumption~\\ref{AssInco} reduces to the\nconstraint $|\\rho| (|\\rho| + 2) < 1$, which holds for all $\\rho \\in\n(-0.414, 0.414)$. The irrepresentability condition for the Lasso on\nthe other hand allows the full range $\\rho \\in (-1,1)$. Thus there is\nagain a regime, $|\\rho| \\in [0.414,1)$, where the Lasso\nirrepresentability condition holds while the log-determinant\ncounterpart fails.\n\n\n\\subsection{Rates in Frobenius and spectral norm}\n\\label{SecFrob}\nWe now derive some corollaries of Theorem~\\ref{ThmMain} concerning\nestimation of $\\ensuremath{\\Theta^*}$ in Frobenius norm, as well as the spectral\nnorm.  Recall that $\\ensuremath{s} = |\\ensuremath{E}(\\ensuremath{\\Theta^*})|$ denotes the total\nnumber of off-diagonal non-zeros in $\\ensuremath{\\Theta^*}$.\n\\begin{cors}\n\\label{CorOperatorNorm} \nUnder the same assumptions as Theorem~\\ref{ThmMain}, with probability\nat least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$, the\nestimator $\\estim{\\Theta}$ satisfies\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{EqnPrecFrob}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{F} & \\leq & \\big\\{2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big(1\n + \\frac{8}{\\mutinco}\\big)\\big\\} \\, \\sqrt{\\ensuremath{s} +\\ensuremath{p}} \\;\n \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}), \\qquad \\mbox{and} \\\\\n\\label{EqnPrecSpectral}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{2} & \\leq & \\big\\{2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big(1\n + \\frac{8}{\\mutinco}\\big) \\big\\}\\, \\min \\{\\sqrt{\\ensuremath{s} +\\ensuremath{p}}, \\,\n \\ensuremath{\\ensuremath{d}} \\} \\; \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray}\n\\end{subequations}\n\\end{cors}\n\\begin{proof}\n\\newcommand{\\ensuremath{\\nu}}{\\ensuremath{\\nu}}\n\nWith the shorthand notation $\\ensuremath{\\nu} \\ensuremath{:  =} 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} (1 + 8\/\\mutinco)\n\\; \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$, Theorem~\\ref{ThmMain}\nguarantees that, with probability at least $1 - 1\/\\ensuremath{p}^{\\tau -\n2}$, $\\|\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}\\|_\\infty \\leq \\ensuremath{\\nu}$.  Since the edge\nset of $\\ensuremath{\\widehat{\\Theta}}$ is a subset of that of $\\ensuremath{\\Theta^*}$, and\n$\\ensuremath{\\Theta^*}$ has at most $\\ensuremath{p} + \\ensuremath{s}$ non-zeros (including the\ndiagonal), we conclude that\n\\begin{eqnarray*}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{F} & = & \\big[ \\sum_{i=1}^\\ensuremath{p}\n(\\ensuremath{\\widehat{\\Theta}}_{ii} - \\ensuremath{\\Theta^*}_{ii})^2 + \\sum_{(i,j) \\in \\ensuremath{E}}\n(\\ensuremath{\\widehat{\\Theta}}_{ij} - \\ensuremath{\\Theta^*}_{ij})^2 \\big]^{1\/2} \\\\\n& \\leq & \\ensuremath{\\nu} \\; \\sqrt{\\ensuremath{s} + \\ensuremath{p}},\n\\end{eqnarray*}\nfrom which the bound~\\eqref{EqnPrecFrob} follows.  On the other hand,\nfor a symmetric matrix, we have\n\\begin{eqnarray}\\label{EqnPrecInftyOp}\n\\matnorm{\\ensuremath{\\widehat{\\Theta}}- \\ensuremath{\\Theta^*}}{2} & \\leq & \\matnorm{\\ensuremath{\\widehat{\\Theta}} -\n\\ensuremath{\\Theta^*}}{\\infty} \\; \\leq \\; \\ensuremath{\\ensuremath{d}} \\ensuremath{\\nu},\n\\end{eqnarray}\nusing the definition of the $\\ensuremath{\\nu}_\\infty$-operator norm, and the fact\nthat $\\ensuremath{\\widehat{\\Theta}}$ and $\\ensuremath{\\Theta^*}$ have at most $\\ensuremath{\\ensuremath{d}}$ non-zeros per\nrow.  Since the Frobenius norm upper bounds the spectral norm, the\nbound~\\eqref{EqnPrecSpectral} follows.\n\n\n\\end{proof}\n\n\\subsubsection{Exponential-type tails}\nFor the exponential tail function case where the rescaled random\nvariables $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are sub-Gaussian with\nparameter $\\ensuremath{\\sigma}$, we can use the expression~\\eqref{EqnExpTailInv}\nfor the inverse function $\\ensuremath{\\widebar{\\delta}_f}$ to derive rates in Frobenius\nand spectral norms.  When the quantities $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}},\n\\mutinco$ remain constant, these bounds can be summarized succinctly\nas a sample size \\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\log \\ensuremath{p})$} is\nsufficient to guarantee the bounds\n\\begin{subequations}\n\\begin{eqnarray}\n\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{F} & = &\n{\\mathcal{O}}\\biggr(\\sqrt{\\frac{(\\ensuremath{s} + \\ensuremath{p})\\,\\log \\ensuremath{p}}{\\ensuremath{n}}}\\,\\biggr),\n\\quad \\mbox{and} \\\\\n\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{2} & = &\n{\\mathcal{O}}\\biggr(\\sqrt{\\frac{\\min\\{\\ensuremath{s} + \\ensuremath{p},\\,\\ensuremath{\\ensuremath{d}}^{2}\\} \\, \\log\n\\ensuremath{p}}{\\ensuremath{n}}}\\,\\biggr),\n\\end{eqnarray}\n\\end{subequations}\nwith probability at least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$.\n\n\\subsubsection{Polynomial-type tails}\n\nSimilarly, let us again consider the polynomial tail case, in which\nthe rescaled variates $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ have bounded $4\nm^{th}$ moments and the samples are drawn i.i.d.  Using the\nexpression~\\eqref{EqnPolyTailInv} for the inverse function we can\nderive rates in the Frobenius and spectral norms.  When the quantities\n$\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\mutinco$ are viewed as constant, we are\nguaranteed that a sample size \\mbox{$\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\,\n\\ensuremath{p}^{\\tau\/m})$} is sufficient to guarantee the bounds\n\\begin{subequations}\n\\begin{eqnarray}\n \\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{F} & = &\n {\\mathcal{O}}\\biggr(\\sqrt{\\frac{(\\ensuremath{s} +\n \\ensuremath{p})\\,\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}\\,\\biggr),\\textrm{ and } \\\\\n\\matnorm{\\estim{\\Theta} - \\ensuremath{\\Theta^*}}{2} & = &\n{\\mathcal{O}}\\biggr(\\sqrt{\\frac{\\min\\{\\ensuremath{s} + \\ensuremath{p},\\,\\ensuremath{\\ensuremath{d}}^{2}\\} \\,\n\\ensuremath{p}^{\\tau\/m}}{\\ensuremath{n}}}\\,\\biggr),\n\\end{eqnarray}\n\\end{subequations}\nwith probability at least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$.\n\n\n\\subsection{Rates for the covariance matrix estimate}\nFinally, we describe some bounds on the estimation of the covariance\nmatrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$.  By Lemma~\\ref{LEM_MLE_CHARAC}, the estimated\nconcentration matrix $\\ensuremath{\\widehat{\\Theta}}$ is positive definite, and hence can\nbe inverted to obtain an estimate of the covariance matrix, which we\ndenote as $\\hat{\\CovHat} \\ensuremath{:  =} (\\ensuremath{\\widehat{\\Theta}})^{-1}$.\n\n\\begin{cors}\n\\label{CorCovBound}\nUnder the same assumptions as Theorem~\\ref{ThmMain}, with probability\nat least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$, the following bounds hold.\n\\begin{enumerate}\n\\item[(a)] The element-wise $\\ell_{\\infty}$ norm of the deviation\n $(\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*})$ satisfies the bound\n\\begin{eqnarray}\n\\label{EqnCovInftyBound}\n\\vecnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} & \\leq & \\ensuremath{C_3},\n\t[\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})] + \\ensuremath{C_4}\n\t\\ensuremath{\\ensuremath{d}}\\, [\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})]^{2}\n\\end{eqnarray}\nwhere $\\ensuremath{C_3} = 2 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}\\Big(1 +\n\\frac{8}{\\mutinco}\\Big)$ and $\\ensuremath{C_4} = 6 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3}\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2}\\Big(1 + \\frac{8}{\\mutinco}\\Big)^{2}$.\n\\item[(b)] The $\\ell_2$ operator-norm of the deviation $(\\hat{\\CovHat} -\n\\ensuremath{\\ensuremath{\\Sigma}^*})$ satisfies the bound\n\\begin{eqnarray}\n\\label{EqnCovSpectralBound}\n\\matnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{2} & \\leq & \\ensuremath{C_3} \\, \\ensuremath{\\ensuremath{d}} \\,\n\t[\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})] + \\ensuremath{C_4} \\ensuremath{\\ensuremath{d}}^{2}\n\t\\, [\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})]^{2}.\n\\end{eqnarray}\n\\end{enumerate}\n\\end{cors}\nThe proof involves certain lemmata and derivations that are parts of\nthe proofs of Theorems~\\ref{ThmMain} and \\ref{ThmModel}, so that we\ndefer it to Section~\\ref{SecCorCovProof}.\n\n\t\n\n\\section{Proofs of main result}\n\\label{SecProof}\n\nIn this section, we work through the proofs of Theorems~\\ref{ThmMain}\nand~\\ref{ThmModel}.  We break down the proofs into a sequence of\nlemmas, with some of the more technical aspects deferred to\nappendices.\n\nOur proofs are based on a technique that we call a \\emph{primal-dual\nwitness method}, used previously in analysis of the\nLasso~\\cite{Wainwright2006_new}.  It involves following a specific\nsequence of steps to construct a pair $(\\ThetaWitness, \\ensuremath{\\widetilde{Z}})$ of\nsymmetric matrices that together satisfy the optimality conditions\nassociated with the convex program~\\eqref{EqnGaussMLE} \\emph{with high\nprobability}.  Thus, when the constructive procedure succeeds,\n$\\ThetaWitness$ is \\emph{equal} to the unique solution\n$\\estim{\\Theta}$ of the convex program~\\eqref{EqnGaussMLE}, and\n$\\ensuremath{\\widetilde{Z}}$ is an optimal solution to its dual.  In this way, the\nestimator $\\estim{\\Theta}$ inherits from $\\ThetaWitness$ various\noptimality properties in terms of its distance to the truth\n$\\ensuremath{\\Theta^*}$, and its recovery of the signed sparsity pattern.  To be\nclear, our procedure for constructing $\\ThetaWitness$ is \\emph{not} a\npractical algorithm for solving the log-determinant\nproblem~\\eqref{EqnGaussMLE}, but rather is used as a proof technique\nfor certifying the behavior of the $M$-estimator~\\eqref{EqnGaussMLE}.\n\n\n\n\\subsection{Primal-dual witness approach}\n\\label{SecPrimalDualWitness}\nAs outlined above, at the core of the primal-dual witness method are the standard convex\noptimality conditions that characterize the optimum $\\ensuremath{\\widehat{\\Theta}}$ of the\nconvex program~\\eqref{EqnGaussMLE}.  For future reference, we note\nthat the sub-differential of the norm $\\ellreg{\\cdot}$ evaluated at\nsome $\\Theta$ consists the set of all symmetric matrices $Z \\in\n\\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times \\ensuremath{p}}$ such that\n\\begin{eqnarray}\\label{EqnSubGradDefn}\nZ_{ij} & = & \\begin{cases} 0 & \\mbox{if $i =j$} \\\\ \\textrm{sign}(\\Theta_{ij})\n& \\mbox{if $i \\neq j$ and $\\Theta_{ij} \\neq 0$} \\\\ \\in [-1, +1] &\n\\mbox{if $i \\neq j$ and $\\Theta_{ij} = 0$.}\n             \\end{cases}\n\\end{eqnarray}\nThe following result is proved in Appendix~\\ref{AppLemMLECharac}:\n\\begin{lems}\n\\label{LEM_MLE_CHARAC}\nFor any $\\ensuremath{\\lambda_\\ensuremath{n}} > 0$ and sample covariance $\\ensuremath{\\widehat{\\Sigma}}$ with strictly\npositive diagonal, the $\\ell_1$-regularized log-determinant\nproblem~\\eqref{EqnGaussMLE} has a unique solution $\\ensuremath{\\widehat{\\Theta}} \\succ 0$\ncharacterized by\n\\begin{eqnarray}\n\\label{EqnZeroSubgrad}\n\\ensuremath{\\widehat{\\Sigma}} - \\ensuremath{\\widehat{\\Theta}}^{-1} + \\ensuremath{\\lambda_\\ensuremath{n}} \\ensuremath{\\hat Z} & = & 0,\n\\end{eqnarray}\nwhere $\\ensuremath{\\hat Z}$ is an element of the subdifferential $\\partial\n\\ellreg{\\ensuremath{\\widehat{\\Theta}}}$.  \n\\end{lems}\n\nBased on this lemma, we construct the primal-dual witness solution\n$(\\ThetaWitness, \\ensuremath{\\widetilde{Z}})$ as follows:\n\\begin{enumerate}\n\\item[(a)]  We determine the\nmatrix $\\ThetaWitness$ by solving the restricted log-determinant\nproblem\n\\begin{eqnarray}\n\\label{EqnRestricted}\n\\ThetaWitness & \\ensuremath{:  =} & \\arg \\min_{\\Theta \\succ 0, \\;\n\\Theta_{\\ensuremath{\\EsetPlus^c}} = 0} \\big \\{\\tracer{\\Theta}{\\ensuremath{\\widehat{\\Sigma}}} - \\log\n\\det(\\Theta) + \\ensuremath{\\lambda_\\ensuremath{n}} \\ellreg{\\Theta} \\big \\}.\n\\end{eqnarray}\nNote that by construction, we have $\\ThetaWitness \\succ 0$, and\nmoreover $\\ThetaWitness_{\\ensuremath{\\EsetPlus^c}} = 0$.\n\\item[(b)] We choose $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{S}}$ as a member of the\nsub-differential of the regularizer $\\ellreg{\\cdot}$, evaluated\nat $\\ensuremath{\\ThetaWitness}$.\n\\item[(c)] We set $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}$ as\n\\begin{eqnarray}\n \\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}} &=& \\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}}\\big\\{-\n \\ensuremath{\\widehat{\\Sigma}}_{\\ensuremath{\\EsetPlus^c}} + [\\invn{\\ensuremath{\\ThetaWitness}}]_{\\ensuremath{\\EsetPlus^c}}\\big\\},\n\\end{eqnarray}\nwhich ensures that constructed matrices $(\\ensuremath{\\ThetaWitness},\\ensuremath{\\widetilde{Z}})$ satisfy\nthe optimality condition~\\eqref{EqnZeroSubgrad}.\n\\item[(d)]  We verify the \\emph{strict dual feasibility} condition\n\\begin{eqnarray*}\n|\\ensuremath{\\widetilde{Z}}_{ij}| & < & 1 \\quad \\mbox{for all $(i,j) \\in \\ensuremath{\\EsetPlus^c}$}.\n\\end{eqnarray*}\n\n\\end{enumerate}\nTo clarify the nature of the construction, steps (a) through (c)\nsuffice to obtain a pair $(\\ensuremath{\\ThetaWitness},\\ensuremath{\\widetilde{Z}})$ that satisfy the\noptimality conditions~\\eqref{EqnZeroSubgrad}, but do \\emph{not}\nguarantee that $\\ensuremath{\\widetilde{Z}}$ is an element of sub-differential $\\partial\n\\ellreg{\\ensuremath{\\ThetaWitness}}$.  By construction, specifically step (b) of the\nconstruction ensures that the entries $\\ensuremath{\\widetilde{Z}}$ in $\\ensuremath{S}$ satisfy\nthe sub-differential conditions, since $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{S}}$ is a member\nof the sub-differential of $\\partial \\ellreg{\\ensuremath{\\ThetaWitness}_{\\ensuremath{S}}}$.\nThe purpose of step (d), then, is to verify that the remaining\nelements of $\\ensuremath{\\widetilde{Z}}$ satisfy the necessary conditions to belong\nto the sub-differential.\n\n\nIf the primal-dual witness construction succeeds, then it acts as a\n\\emph{witness} to the fact that the solution $\\ensuremath{\\ThetaWitness}$ to the\nrestricted problem~\\eqref{EqnRestricted} is equivalent to the solution\n$\\ensuremath{\\widehat{\\Theta}}$ to the original (unrestricted)\nproblem~\\eqref{EqnGaussMLE}.  We exploit this fact in our proofs of\nTheorems~\\ref{ThmMain} and \\ref{ThmModel} that build on this: we first show\nthat the primal-dual witness technique succeeds with high-probability,\nfrom which we can conclude that the support of the optimal solution\n$\\ensuremath{\\widehat{\\Theta}}$ is contained within the support of the true $\\ensuremath{\\Theta^*}$.\nIn addition, we exploit the characterization of $\\ensuremath{\\widehat{\\Theta}}$ provided\nby the primal-dual witness construction to establish the elementwise\n$\\ell_\\infty$ bounds claimed in Theorem~\\ref{ThmMain}.\nTheorem~\\ref{ThmModel} requires checking, in addition, that certain\nsign consistency conditions hold, for which we require lower bounds on\nthe value of the minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$.\n\nIn the analysis to follow, some additional notation is useful.  We let\n$\\ensuremath{W}$ denote the ``effective noise'' in the sample covariance\nmatrix $\\ensuremath{\\widehat{\\Sigma}}$, namely\n\\begin{eqnarray}\n\\label{EqnWdefn}\n\\ensuremath{W} & \\ensuremath{:  =} & \\ensuremath{\\widehat{\\Sigma}} - (\\ensuremath{\\Theta^*})^{-1}.\n\\end{eqnarray}\nSecond, we use $\\Delta = \\ThetaWitness - \\ensuremath{\\Theta^*}$ to measure the\ndiscrepancy between the primal witness matrix $\\ensuremath{\\ThetaWitness}$ and the\ntruth $\\ensuremath{\\Theta^*}$.  Finally, recall the log-determinant barrier $g$\nfrom equation~\\eqref{EqnDefnLogDet}. We let $\\ensuremath{R}(\\Delta)$ denote the\ndifference of the gradient $\\nabla g(\\ensuremath{\\ThetaWitness}) =\n\\invn{\\ThetaWitness}$ from its first-order Taylor expansion around\n$\\ensuremath{\\Theta^*}$. Using known results on the first and second derivatives\nof the log-determinant function (see p. 641 in Boyd and\nVandenberghe~\\cite{Boyd02}), this remainder takes the form\n\\begin{eqnarray}\n\\label{EqnRdefn}\n\\ensuremath{R}(\\Delta) & = & \\invn{\\ThetaWitness} - \\invn{\\ensuremath{\\Theta^*}} +\n{\\ensuremath{\\Theta^*}}^{-1} \\Delta {\\ensuremath{\\Theta^*}}^{-1}.\n\\end{eqnarray}\n\n\\subsection{Auxiliary results}\n\nWe begin with some auxiliary lemmata, required in the proofs of our\nmain theorems.  In Section~\\ref{SecStrictDual}, we provide sufficient\nconditions on the quantities $\\ensuremath{W}$ and $\\ensuremath{R}$ for the strict dual\nfeasibility condition to hold. In Section~\\ref{SecRemainder}, we\ncontrol the remainder term $\\ensuremath{R}(\\Delta)$ in terms of $\\Delta$, while\nin Section~\\ref{SecEllinfBound}, we control $\\Delta$ itself, providing\nelementwise $\\ell_\\infty$ bounds on $\\Delta$.  In\nSection~\\ref{SecSignConsis}, we show that under appropriate conditions\non the minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$, the bounds in the earlier lemmas\nguarantee that the sign consistency condition holds.  All of the\nanalysis in these sections is \\emph{deterministic} in nature.  In\nSection~\\ref{SecNoise}, we turn to the probabilistic component of the\nanalysis, providing control of the noise $\\ensuremath{W}$ in the sample\ncovariance matrix.  Finally, the proofs of Theorems~\\ref{ThmMain}\nand~\\ref{ThmModel} follows by using this probabilistic control of\n$\\ensuremath{W}$ and the stated conditions on the sample size to show that\nthe deterministic conditions hold with high probability.\n\n\n\\subsubsection{Sufficient conditions for strict dual feasibility}\n\\label{SecStrictDual}\n\nWe begin by stating and proving a lemma that provides sufficient\n(deterministic) conditions for strict dual feasibility to hold, so\nthat $\\|\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}\\|_\\infty < 1$.\n\\begin{lems}[Strict dual feasibility]\n\\label{LemStrictDual}\nSuppose that\n\\begin{eqnarray}\n\\label{EqnDetSuff}\n\\max \\big \\{ \\|\\ensuremath{W}\\|_\\infty, \\; \\|\\ensuremath{R}(\\Delta)\\|_\\infty \\big \\}  &\n\\leq & \\frac{\\mutinco \\, \\ensuremath{\\lambda_\\ensuremath{n}}}{8}.\n\\end{eqnarray}\nThen the matrix $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}$ constructed in step (c)\nsatisfies $\\|\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}\\|_\\infty < 1$, and therefore\n$\\ThetaWitness = \\ensuremath{\\widehat{\\Theta}}$.\n\\end{lems}\n\\begin{proof}\nUsing the definitions~\\eqref{EqnWdefn} and~\\eqref{EqnRdefn}, we can\nre-write the stationary condition~\\eqref{EqnZeroSubgrad} in an\nalternative but equivalent form\n\\begin{eqnarray}\n\\label{EqnLME}\n\\invn{\\ensuremath{\\Theta^*}} \\Delta \\invn{\\ensuremath{\\Theta^*}} + \\ensuremath{W} - \\ensuremath{R}(\\Delta) +\n\\ensuremath{\\lambda_\\ensuremath{n}} \\ensuremath{\\wtil{Z}} & = 0.\n\\end{eqnarray}\nThis is a linear-matrix equality, which can be re-written as an\nordinary linear equation by ``vectorizing'' the matrices.  We use the\nnotation $\\ensuremath{\\operatorname{vec}}(A)$, or equivalently $\\myvec{A}$ for the\n$\\ensuremath{p}^2$-vector version of the matrix $A \\in \\ensuremath{{\\mathbb{R}}}^{\\ensuremath{p} \\times\n\\ensuremath{p}}$, obtained by stacking up the rows into a single column vector.\nIn vectorized form, we have\n\\begin{equation*}\n\\ensuremath{\\operatorname{vec}} \\big(\\invn{\\ensuremath{\\Theta^*}} \\Delta \\invn{\\ensuremath{\\Theta^*}} \\big) = \\big\n(\\invn{\\ensuremath{\\Theta^*}} \\otimes \\invn{\\ensuremath{\\Theta^*}} \\big) \\myvec{\\Delta} \\; =\n\\; \\ensuremath{\\Gamma^*} \\myvec{\\Delta}.\n\\end{equation*}\nIn terms of the disjoint decomposition $\\ensuremath{S}$ and\n$\\ensuremath{\\EsetPlus^c}$, equation~\\eqref{EqnLME} can be re-written as two\nblocks of linear equations as follows:\n\\begin{subequations}\n\\begin{eqnarray}\n\\label{EqnStatBlockS} \\ensuremath{\\Gamma^*}_{\\ensuremath{S}\n\\ensuremath{S}} \\myvec{\\Delta}_{\\ensuremath{S}} + \\myvec{\\ensuremath{W}}_{\\ensuremath{S}} -\n\\myvec{\\ensuremath{R}}_{\\ensuremath{S}} + \\ensuremath{\\lambda_\\ensuremath{n}} \\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{S}} & = & 0\n\\\\\n\\label{EqnStatBlockScomp}\n\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c} \\ensuremath{S}} \\myvec{\\Delta}_{\\ensuremath{S}} +\n\\myvec{\\ensuremath{W}}_{\\ensuremath{\\EsetPlus^c}} - \\myvec{\\ensuremath{R}}_{\\ensuremath{\\EsetPlus^c}} +\n\\ensuremath{\\lambda_\\ensuremath{n}} \\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}} & = & 0.\n\\end{eqnarray}\n\\end{subequations}\nHere we have used the fact that $\\Delta_{\\ensuremath{\\EsetPlus^c}} = 0$ by\nconstruction.\n\nSince $\\ensuremath{\\Gamma^*}_{\\ensuremath{S} \\ensuremath{S}}$ is invertible, we can solve for\n$\\myvec{\\Delta}_{\\ensuremath{S}}$ from equation~\\eqref{EqnStatBlockS} as\nfollows:\n\\begin{eqnarray*}\n\\myvec{\\Delta}_{\\ensuremath{S}} & = & \\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}\n\\big[-\\myvec{\\ensuremath{W}}_{\\ensuremath{S}} + \\myvec{\\ensuremath{R}}_{\\ensuremath{S}} - \\ensuremath{\\lambda_\\ensuremath{n}}\n\\myvec{\\ensuremath{\\widetilde{Z}}_\\ensuremath{S}} \\big].\n\\end{eqnarray*}\nSubstituting this expression into equation~\\eqref{EqnStatBlockScomp},\nwe can solve for $\\ensuremath{\\widetilde{Z}}_{\\ensuremath{\\EsetPlus^c}}$ as follows:\n\\begin{eqnarray}\n\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}} &= & -\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}}\n\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\\myvec{\\Delta}_{\\ensuremath{S}} +\n\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\myvec{\\ensuremath{R}}_{\\ensuremath{\\EsetPlus^c}} - \\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}}\n\\myvec{\\ensuremath{W}}_{\\ensuremath{\\EsetPlus^c}} \\nonumber\\\\\n& = & -\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}(\\myvec{\\ensuremath{W}}_{\\ensuremath{S}} -\n\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}) + \\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}} \\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{S}} -\n\\frac{1}{\\ensuremath{\\lambda_\\ensuremath{n}}} (\\myvec{\\ensuremath{W}}_{\\ensuremath{\\EsetPlus^c}} -\n\\myvec{\\ensuremath{R}}_{\\ensuremath{\\EsetPlus^c}}).\n\\end{eqnarray}\nTaking the $\\ell_\\infty$ norm of both sides yields\n\\begin{multline*}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}}\\|_{\\infty} \\leq \\frac{1}{\\lambda_n}\n\\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}}{\\infty}\n(\\|\\myvec{\\ensuremath{W}}_{\\ensuremath{S}}\\|_{\\infty} +\n\\|\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}\\|_{\\infty}) \\\\\n+ \\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}}{\\infty}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{S}}\\|_{\\infty} + \\frac{1}{\\lambda_n}\n(\\|\\myvec{\\ensuremath{W}}_{\\ensuremath{S}}\\|_{\\infty} +\n\\|\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}\\|_{\\infty}).\n\\end{multline*}\nRecalling Assumption~\\ref{AssInco}---namely, that\n$\\matnorm{\\ensuremath{\\Gamma^*}_{\\ensuremath{\\EsetPlus^c}\\ensuremath{S}}\n\\inv{\\ensuremath{\\Gamma^*}_{\\ensuremath{S}\\EsetPlus}}}{\\infty} \\le (1 - \\mutinco)$---we\nhave\n\\begin{eqnarray*}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}}\\|_{\\infty} & \\leq &\n\\frac{2-\\mutinco}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\,\n(\\|\\myvec{\\ensuremath{W}}_{\\ensuremath{S}}\\|_{\\infty} +\n\\|\\myvec{\\ensuremath{R}}_{\\ensuremath{S}}\\|_{\\infty}) + (1 - \\mutinco),\n\\end{eqnarray*}\nwhere we have used the fact that $\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_\\ensuremath{S}\\|_{\\infty}\n\\leq 1$, since $\\ensuremath{\\widetilde{Z}}$ belongs to the sub-differential of the norm\n$\\ellreg{\\cdot}$ by construction.  Finally, applying\nassumption~\\eqref{EqnDetSuff} from the lemma statement, we have\n\\begin{eqnarray*}\n\\|\\myvec{\\ensuremath{\\widetilde{Z}}}_{\\ensuremath{\\EsetPlus^c}}\\|_{\\infty} & \\leq &\n\\frac{(2-\\mutinco)}{\\ensuremath{\\lambda_\\ensuremath{n}}} \\, \\big( \\frac{\\mutinco \\ensuremath{\\lambda_\\ensuremath{n}}}{4}) +\n(1-\\mutinco) \\\\\n& \\leq &  \\frac{\\mutinco}{2} + (1-\\mutinco)  \\: < \\; 1,\n\\end{eqnarray*}\nas claimed.\n\n\n\\end{proof}\n\n\n\\subsubsection{Control of remainder term}\n\\label{SecRemainder}\n\nOur next step is to relate the behavior of the remainder\nterm~\\eqref{EqnRdefn} to the deviation $\\Delta = \\ensuremath{\\ThetaWitness} -\n\\ensuremath{\\Theta^*}$.\n\\begin{lems}[Control of remainder]\n\\label{LEM_R_CONV}\nSuppose that the elementwise $\\ell_\\infty$ bound $\\|\\Delta\\|_\\infty\n\\leq \\frac{1}{3 \\, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}$ holds.  Then:\n\\begin{eqnarray}\n\\label{EqnRemExpand}\n\\ensuremath{R}(\\Delta) & = & \\invn{\\opt{\\Theta}} \\Delta \\invn{\\opt{\\Theta}} \\Delta\nJ \\invn{\\opt{\\Theta}},\n\\end{eqnarray}\nwhere $J \\ensuremath{:  =} \\sum_{k=0}^{\\infty} (-1)^{k} \\big(\\invn{\\opt{\\Theta}}\n\\Delta\\big)^{k}$ has norm $\\matnorm{J^T}{\\infty} \\leq 3\/2$.  Moreover,\nin terms of the elementwise $\\ell_\\infty$-norm, we have\n\\begin{eqnarray}\n\\label{EqnRemBound}\n\\| \\ensuremath{R}(\\Delta)\\|_\\infty & \\leq & \\frac{3}{2} \\ensuremath{\\ensuremath{d}}\n\\|\\Delta\\|_\\infty^2 \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3.\n\\end{eqnarray}\n\\end{lems}\nWe provide the proof of this lemma in Appendix~\\ref{APP_LEM_R_CONV}.  It\nis straightforward, based on standard matrix expansion techniques.\n\n\\subsubsection{Sufficient conditions for $\\ell_\\infty$ bounds}\n\\label{SecEllinfBound}\n\nOur next lemma provides control on the deviation \\mbox{$\\Delta =\n\\ThetaWitness - \\ensuremath{\\Theta^*}$,} measured in elementwise $\\ell_\\infty$\nnorm.\n\\begin{lems}[Control of $\\Delta$]\n\\label{LEM_D_CONV}\nSuppose that\n\\begin{eqnarray}\n\\label{EqnDconvAss}\nr \\ensuremath{:  =} 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big( \\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}} \\big) & \\leq & \\min\n\\big \\{ \\frac{1}{3 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}, \\; \\frac{1}{3 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3\n\\;\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\ensuremath{\\ensuremath{d}}} \\big \\}.\n\\end{eqnarray}\nThen we have the elementwise $\\ell_\\infty$ bound\n\\begin{eqnarray}\n\\label{EqnDconvBound}\n\\|\\Delta \\|_\\infty = \\| \\ensuremath{\\ThetaWitness} - \\ensuremath{\\Theta^*} \\|_\\infty & \\leq & r.\n\\end{eqnarray}\n\\end{lems}\n\nWe prove the lemma in Appendix~\\ref{APP_LEM_D_CONV}; at a high level,\nthe main steps involved are the following.  We begin by noting that\n$\\ThetaWitness_{\\ensuremath{\\EsetPlus^c}} = \\ensuremath{\\Theta^*}_{\\ensuremath{\\EsetPlus^c}} = 0$, so\nthat $\\vecnorm{\\Delta}{\\infty} =\n\\vecnorm{\\Delta_{\\ensuremath{S}}}{\\infty}$.  Next, we characterize\n$\\ensuremath{\\ThetaWitness}_{\\ensuremath{S}}$ in terms of the zero-gradient condition\nassociated with the restricted problem~\\eqref{EqnRestricted}.  We then\ndefine a continuous map $F: \\Delta_{\\ensuremath{S}} \\mapsto\nF(\\Delta_{\\ensuremath{S}})$ such that its fixed points are equivalent to\nzeros of this gradient expression in terms of $\\Delta_\\ensuremath{S} =\n\\ensuremath{\\ThetaWitness}_\\ensuremath{S} - \\ensuremath{\\Theta^*}_{\\ensuremath{S}}$.  We then show that the\nfunction $F$ maps the $\\ell_\\infty$-ball\n\\begin{eqnarray}\n\\label{EqnDefnRad}\n\\ensuremath{\\mathbb{B}}(\\ensuremath{r}) & \\ensuremath{:  =} & \\{ \\Theta_{\\ensuremath{S}} \\mid \\|\\Theta_\\ensuremath{S}\n\\|_\\infty \\leq \\ensuremath{r} \\}, \\qquad \\mbox{with $\\ensuremath{r} \\ensuremath{:  =} 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big\n(\\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}} \\big)$},\n\\end{eqnarray}\nonto itself.  Finally, with these results in place, we can apply\nBrouwer's fixed point theorem (e.g., p. 161; Ortega and\nRheinboldt~\\cite{OrtegaR70}) to conclude that $F$ does indeed have a\nfixed point inside $\\ensuremath{\\mathbb{B}}(\\ensuremath{r})$.  \n\n\\subsubsection{Sufficient conditions for sign consistency}\n\\label{SecSignConsis}\n\n\nWe now show how a lower bound on the minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$, when\ncombined with Lemma~\\ref{LEM_D_CONV}, allows us to guarantee\n\\emph{sign consistency} of the primal witness matrix\n$\\ensuremath{\\ThetaWitness}_{\\ensuremath{S}}$.\n\\begin{lems}[Sign Consistency]\n\\label{LemSignConsis}\nSuppose the minimum absolute value $\\ensuremath{\\theta_{\\operatorname{min}}}$ of non-zero entries\nin the true concentration matrix $\\ensuremath{\\Theta^*}$ is lower bounded as\n\\begin{eqnarray}\n\\label{EqnThetaminLow}\n\\ensuremath{\\theta_{\\operatorname{min}}} & \\geq & 4 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big (\\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}}\n\\big),\n\\end{eqnarray}\nthen \\mbox{$\\textrm{sign}(\\ensuremath{\\ThetaWitness}_\\ensuremath{S}) = \\textrm{sign}(\\ensuremath{\\Theta^*}_\\ensuremath{S})$}\nholds.  \n\\end{lems}\nThis claim follows from the bound~\\eqref{EqnThetaminLow} combined with\nthe bound~\\eqref{EqnDconvBound} ,which together imply that for all\n$(i,j) \\in \\ensuremath{S}$, the estimate $\\ensuremath{\\ThetaWitness}_{ij}$ cannot differ\nenough from $\\ensuremath{\\Theta^*}_{ij}$ to change sign.\n\n\n\n\n\n\n\n\\subsubsection{Control of noise term}\n\\label{SecNoise}\n\nThe final ingredient required for the proofs of Theorems~\\ref{ThmMain}\nand \\ref{ThmModel} is control on the sampling noise $\\ensuremath{W} = \\ensuremath{\\estim{\\ensuremath{\\Sigma}}}\n- \\ensuremath{\\ensuremath{\\Sigma}^*}$.  This control is specified in terms of the decay\nfunction $f$ from equation~\\eqref{EqnSamTail}.\n\\begin{lems}[Control of Sampling Noise]\n\\label{LemWconv}\nFor any $\\tau > 2$ and sample size $\\ensuremath{n}$ such that\n$\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^\\tau) \\le 1\/v_*$, we have\n\\begin{eqnarray}\n\\ensuremath{\\mathbb{P}}\\biggr[ \\|\\ensuremath{W}\\|_\\infty \\geq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\biggr] & \\leq &\n\\frac{1}{\\ensuremath{p}^{\\tau - 2}} \\; \\rightarrow \\; 0.\n\\end{eqnarray}\n\n\\end{lems}\n\\begin{proof}\nUsing the definition~\\eqref{EqnSamTail} of the decay function\n$f$, and applying the union bound over all $\\ensuremath{p}^2$ entries of\nthe noise matrix, we obtain that for all $\\delta \\leq 1\/v_*$,\n\\begin{eqnarray*}\n\\ensuremath{\\mathbb{P}} \\big[\\max_{i, j} |\\ensuremath{W}_{ij}| \\geq \\delta \\big] & \\leq &\n\\ensuremath{p}^2\/f(\\ensuremath{n},\\delta).\n\\end{eqnarray*}\nSetting $\\delta = \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$ yields that\n\\begin{eqnarray*}\n\\ensuremath{\\mathbb{P}} \\big[\\max_{i, j} |\\ensuremath{W}_{ij}| \\geq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\big] & \\leq & \\ensuremath{p}^2\/\n\\big[f(\\ensuremath{n},\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})) \\big] \\; =\n\\; 1\/\\ensuremath{p}^{\\tau - 2},\n\\end{eqnarray*}\nas claimed.  Here the last equality follows since\n$f(\\ensuremath{n},\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})) =\n\\ensuremath{p}^\\tau$, using the definition~\\eqref{EqnSamTailT} of the\ninverse function $\\ensuremath{\\widebar{\\delta}_f}$.\n\\end{proof}\n\n\\def\\delta'{\\delta'}\n\\def\\numobsNew{\\ensuremath{n}'} \n\n\n\n\\subsection{Proof of Theorem~\\ref{ThmMain}}\n\\label{SecThmMain}\n\nWe now have the necessary ingredients to prove\nTheorem~\\ref{ThmMain}. We first show that with high probability the\nwitness matrix $\\ensuremath{\\ThetaWitness}$ is equal to the solution $\\ensuremath{\\widehat{\\Theta}}$ to the\noriginal log-determinant problem~\\eqref{EqnGaussMLE}, in particular by\nshowing that the primal-dual witness construction (described in in\nSection~\\ref{SecPrimalDualWitness}) succeeds with high probability.\nLet $\\mathcal{A}$ denote the event that $\\|\\ensuremath{W}\\|_\\infty \\leq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$.  Using the monotonicity of the\ninverse tail function~\\eqref{EqnMonot}, the lower lower\nbound~\\eqref{EqnSampleBound} on the sample size $\\ensuremath{n}$ implies that\n$\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\le 1\/v_*$.  Consequently,\nLemma~\\ref{LemWconv} implies that $\\ensuremath{\\mathbb{P}}(\\mathcal{A}) \\ge 1 -\n\\frac{1}{\\ensuremath{p}^{\\tau - 2}}$.  Accordingly, we condition on the\nevent $\\mathcal{A}$ in the analysis to follow.\n\nWe proceed by verifying that assumption~\\eqref{EqnDetSuff} of\nLemma~\\ref{LemStrictDual} holds.  Recalling the choice of\nregularization penalty \\mbox{$\\ensuremath{\\lambda_\\ensuremath{n}} = (8\/\\mutinco)\\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$,} we have $\\|\\ensuremath{W}\\|_\\infty\n\\leq (\\mutinco\/8) \\ensuremath{\\lambda_\\ensuremath{n}}$.  In order to establish\ncondition~\\eqref{EqnDetSuff} it remains to establish the bound\n$\\|\\ensuremath{R}(\\Delta)\\|_\\infty \\leq \\frac{\\mutinco \\,\\ensuremath{\\lambda_\\ensuremath{n}}}{8}$.  We do so\nin two steps, by using Lemmas~\\ref{LEM_D_CONV} and~\\ref{LEM_R_CONV}\nconsecutively.  First, we show that the\nprecondition~\\eqref{EqnDconvAss} required for Lemma~\\ref{LEM_D_CONV}\nto hold is satisfied under the specified conditions on $\\ensuremath{n}$ and\n$\\ensuremath{\\lambda_\\ensuremath{n}}$. From Lemma~\\ref{LemWconv} and our choice of regularization\nconstant \\mbox{$\\ensuremath{\\lambda_\\ensuremath{n}} = (8\/\\mutinco)\\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})$,}\n\\begin{eqnarray*}\n2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big( \\,\\|\\ensuremath{W}\\|_\\infty + \\ensuremath{\\lambda_\\ensuremath{n}} \\big) & \\leq & 2\n\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 + \\frac{8}{\\mutinco}\\Big) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}),\n\\end{eqnarray*}\nprovided $\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\le 1\/v_*$.  From\nthe lower bound~\\eqref{EqnSampleBound} and the\nmonotonicity~\\eqref{EqnMonot} of the tail inverse functions, we have\n\\begin{eqnarray}\n\\label{EqnDelPreqBound}\n2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 + \\frac{8}{\\mutinco}\\Big) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) & \\leq & \\min \\big\\{ \\frac{1}{3\n\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}, \\; \\frac{1}{3 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\;\\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\ensuremath{\\ensuremath{d}}}\n\\big\\},\n\\end{eqnarray}\nshowing that the assumptions of Lemma~\\ref{LEM_D_CONV} are satisfied.\nApplying this lemma, we conclude that\n\\begin{eqnarray}\n\\label{EqnDelBound}\n\\| \\Delta \\|_\\infty & \\leq & 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\big( \\, \\|\\ensuremath{W}\\|_\\infty +\n\\ensuremath{\\lambda_\\ensuremath{n}} \\big) \\; \\leq \\; 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 + \\frac{8}{\\mutinco}\\Big)\n\\, \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray}\n\nTurning next to Lemma~\\ref{LEM_R_CONV}, we see that its assumption\n$\\|\\Delta\\|_{\\infty} \\leq \\frac{1}{3 \\, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{\\ensuremath{d}}}$ holds, by\napplying equations~\\eqref{EqnDelPreqBound} and \\eqref{EqnDelBound}.\nConsequently, we have\n\\begin{eqnarray*}\n\\|\\ensuremath{R}(\\Delta)\\|_\\infty & \\leq & \\frac{3}{2} \\,\\ensuremath{\\ensuremath{d}}\\;\n\\|\\Delta\\|_\\infty^2 \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\\\\n& \\leq & 6 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^2 \\, \\ensuremath{\\ensuremath{d}} \\, \\Big(1 +\n\\frac{8}{\\mutinco}\\Big)^2 [\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})]^{2}\n\\\\\n& = & \\Biggr \\{ 6 \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^2 \\, \\ensuremath{\\ensuremath{d}} \\, \\Big(1 +\n\\frac{8}{\\mutinco}\\Big)^2 \\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\Biggr\n\\} \\frac{\\mutinco \\ensuremath{\\lambda_\\ensuremath{n}}}{8} \\\\\n& \\leq & \\frac{\\mutinco \\ensuremath{\\lambda_\\ensuremath{n}}}{8},\n\\end{eqnarray*}\nas required, where the final inequality follows from our\ncondition~\\eqref{EqnSampleBound} on the sample size, and the\nmonotonicity property~\\eqref{EqnMonot}.  \n\nOverall, we have shown that the assumption~\\eqref{EqnDetSuff} of\nLemma~\\ref{LemStrictDual} holds, allowing us to conclude that\n$\\ensuremath{\\ThetaWitness} = \\ensuremath{\\widehat{\\Theta}}$. The estimator $\\ensuremath{\\widehat{\\Theta}}$ then satisfies the\n$\\ell_\\infty$-bound~\\eqref{EqnDelBound} of $\\ensuremath{\\ThetaWitness}$, as claimed in\nTheorem~\\ref{ThmMain}(a), and moreover, we have\n$\\ensuremath{\\widehat{\\Theta}}_{\\ensuremath{\\EsetPlus^c}} = \\ensuremath{\\ThetaWitness}_{\\ensuremath{\\EsetPlus^c}} = 0$, as\nclaimed in Theorem~\\ref{ThmMain}(b).  Since the above was conditioned\non the event $\\mathcal{A}$, these statements hold with probability\n$\\ensuremath{\\mathbb{P}}(\\mathcal{A}) \\ge 1 - \\frac{1}{\\ensuremath{p}^{\\tau - 2}}$.\n\n\\subsection{Proof of Theorem~\\ref{ThmModel}}\n\\label{SecThmModel}\n\nWe now turn to the proof of Theorem~\\ref{ThmModel}. A little\ncalculation shows that the assumed lower bound~\\eqref{EqnNumobsModel}\non the sample size $\\ensuremath{n}$ and the monotonicity\nproperty~\\eqref{EqnMonot} together guarantee that\n\\begin{eqnarray*}\n\\ensuremath{\\theta_{\\operatorname{min}}} & > & 4 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 +\\frac{8}{\\mutinco}\\Big) \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau})\n\\end{eqnarray*}\nProceeding as in the proof of Theorem~\\ref{ThmMain}, with probability\nat least $1 - 1\/\\ensuremath{p}^{\\tau - 2}$, we have the equality\n\\mbox{$\\ensuremath{\\ThetaWitness} = \\ensuremath{\\widehat{\\Theta}}$,} and also that $\\|\\ensuremath{\\ThetaWitness} -\n\\ensuremath{\\Theta^*}\\|_\\infty \\leq \\ensuremath{\\theta_{\\operatorname{min}}}\/2$.  Consequently,\nLemma~\\ref{LemSignConsis} can be applied, guaranteeing that\n$\\textrm{sign}(\\ensuremath{\\Theta^*}_{ij}) = \\textrm{sign}(\\ensuremath{\\ThetaWitness}_{ij})$ for all $(i,j) \\in\n\\ensuremath{E}$.  Overall, we conclude that with probability at least $1 -\n1\/\\ensuremath{p}^{\\tau - 2}$, the sign consistency condition\n$\\textrm{sign}(\\ensuremath{\\Theta^*}_{ij}) = \\textrm{sign}(\\ensuremath{\\widehat{\\Theta}}_{ij})$ holds for all $(i,j)\n\\in \\ensuremath{E}$, as claimed.\n\n\n\n\\def\\hat{\\Delta}} \\def\\Lin{L{\\hat{\\Delta}} \\def\\Lin{L}\n\n\\subsection{Proof of Corollary~\\ref{CorCovBound}}\n\\label{SecCorCovProof}\nWith the shorthand $\\hat{\\Delta}} \\def\\Lin{L = \\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}$, we have\n\\begin{equation*}\n\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*} = (\\ensuremath{\\Theta^*} + \\hat{\\Delta}} \\def\\Lin{L)^{-1} -\n\\inv{\\ensuremath{\\Theta^*}}.\n\\end{equation*}\nFrom the definition~\\eqref{EqnRdefn} of the residual $R(\\cdot)$, this\ndifference can be written as\n\\begin{eqnarray}\n\\label{EqnCovDev}\n\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*} & = & - \\invn{\\ensuremath{\\Theta^*}} \\hat{\\Delta}} \\def\\Lin{L\n\\invn{\\ensuremath{\\Theta^*}} + R(\\hat{\\Delta}} \\def\\Lin{L).\n\\end{eqnarray}\n\nProceeding as in the proof of Theorem~\\ref{ThmMain} we condition on\nthe event $\\mathcal{A} = \\{ \\|\\ensuremath{W}\\|_\\infty \\leq\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}) \\}$, and which holds with\nprobability $\\ensuremath{\\mathbb{P}}(\\mathcal{A}) \\ge 1 - \\frac{1}{\\ensuremath{p}^{\\tau - 2}}$.\nAs in the proof of that theorem, we are guaranteed that the\nassumptions of Lemma~\\ref{LEM_R_CONV} are satisfied, allowing us to\nconclude\n\\begin{align}\\label{EqnRequiv}\n\\ensuremath{R}(\\hat{\\Delta}} \\def\\Lin{L) = \\invn{\\opt{\\Theta}} \\hat{\\Delta}} \\def\\Lin{L \\invn{\\opt{\\Theta}}\n\\hat{\\Delta}} \\def\\Lin{L J \\invn{\\opt{\\Theta}},\n\\end{align}\nwhere $J \\ensuremath{:  =} \\sum_{k=0}^{\\infty} (-1)^{k} \\big(\\invn{\\opt{\\Theta}}\n\\hat{\\Delta}} \\def\\Lin{L\\big)^{k}$ has norm $\\matnorm{J^T}{\\infty} \\leq 3\/2$.\n\nWe begin by proving the bound~\\eqref{EqnCovInftyBound}.  From\nequation~\\eqref{EqnCovDev}, we have $\\vecnorm{\\hat{\\CovHat} -\n\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} \\leq \\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} +\n\\vecnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty}$.  From Lemma~\\ref{LEM_R_CONV}, we have\nthe elementwise $\\ell_\\infty$-norm bound\n\\begin{eqnarray*}\n\\| \\ensuremath{R}(\\hat{\\Delta}} \\def\\Lin{L)\\|_\\infty & \\leq & \\frac{3}{2} \\ensuremath{\\ensuremath{d}}\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_\\infty^2 \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3.\n\\end{eqnarray*}\nThe quantity $L(\\hat{\\Delta}} \\def\\Lin{L)$ in turn can be bounded as follows,\n\\begin{eqnarray*}\n\\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} & = &\n\\max_{ij}\\big|e_{i}^{T}\\invn{\\ensuremath{\\Theta^*}} \\hat{\\Delta}} \\def\\Lin{L \\invn{\\ensuremath{\\Theta^*}}\ne_{j}\\big|\\\\ &\\le& \\max_{i} \\|e_{i}^{T}\\invn{\\ensuremath{\\Theta^*}}\\|_{1}\n\\max_{j} \\|\\hat{\\Delta}} \\def\\Lin{L \\invn{\\ensuremath{\\Theta^*}} e_j\\|_{\\infty}\\\\ &\\le& \\max_{i}\n\\|e_{i}^{T}\\invn{\\ensuremath{\\Theta^*}}\\|_{1} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} \\|\\max_{j}\n\\|\\invn{\\ensuremath{\\Theta^*}} e_j\\|_{1}\n\\end{eqnarray*}\nwhere we used the inequality that $\\|\\hat{\\Delta}} \\def\\Lin{L u \\|_{\\infty} \\le\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} \\|u\\|_{1}$.  Simplifying further, we obtain\n\\begin{eqnarray*}\n\\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} & \\leq &\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty}\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{1}\\\\ \n& \\leq & \\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty}^{2}\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} \\\\ \n& \\leq & \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty},\n\\end{eqnarray*}\nwhere we have used the fact that $\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{1} =\n\\matnorm{[\\invn{\\ensuremath{\\Theta^*}}]^{T}}{\\infty} =\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty}$, which follows from the symmetry\nof $\\invn{\\ensuremath{\\Theta^*}}$.  Combining the pieces, we obtain\n\\begin{eqnarray}\n\\label{EqnCovDevInftya}\n\\vecnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} &\\le &\n\\vecnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} + \\vecnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty}\\\\\n& \\leq & \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\|\\hat{\\Delta}} \\def\\Lin{L\\|_{\\infty} + \\frac{3}{2} \\ensuremath{\\ensuremath{d}}\n\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3 \\|\\hat{\\Delta}} \\def\\Lin{L\\|_\\infty^2. \\nonumber\n\\end{eqnarray}\nThe claim then follows from the elementwise $\\ell_\\infty$-norm\nbound~\\eqref{EqnEllinfBound} from Theorem~\\ref{ThmMain}.\n\n\nNext, we establish the bound~\\eqref{EqnCovSpectralBound} in spectral\nnorm.  Taking the $\\ell_{\\infty}$ operator norm of both sides of\nequation~\\eqref{EqnCovDev} yields the inequality $\\matnorm{\\hat{\\CovHat} -\n\\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} \\leq \\matnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} +\n\\matnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty}$. Using the\nexpansion~\\eqref{EqnRequiv}, and the sub-multiplicativity of the\n$\\ell_{\\infty}$ operator norm, we obtain\n\\begin{eqnarray*}\n\\matnorm{\\ensuremath{R}(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} & \\leq &\n\\matnorm{\\invn{\\opt{\\Theta}}}{\\infty} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}\n\\matnorm{\\invn{\\opt{\\Theta}}}{\\infty} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}\n\\matnorm{J}{\\infty} \\matnorm{\\invn{\\opt{\\Theta}}}{\\infty}\\\\ &\\le&\n\\matnorm{\\invn{\\opt{\\Theta}}}{\\infty}^{3} \\matnorm{J}{\\infty}\n\\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}^{2}\\\\ \n& \\leq & \\frac{3}{2} \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}^{2},\n\\end{eqnarray*}\nwhere the last inequality uses the bound $\\matnorm{J}{\\infty} \\leq\n3\/2$.  (Proceeding as in the proof of Lemma~\\ref{LEM_R_CONV}, this\nbound holds conditioned on $\\mathcal{A}$, and for the sample size\nspecified in the theorem statement.)  In turn, the term $L(\\hat{\\Delta}} \\def\\Lin{L)$\ncan be bounded as\n\\begin{eqnarray*}\n\\matnorm{L(\\Delta)}{\\infty} & \\leq & \\matnorm{\\invn{\\ensuremath{\\Theta^*}}\n\\hat{\\Delta}} \\def\\Lin{L \\invn{\\ensuremath{\\Theta^*}}}{\\infty}\\\\ &\\le&\n\\matnorm{\\invn{\\ensuremath{\\Theta^*}}}{\\infty}^{2} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty}\\\\\n& \\leq & \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty},\n\\end{eqnarray*}\nwhere the second inequality uses the sub-multiplicativity of the\n$\\ell_\\infty$-operator norm.  Combining the pieces yields\n\\begin{eqnarray}\n\\label{EqnCovDevInftyOp}\n\\matnorm{\\hat{\\CovHat} - \\ensuremath{\\ensuremath{\\Sigma}^*}}{\\infty} & \\leq &\n\\matnorm{L(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} + \\matnorm{R(\\hat{\\Delta}} \\def\\Lin{L)}{\\infty} \\;\n\\leq \\; \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{2} \\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty} + \\frac{3}{2} \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^3\n\\|\\hat{\\Delta}} \\def\\Lin{L\\|_\\infty^2.\n\\end{eqnarray}\nConditioned on the event $\\mathcal{A}$, we have the\nbound~\\eqref{EqnPrecInftyOp} on the $\\ell_\\infty$-operator norm\n\\begin{eqnarray*}\n\\matnorm{\\hat{\\Delta}} \\def\\Lin{L}{\\infty} & \\leq & 2 \\ensuremath{K_{\\ensuremath{\\Gamma^*}}} \\Big(1 +\n\\frac{8}{\\mutinco}\\Big) \\, \\ensuremath{\\ensuremath{d}} \\,\n\\ensuremath{\\widebar{\\delta}_f}(\\ensuremath{n},\\ensuremath{p}^{\\tau}).\n\\end{eqnarray*}\nSubstituting this bound, as well as the elementwise $\\ell_\\infty$-norm\nbound~\\eqref{EqnEllinfBound} from Theorem~\\ref{ThmMain}, into the\nbound~\\eqref{EqnCovDevInftyOp} yields the stated claim.\n\n\n\n\\section{Experiments}\n\\label{SecExperiments}\n\nIn this section, we illustrate our results with various experimental\nsimulations, reporting results in terms of the probability of correct\nmodel selection (Theorem~\\ref{ThmModel}) or the $\\ell_\\infty$-error\n(Theorem~\\ref{ThmMain}).  For these illustrations, we study the case\nof Gaussian graphical models, and results for three different classes\nof graphs, namely chains, grids, and star-shaped graphs. We also\nconsider various scalings of the quantities which affect the\nperformance of the estimator: in addition the triple $(\\ensuremath{n}, \\ensuremath{p},\n\\ensuremath{\\ensuremath{d}})$, we also report some results concerning the role of the\nparameters $\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ and $\\ensuremath{\\theta_{\\operatorname{min}}}$ that we have identified in the\nmain theorems. For all results\nreported here, we solved the resulting $\\ell_1$-penalized\nlog-determinant program~\\eqref{EqnGaussMLE} using the \\texttt{glasso}\nprogram of \\citet{FriedHasTib2007}, which builds on the block\nco-ordinate descent algorithm of~\\citet{AspreBanG2008}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\raisebox{.6in}{\\widgraph{0.25\\textwidth}{chain.eps}}& &\n\\raisebox{.0in}{\\widgraph{0.28\\textwidth}{grid4.eps}}& &\n\\raisebox{.1in}{\\widgraph{0.25\\textwidth}{stargraph.eps}}\\\\\n(a) && (b) && (c)\n\\end{tabular}\n\\caption{Illustrations of different graph classes used in simulations.\n(a) Chain ($\\ensuremath{\\ensuremath{d}} = 2$).  (b) Four-nearest neighbor grid ($\\ensuremath{\\ensuremath{d}} =\n4$) and (c) Star-shaped graph ($\\ensuremath{\\ensuremath{d}} \\in \\{1,\\hdots,\\ensuremath{p} - 1\\}$).}\n\\label{FigGraphs}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{FigGraphs} illustrates the three types of graphs used in\nour simulations: chain graphs (panel (a)), four-nearest neighbor\nlattices or grids (panel (b)), and star-shaped graphs (panel (c)).\nFor the chain and grid graphs, the maximal node degree $\\ensuremath{\\ensuremath{d}}$ is\nfixed by definition, to $\\ensuremath{\\ensuremath{d}} =2$ for chains, and $\\ensuremath{\\ensuremath{d}} =4$ for\nthe grids.  Consequently, these graphs can capture the dependence of\nthe required sample size $\\ensuremath{n}$ only as a function of the graph\nsize $\\ensuremath{p}$, and the parameters $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$,\n$\\ensuremath{\\theta_{\\operatorname{min}}}$).  The star graph allows us to vary both $\\ensuremath{\\ensuremath{d}}$ and\n$\\ensuremath{p}$, since the degree of the central hub can be varied between $1$\nand $\\ensuremath{p} -1$.  For each graph type, we varied the size of the graph\n$\\ensuremath{p}$ in different ranges, from $\\ensuremath{p} =64$ upwards to $\\ensuremath{p} =\n375$.\n\n\nFor the chain and star graphs, we define a covariance matrix\n$\\ensuremath{\\ensuremath{\\Sigma}^*}$ with entries $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii} = 1$ for all $i =1,\n\\ldots, \\ensuremath{p}$, and $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = \\rho$ for all $(i,j) \\in\n\\ensuremath{E}$ for specific values of $\\rho$ specified below.  Note that these\ncovariance matrices are sufficient to specify the full model.  For the\nfour-nearest neighbor grid graph, we set the entries of the\nconcentration matrix $\\ensuremath{\\Theta^*}_{ij} = \\omega$ for $(i,j) \\in \\ensuremath{E}$,\nwith the value $\\omega$ specified below. In all cases, we set the\nregularization parameter $\\ensuremath{\\lambda_\\ensuremath{n}}$ proportional to\n$\\sqrt{\\log(\\ensuremath{p})\/\\ensuremath{n}}$, as suggested by Theorems~\\ref{ThmMain}\nand~\\ref{ThmModel}, which is reasonable since the main purpose of\nthese simulations is to illustrate our theoretical results.  However,\nfor general data sets, the relevant theoretical parameters cannot be\ncomputed (since the true model is unknown), so that a data-driven\napproach such as cross-validation might be required for selecting the\nregularization parameter $\\ensuremath{\\lambda_\\ensuremath{n}}$.\n\n\n\nGiven a Gaussian graphical model instance, and the number of samples\n$\\ensuremath{n}$, we drew $N = 100$ batches of $\\ensuremath{n}$ independent samples\nfrom the associated multivariate Gaussian distribution. We estimated\nthe probability of correct model selection as the fraction of the $N=\n100$ trials in which the estimator recovers the signed-edge set\nexactly.\n\n\\defK{K}\nNote that any multivariate Gaussian random vector is sub-Gaussian; in\nparticular, the rescaled variates $X_i\/\\sqrt{\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii}}$ are\nsub-Gaussian with parameter $\\ensuremath{\\sigma} = 1$, so that the elementwise\n$\\ell_\\infty$-bound from Corollary~\\ref{CorEllinfSubg} applies.\nSuppose we collect relevant parameters such as $\\ensuremath{\\theta_{\\operatorname{min}}}$ and the\ncovariance and Hessian-related terms $\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ and\n$\\mutinco$ into a single ``model-complexity'' term $K$ defined\nas\n\\begin{eqnarray}\\label{EqnKdefn}\nK & \\ensuremath{:  =} & \\left[(1 + 8 \\mutinco^{-1}) (\\max_{i}\\ensuremath{\\ensuremath{\\Sigma}^*}_{ii})\n\\max\\{\\ensuremath{K_{\\ensuremath{\\Sigma}^*}} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}}^{3} \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}^{2},\\frac{\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}}{\n\\ensuremath{\\ensuremath{d}}\\, \\ensuremath{\\theta_{\\operatorname{min}}}}\\}\\right].\n\\end{eqnarray}\nThen, as a corollary of Theorem~\\ref{ThmModel}, a sample size of order \n\\begin{eqnarray}\n\\label{EqnCrudeBound}\n\\ensuremath{n} & = & \\Omega\\left( K^{2} \\; \\ensuremath{\\ensuremath{d}}^2 \\, \\tau \\log \\ensuremath{p}\n\\right),\n\\end{eqnarray}\nis sufficient for model selection consistency with probability greater\nthan $1-1\/\\ensuremath{p}^{\\tau-2}$. In the subsections to follow, we\ninvestigate how the empirical sample size $\\ensuremath{n}$ required for model\nselection consistency scales in terms of graph size $\\ensuremath{p}$, maximum\ndegree $\\ensuremath{\\ensuremath{d}}$, as well as the ``model-complexity'' term $K$ defined\nabove.\n \n\\newcommand{.48\\textwidth}{.48\\textwidth}\n\\begin{figure}[h]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\widgraph{.48\\textwidth}{figures\/chainprobvsn.eps} & \\hspace*{.1in} &\n\\widgraph{.48\\textwidth}{figures\/chainprobvsnlogp.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{Simulations for chain graphs with varying number of nodes\n$\\ensuremath{p}$, edge covariances $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 0.10$. Plots of\nprobability of correct signed edge-set recovery plotted versus the\nordinary sample size $\\ensuremath{n}$ in panel (a), and versus the rescaled\nsample size $\\ensuremath{n}\/\\log \\ensuremath{p}$ in panel (b). Each point corresponds\nto the average over $100$ trials. }\n\\label{FigChainProbvsNP}\n\\end{figure}\n\n\n\n\\subsection{Dependence on graph size}\n\nPanel (a) of Figure~\\ref{FigChainProbvsNP} plots the probability of\ncorrect signed edge-set recovery against the sample size $\\ensuremath{n}$ for\na chain-structured graph of three different sizes.  For these chain\ngraphs, regardless of the number of nodes $\\ensuremath{p}$, the maximum node\ndegree is constant $\\ensuremath{\\ensuremath{d}} = 2$, while the edge covariances are set\nas $\\ensuremath{\\Sigma}_{ij} = 0.2$ for all $(i,j) \\in \\ensuremath{E}$, so that the\nquantities $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\mutinco)$ remain constant.  Each of\nthe curve in panel (a) corresponds to a different graph size $\\ensuremath{p}$.\nFor each curve, the probability of success starts at zero (for small\nsample sizes $\\ensuremath{n}$), but then transitions to one as the sample\nsize is increased.  As would be expected, it is more difficult to\nperform model selection for larger graph sizes, so that (for instance)\nthe curve for $\\ensuremath{p} = 375$ is shifted to the right relative to the\ncurve for $\\ensuremath{p} = 64$.  Panel (b) of Figure~\\ref{FigChainProbvsNP}\nreplots the same data, with the horizontal axis rescaled by $(1\/\\log\n\\ensuremath{p})$.  This scaling was chosen because for sub-Gaussian tails, our\ntheory predicts that the sample size should scale logarithmically with\n$\\ensuremath{p}$ (see equation~\\eqref{EqnCrudeBound}).  Consistent with this\nprediction, when plotted against the rescaled sample size\n$\\ensuremath{n}\/\\log \\ensuremath{p}$, the curves in panel (b) all stack up.\nConsequently, the ratio $(\\ensuremath{n}\/\\log\\ensuremath{p})$ acts as an effective\nsample size in controlling the success of model selection, consistent\nwith the predictions of Theorem~\\ref{ThmModel} for sub-Gaussian\nvariables.\n\nFigure~\\ref{FigStarProbvsNP} shows the same types of plots for a\nstar-shaped graph with fixed maximum node degree $\\ensuremath{\\ensuremath{d}} = 40$, and\nFigure~\\ref{FigGridProbvsNP} shows the analogous plots for a grid\ngraph with fixed degree $\\ensuremath{\\ensuremath{d}} = 4$.  As in the chain case, these\nplots show the same type of stacking effect in terms of the scaled\nsample size $\\ensuremath{n}\/\\log \\ensuremath{p}$, when the degree $\\ensuremath{\\ensuremath{d}}$ and other\nparameters ($(\\mutinco, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\ensuremath{K_{\\ensuremath{\\Sigma}^*}})$) are held fixed.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\widgraph{.48\\textwidth}{figures\/starprobvsn.eps} & \\hspace*{.1in} &\n\\widgraph{.48\\textwidth}{figures\/starprobvsnlogp.eps} \\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{Simulations for a star graph with varying number of nodes\n$\\ensuremath{p}$, fixed maximal degree $\\ensuremath{\\ensuremath{d}} = 40$, and edge covariances\n$\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 1\/16$ for all edges. Plots of probability of\ncorrect signed edge-set recovery versus the sample size $\\ensuremath{n}$ in\npanel (a), and versus the rescaled sample size $\\ensuremath{n}\/\\log \\ensuremath{p}$ in\npanel (b).  Each point corresponds to the average over $N = 100$\ntrials.}\n\\label{FigStarProbvsNP}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\widgraph{.48\\textwidth}{figures\/gridprobvsn.eps} &\n\\widgraph{.48\\textwidth}{figures\/gridprobvsnlogp.eps} \\\\\n(a) & (b)\n\\end{tabular}\n\\end{center}\n\\caption{Simulations for $2$-dimensional lattice with\n$4$-nearest-neighbor interaction, edge strength interactions\n$\\ensuremath{\\Theta^*}_{ij} = 0.1$, and a varying number of nodes $\\ensuremath{p}$. Plots\nof probability of correct signed edge-set recovery versus the sample\nsize $\\ensuremath{n}$ in panel (a), and versus the rescaled sample size\n$\\ensuremath{n}\/\\log \\ensuremath{p}$ in panel (b).  Each point corresponds to the\naverage over $N = 100$ trials.}\n\\label{FigGridProbvsNP}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tabular}{cc}\n\\widgraph{.48\\textwidth}{figures\/dvarstarprobvsn.eps} &\n\\widgraph{.48\\textwidth}{figures\/dvarstarprobvsnd.eps} \\\\\n(a) & (b)\n\\end{tabular}\n\\end{center}\n\\caption{ Simulations for star graphs with fixed number of nodes\n$\\ensuremath{p} = 200$, varying maximal (hub) degree $\\ensuremath{\\ensuremath{d}}$, edge\ncovariances $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 2.5\/\\ensuremath{\\ensuremath{d}}$.  Plots of probability of\ncorrect signed edge-set recovery versus the sample size $\\ensuremath{n}$ in\npanel (a), and versus the rescaled sample size $\\ensuremath{n}\/\\ensuremath{\\ensuremath{d}}$ in\npanel (b). }\n\\label{FigStarProbvsND}\n\\end{figure}\n\n\\subsection{Dependence on the maximum node degree} \n\nPanel (a) of Figure~\\ref{FigStarProbvsND} plots the probability of\ncorrect signed edge-set recovery against the sample size $\\ensuremath{n}$ for\nstar-shaped graphs; each curve corresponds to a different choice of\nmaximum node degree $\\ensuremath{\\ensuremath{d}}$, allowing us to investigate the\ndependence of the sample size on this parameter.  So as to control\nthese comparisons, the models are chosen such that quantities other\nthan the maximum node-degree $\\ensuremath{\\ensuremath{d}}$ are fixed: in particular, we\nfix the number of nodes $\\ensuremath{p} = 200$, and the edge covariance entries\nare set as $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 2.5\/\\ensuremath{\\ensuremath{d}}$ for $(i,j) \\in \\ensuremath{E}$ so\nthat the quantities $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\mutinco)$ remain constant.\nThe minimum value $\\ensuremath{\\theta_{\\operatorname{min}}}$ in turn scales as $1\/\\ensuremath{\\ensuremath{d}}$.  Observe\nhow the plots in panel (a) shift to the right as the maximum node\ndegree $\\ensuremath{\\ensuremath{d}}$ is increased, showing that star-shaped graphs with\nhigher degrees are more difficult.  In panel (b) of\nFigure~\\ref{FigStarProbvsND}, we plot the same data versus the\nrescaled sample size $\\ensuremath{n}\/\\ensuremath{\\ensuremath{d}}$.  Recall that if all the curves\nwere to stack up under this rescaling, then it means the required\nsample size $\\ensuremath{n}$ scales linearly with $\\ensuremath{\\ensuremath{d}}$.  These plots are\ncloser to aligning than the unrescaled plots, but the agreement is not\nperfect.  In particular, observe that the curve $\\ensuremath{\\ensuremath{d}}$ (right-most\nin panel (a)) remains a bit to the right in panel (b), which suggests\nthat a somewhat more aggressive rescaling---perhaps\n$\\ensuremath{n}\/\\ensuremath{\\ensuremath{d}}^\\gamma$ for some $\\gamma \\in (1,2)$---is appropriate.\n\nNote that for $\\ensuremath{\\theta_{\\operatorname{min}}}$ scaling as $1\/\\ensuremath{\\ensuremath{d}}$, the sufficient\ncondition from Theorem~\\ref{ThmModel}, as summarized in\nequation~\\eqref{EqnCrudeBound}, is $\\ensuremath{n} = \\Omega(\\ensuremath{\\ensuremath{d}}^2 \\log\n\\ensuremath{p})$, which appears to be overly conservative based on these data.\nThus, it might be possible to tighten our theory under certain\nregimes.\n\n\n\n\\subsection{Dependence on covariance and Hessian terms}\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\nNext, we study the dependence of the sample size required for model\nselection consistency on the model complexity term $K$ defined\nin \\eqref{EqnKdefn}, which is a collection of the quantities\n$\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}$, $\\ensuremath{K_{\\ensuremath{\\Gamma^*}}}$ and $\\mutinco$ defined by the covariance\nmatrix and Hessian, as well as the minimum value\n$\\ensuremath{\\theta_{\\operatorname{min}}}$. Figure~\\ref{FigChainNstarvsGamma} plots the probability\nof correct signed edge-set recovery versus the sample size $\\ensuremath{n}$\nfor chain graphs.  Here each curve corresponds to a different setting\nof the model complexity factor $K$, but with a fixed\nnumber of nodes $\\ensuremath{p} = 120$, and maximum node-degree $\\ensuremath{\\ensuremath{d}} = 2$.\nWe varied the actor $K$ by varying the value $\\rho$ of the\nedge covariances $\\ensuremath{\\Sigma}_{ij} = \\rho,\\, (i,j) \\in \\ensuremath{E}$.  Notice how\nthe curves, each of which corresponds to a different model complexity\nfactor, shift rightwards as $K$ is increased so that\nmodels with larger values of $K$ require greater number of\nsamples $\\ensuremath{n}$ to achieve the same probability of correct model\nselection.  These rightward-shifts are in qualitative agreement with\nthe prediction of Theorem~\\ref{ThmMain}, but we suspect that our\nanalysis is not sharp enough to make accurate quantitative predictions\nregarding this scaling.\n\n\\newcommand{.5\\textwidth}{.5\\textwidth}\n\n\\begin{figure}\n\\begin{center}\n\\widgraph{.5\\textwidth}{figures\/gammavarchainprobvsn.eps} \\\\\n\\end{center}\n\\caption{Simulations for chain graph with fixed number of nodes $\\ensuremath{p}\n= 120$, and varying model complexity $K$. Plot of\nprobability of correct signed edge-set recovery versus the sample size\n$\\ensuremath{n}$.}\n\\label{FigChainNstarvsGamma}\n\\end{figure}\n\n\\subsection{Convergence rates in elementwise $\\ell_\\infty$-norm}\n\nFinally, we report some simulation results on the convergence rate in\nelementwise $\\ell_\\infty$-norm.  According to\nCorollary~\\ref{CorEllinfSubg}, in the case of sub-Gaussian tails, if\nthe elementwise $\\ell_\\infty$-norm should decay at rate\n${\\mathcal{O}}(\\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}})$, once the sample size\n$\\ensuremath{n}$ is sufficiently large.  Figure~\\ref{FigStarNormvsBaserate}\nshows the behavior of the elementwise $\\ell_\\infty$-norm for\nstar-shaped graphs of varying sizes $\\ensuremath{p}$.  The results reported\nhere correspond to the maximum degree $\\ensuremath{\\ensuremath{d}} = \\lceil 0.1 \\ensuremath{p}\n\\rceil$; we also performed analogous experiments for $\\ensuremath{\\ensuremath{d}} =\n{\\mathcal{O}}(\\log \\ensuremath{p})$ and $\\ensuremath{\\ensuremath{d}} = {\\mathcal{O}}(1)$, and observed\nqualitatively similar behavior.  The edge correlations were set as\n$\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij} = 2.5\/\\ensuremath{\\ensuremath{d}}$ for all $(i,j) \\in \\ensuremath{E}$ so that the\nquantities $(\\ensuremath{K_{\\ensuremath{\\Sigma}^*}}, \\ensuremath{K_{\\ensuremath{\\Gamma^*}}}, \\mutinco)$ remain constant.  With\nthese settings, each curve in Figure~\\ref{FigStarNormvsBaserate}\ncorresponds to a different problem size, and plots the elementwise\n$\\ell_\\infty$-error versus the rescaled sample size $\\ensuremath{n}\/\\log\n\\ensuremath{p}$, so that we expect to see curves of the form $f(t) =\n1\/\\sqrt{t}$.  The curves show that when the rescaled sample size\n$(\\ensuremath{n}\/\\log \\ensuremath{p})$ is larger than some threshold (roughly $40$ in\nthe plots shown), the elementwise $\\ell_\\infty$ norm decays at the\nrate $\\sqrt{\\frac{\\log \\ensuremath{p}}{\\ensuremath{n}}}$, which is consistent with\nCorollary~\\ref{CorEllinfSubg}.\n\n\\begin{figure}\n\\begin{center}\n\\widgraph{.5\\textwidth}{figures\/starcvgrate.eps} \\\\\n\\end{center}\n\\caption{Simulations for a star graph with varying number of nodes\n$\\ensuremath{p}$, maximum node degree $\\ensuremath{\\ensuremath{d}} = \\lceil 0.1 \\ensuremath{p} \\rceil$, edge\ncovariances $\\ensuremath{\\ensuremath{\\Sigma}^*}_{ij}= 2.5\/\\ensuremath{\\ensuremath{d}}$. Plot of the element-wise\n$\\ell_\\infty$ norm of the concentration matrix estimate error\n$\\vecnorm{\\ensuremath{\\widehat{\\Theta}} - \\ensuremath{\\Theta^*}}{\\infty}$ versus the rescaled sample\nsize $\\ensuremath{n}\/\\log (\\ensuremath{p})$.}\n\\label{FigStarNormvsBaserate}\n\\end{figure}\n\n\\section{Discussion}\n\nThe focus of this paper is the analysis of the high-dimensional\nscaling of the $\\ell_1$-regularized log determinant\nproblem~\\eqref{EqnGaussMLE} as an estimator of the concentration\nmatrix of a random vector.  Our main contributions were to derive\nsufficient conditions for its model selection consistency as well as\nconvergence rates in both elementwise $\\ell_\\infty$-norm, as well as\nFrobenius and spectral norms.  Our results allow for a range of tail\nbehavior, ranging from the exponential-type decay provided by Gaussian\nrandom vectors (and sub-Gaussian more generally), to polynomial-type\ndecay guaranteed by moment conditions.  In the Gaussian case, our\nresults have natural interpretations in terms of Gaussian Markov\nrandom fields.\n\nOur main results relate the i.i.d. sample size $\\ensuremath{n}$ to various\nparameters of the problem required to achieve consistency.  In\naddition to the dependence on matrix size $\\ensuremath{p}$, number of edges\n$\\ensuremath{s}$ and graph degree $\\ensuremath{\\ensuremath{d}}$, our analysis also illustrates\nthe role of other quantities, related to the structure of the\ncovariance matrix $\\ensuremath{\\ensuremath{\\Sigma}^*}$ and the Hessian of the objective\nfunction, that have an influence on consistency rates.  Our main\nassumption is an irrepresentability or mutual incoherence condition,\nsimilar to that required for model selection consistency of the Lasso,\nbut involving the Hessian of the log-determinant objective\nfunction~\\eqref{EqnGaussMLE}, evaluated at the true model. When the\ndistribution of $X$ is multivariate Gaussian, this Hessian is the\nFisher information matrix of the model, and thus can be viewed as an\nedge-based counterpart to the usual node-based covariance matrix We\nreport some examples where irrepresentability condition for the Lasso\nhold and the log-determinant condition fails, but we do not know in\ngeneral if one requirement dominates the other.  In addition to these\ntheoretical results, we provided a number of simulation studies\nshowing how the sample size required for consistency scales with\nproblem size, node degrees, and the other complexity parameters\nidentified in our analysis.\n\nThere are various interesting questions and possible extensions to\nthis paper.  First, in the current paper, we have only derived\nsufficient conditions for model selection consistency. As in past work\non the Lasso~\\cite{Wainwright2006_new}, it would also be interesting\nto derive a \\emph{converse result}---namely, to prove that if the\nsample size $\\ensuremath{n}$ is smaller than some function of $(\\ensuremath{p},\n\\ensuremath{\\ensuremath{d}}, \\ensuremath{s})$ and other complexity parameters, then regardless\nof the choice of regularization constant, the log-determinant method\nfails to recover the correct graph structure.  Second, while this\npaper studies the problem of estimating a fixed graph or concentration\nmatrix, a natural extension would allow the graph to vary over time, a\nproblem setting which includes the case where the observations are\ndependent. For instance, \\citet{ZhoLafWas08} study the estimation of\nthe covariance matrix of a Gaussian distribution in a time-varying\nsetting, and it would be interesting to extend results of this paper\nto this more general setting.\n\n\n\n\\subsection*{Acknowledgements}  We thank Shuheng Zhou for helpful comments\non an earlier draft of this work.  Work was partially supported by NSF\ngrant DMS-0605165.  Yu also acknowledges support from ARO\nW911NF-05-1-0104, NSFC-60628102, and a grant from MSRA.\n\n\n\n\\subsection*{#1} \\end{center}}\n\n\\newcommand{\\ensuremath{\\mathbb{P}}}{\\ensuremath{\\mathbb{P}}}\n\n\n\n\\newcommand{\\ensuremath{\\mathcal{Q}}}{\\ensuremath{\\mathcal{Q}}}\n\n\\newcommand{\\tracer}[2]{\\ensuremath{\\langle \\!\\langle {#1}, \\; {#2}\n\\rangle \\!\\rangle}}\n\n\\newcommand{\\Symcone}[1]{\\ensuremath{\\mathcal{S}^{#1}}}\n\n\\newcommand{\\pvert}[2]{\\ensuremath{#1_{#2}}}\n\n\\newcommand{\\ensuremath{{\\boldsymbol{\\nu}}}}{\\ensuremath{{\\boldsymbol{\\nu}}}}\n\\newcommand{\\ensuremath{\\mathcal{M}}}{\\ensuremath{\\mathcal{M}}}\n\n\n\n\n\\newcommand{\\ensuremath{\\frac{1}{2}}}{\\ensuremath{\\frac{1}{2}}}\n\\newcommand{\\ensuremath{\\operatorname{ext}}}{\\ensuremath{\\operatorname{ext}}}\n\\newcommand{\\ensuremath{\\operatorname{co}}}{\\ensuremath{\\operatorname{co}}}\n\n\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\\newcommand{\\ensuremath{\\ensuremath{\\Partc^*}_{app}}}{\\ensuremath{\\ensuremath{\\Partc^*}_{app}}}\n\\newcommand{\\ensuremath{B}}{\\ensuremath{B}}\n\\newcommand{\\ensuremath{\\Partapp^*}}{\\ensuremath{\\ensuremath{B}^*}}\n\\newcommand{\\ensuremath{APP}}{\\ensuremath{APP}}\n\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\n\\newcommand{\\ensuremath{A}}{\\ensuremath{A}}\n\\newcommand{\\ensuremath{B}}{\\ensuremath{B}}\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\n\\newcommand{\\ensuremath{{\\boldsymbol{\\ensuremath{\\theta}}}}}{\\ensuremath{{\\boldsymbol{\\ensuremath{\\theta}}}}}\n\\newcommand{\\ensuremath{{\\boldsymbol{\\ensuremath{\\theta}}}^*}}{\\ensuremath{{\\boldsymbol{\\ensuremath{\\theta}}}^*}}\n\n\\newcommand{\\myproj}[2]{\\ensuremath{{#1}({#2})}}\n\n\\newcommand{\\ensuremath{\\rho}}{\\ensuremath{\\rho}}\n\\newcommand{\\ensuremath{{\\boldsymbol{\\rho}}}}{\\ensuremath{{\\boldsymbol{\\rho}}}}\n\\newcommand{\\ensuremath{{\\boldsymbol{\\rho_e}}}}{\\ensuremath{{\\boldsymbol{\\rho_e}}}}\n\n\\newcommand{\\ensuremath{\\nu}}{\\ensuremath{\\nu}}\n\\newcommand{\\ensuremath{\\nu}}{\\ensuremath{\\nu}}\n\n\\newcommand{\\support}[2]{\\ensuremath{\\delta^*(#1 \\, | \\, #2)}}\n\\newcommand{\\indicate}[2]{\\ensuremath{\\delta(#1 \\, | \\, #2)}}\n\n\\newcommand{\\ensuremath{\\delta^*}}{\\ensuremath{\\delta^*}}\n\\newcommand{\\ensuremath{\\delta}}{\\ensuremath{\\delta}}\n\n\n\\newcommand{\\ensuremath{x}}{\\ensuremath{x}}\n\\newcommand{\\ensuremath{x^*}}{\\ensuremath{x^*}}\n\n\n\\newcommand{\\ensuremath{y}}{\\ensuremath{y}}\n\\newcommand{\\ensuremath{y^*}}{\\ensuremath{y^*}}\n\n\\newcommand{\\ensuremath{f}}{\\ensuremath{f}}\n\\newcommand{\\ensuremath{f^*}}{\\ensuremath{f^*}}\n\n\n\n\n\n\\newcommand{\\ensuremath{\\ensuremath{{\\mathbb{R}}}^n}}{\\ensuremath{\\ensuremath{{\\mathbb{R}}}^n}}\n\\newcommand{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_+^n}}{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_+^n}}\n\\newcommand{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_{++}^n}}{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_{++}^n}}\n\n\n\\newcommand{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_+}}{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_+}}\n\\newcommand{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_{++}}}{\\ensuremath{\\ensuremath{{\\mathbb{R}}}_{++}}}\n\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\\newcommand{\\ensuremath{\\operatorname{aff}}}{\\ensuremath{\\operatorname{aff}}}\n\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\\newcommand{\\ensuremath{\\mu}}{\\ensuremath{\\mu}}\n\n\n\\newcommand{\\ensuremath{K}}{\\ensuremath{K}}\n\\newcommand{\\ensuremath{K^*}}{\\ensuremath{K^*}}\n\n\\newcommand{\\ensuremath{X}}{\\ensuremath{X}}\n\n\\newcommand{\\ensuremath{c}}{\\ensuremath{c}}\n\n\\newcommand{\\ensuremath{\\preceq_{\\cone}}}{\\ensuremath{\\preceq_{\\ensuremath{K}}}}\n\\newcommand{\\ensuremath{\\prec_{\\cone}}}{\\ensuremath{\\prec_{\\ensuremath{K}}}}\n\\newcommand{\\ensuremath{\\succeq_{\\cone}}}{\\ensuremath{\\succeq_{\\ensuremath{K}}}}\n\\newcommand{\\ensuremath{\\succ_{\\cone}}}{\\ensuremath{\\succ_{\\ensuremath{K}}}}\n\n\\newcommand{\\ensuremath{\\gamma}}{\\ensuremath{\\gamma}}\n\n\n\\newcommand{\\ensuremath{m}}{\\ensuremath{m}}\n\\newcommand{\\ensuremath{\\mathcal{P}}}{\\ensuremath{\\mathcal{P}}}\n\\newcommand{\\mathcal{M}}{\\mathcal{M}}\n\\newcommand{\\mathcal{A}}{\\mathcal{A}}\n\\newcommand{\\mathcal{B}}{\\mathcal{B}}\n\\newcommand{\\mathcal{F}}{\\mathcal{F}}\n\n\n\n\n\\newcommand{\\op}[1]{\\ensuremath{\\operatorname{#1}}}\n\n\\newcommand{\\ensuremath{\\operatorname{SDEF}}}{\\ensuremath{\\operatorname{SDEF}}}\n\n\\newcommand{\\ensuremath{M}}{\\ensuremath{M}}\n\n\\newcommand{\\ensuremath{n}}{\\ensuremath{n}}\n\n\\newcommand{\\ensuremath{\\operatorname{dom}}}{\\ensuremath{\\operatorname{dom}}}\n\n\\newcommand{\\ensuremath{\\Lambda}}{\\ensuremath{\\Lambda}}\n\n\n\\newcommand{\\ensuremath{\\ensuremath{{\\scr{X}}}^{\\spnodenum}}}{\\ensuremath{\\ensuremath{{\\scr{X}}}^{\\ensuremath{n}}}}\n\n\n\\newcommand{\\ensuremath{\\operatorname{LOCAL}}}{\\ensuremath{\\operatorname{LOCAL}}}\n\\newcommand{\\ensuremath{\\operatorname{LOCAL}}}{\\ensuremath{\\operatorname{LOCAL}}}\n\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\n\n\\newcommand{\\ensuremath{\\tau}}{\\ensuremath{\\tau}}\n\\newcommand{\\ensuremath{\\tau}}{\\ensuremath{\\tau}}\n\n\\newcommand{\\ensuremath{\\gamma^*}}{\\ensuremath{\\gamma^*}}\n\\newcommand{\\ensuremath{\\ell^*}}{\\ensuremath{\\ell^*}}\n\n\\newcommand{\\operatorname{maximize}}{\\operatorname{maximize}}\n\\newcommand{\\operatorname{minimize}}{\\operatorname{minimize}}\n\\newcommand{\\operatorname{subject} \\; \\operatorname{to}}{\\operatorname{subject} \\; \\operatorname{to}}\n\n\\newcommand{\\ensuremath{B}}{\\ensuremath{B}}\n\\newcommand{\\ensuremath{\\Partcapp^*}}{\\ensuremath{\\ensuremath{B}^*}}\n\n\\newcommand{\\ensuremath{\\wtil{\\ensuremath{A}}}}{\\ensuremath{\\wtil{\\ensuremath{A}}}}\n\\newcommand{\\ensuremath{\\wtil{\\ensuremath{A}}^*}}{\\ensuremath{\\wtil{\\ensuremath{A}}^*}}\n\n\\newcommand{\\ensuremath{\\mu}}{\\ensuremath{\\mu}}\n\\newcommand{\\ensuremath{\\mu}}{\\ensuremath{\\mu}}\n\\newcommand{\\ensuremath{\\wtil{\\meanpar}}}{\\ensuremath{\\wtil{\\ensuremath{\\mu}}}}\n\\newcommand{\\ensuremath{\\meanpar^{\\operatorname{MF}}}}{\\ensuremath{\\ensuremath{\\mu}^{\\operatorname{MF}}}}\n\n\n\n\n\\newcommand{\\lp}[2]{\\ensuremath{\\|#2\\|_{#1}}}\n\n\\newcommand{\\ensuremath{\\wedge}}{\\ensuremath{\\wedge}}\n\\newcommand{\\ensuremath{\\vee}}{\\ensuremath{\\vee}}\n\n\\newcommand{\\ensuremath{\\operatorname{OUT}}}{\\ensuremath{\\operatorname{OUT}}}\n\\newcommand{\\ensuremath{\\tau}}{\\ensuremath{\\tau}}\n\n\n\n\\makeatletter\n\\long\\def\\@makecaption#1#2{\n        \\vskip 0.8ex\n        \\setbox\\@tempboxa\\hbox{\\small {\\bf #1:} #2}\n        \\parindent 1.5em \n        \\dimen0=\\hsize\n        \\advance\\dimen0 by -3em\n        \\ifdim \\wd\\@tempboxa >\\dimen0\n                \\hbox to \\hsize{\n                        \\parindent 0em\n                        \\hfil \n                        \\parbox{\\dimen0}{\\def\\baselinestretch{0.96}\\small\n                                {\\bf #1.} #2\n                               \n                                } \n                        \\hfil}\n        \\else \\hbox to \\hsize{\\hfil \\box\\@tempboxa \\hfil}\n        \\fi\n        }\n\\makeatother\n\n\n\n\\long\\def\\comment#1{}\n\n\\makeatletter\n\\def\\@cite#1#2{[\\if@tempswa #2 \\fi #1]}\n\\makeatother\n\n\n\\makeatletter\n\\long\\def\\barenote#1{\n    \\insert\\footins{\\footnotesize\n    \\interlinepenalty\\interfootnotelinepenalty \n    \\splittopskip\\footnotesep\n    \\splitmaxdepth \\dp\\strutbox \\floatingpenalty \\@MM\n    \\hsize\\columnwidth \\@parboxrestore\n    {\\rule{z@}{\\footnotesep}\\ignorespaces\n             \n      #1\\strut}}}\n\\makeatother\n\n\n\\newcommand{\\ensuremath{\\operatorname{TRACT}}}{\\ensuremath{\\operatorname{TRACT}}}\n\\newcommand{\\ensuremath{\\operatorname{FACT}}}{\\ensuremath{\\operatorname{FACT}}}\n\\newcommand{\\ensuremath{{\\mathcal{M}}_e}}{\\ensuremath{{\\mathcal{M}}_e}}\n\n\\newcommand{\\ensuremath{\\ensuremath{H}_{0}}}{\\ensuremath{\\ensuremath{H}_{0}}}\n\n\n\\newcommand{\\ensuremath{\\ensuremath{\\Phi}^*}}{\\ensuremath{\\ensuremath{\\Phi}^*}}\n\n\\newcommand{\\ensuremath{A}}{\\ensuremath{A}}\n\\newcommand{\\ensuremath{\\Partc^*}}{\\ensuremath{\\ensuremath{A}^*}}\n\n\\newcommand{\\ensuremath{\\operatorname{ri}}}{\\ensuremath{\\operatorname{ri}}}\n\n\\newcommand{\\ensuremath{\\operatorname{int}}}{\\ensuremath{\\operatorname{int}}}\n\\newcommand{\\ensuremath{\\operatorname{int}}}{\\ensuremath{\\operatorname{int}}}\n\n\\newcommand{\\ensuremath{\\mathbb{I\\,}}}{\\ensuremath{\\mathbb{I\\,}}}\n\n\\newcommand{\\ensuremath{\\mathcal{M}}}{\\ensuremath{\\mathcal{M}}}\n\n\n\\newcommand{\\ensuremath{\\operatorname{HYPER}}}{\\ensuremath{\\operatorname{HYPER}}}\n\n\n\\newcommand{\\ensuremath{\\begin{bmatrix} 0.4 & 0.1 \\\\ 0.1 & 0.4 \\end{bmatrix}}}{\\ensuremath{\\begin{bmatrix} 0.4 & 0.1 \\\\ 0.1 & 0.4 \\end{bmatrix}}}\n\n\n\\newcommand{\\ensuremath{\\begin{bmatrix} 0.1 & 0.4 \\\\ 0.4 & 0.1 \\end{bmatrix}}}{\\ensuremath{\\begin{bmatrix} 0.1 & 0.4 \\\\ 0.4 & 0.1 \\end{bmatrix}}}\n\n\\newcommand{\\ensuremath{\\begin{bmatrix} 0.5 \\\\ 0.5 \\end{bmatrix}}}{\\ensuremath{\\begin{bmatrix} 0.5 \\\\ 0.5 \\end{bmatrix}}}\n\n\n\n\n\n\\newcommand{\\ensuremath{{\\mathcal{G}}}}{\\ensuremath{{\\mathcal{G}}}}\n\n\\newcommand{\\ensuremath{{\\mathcal{D}}}}{\\ensuremath{{\\mathcal{D}}}}\n\n\\newcommand{\\ensuremath{\\ensuremath{\\theta}^{BP}}}{\\ensuremath{\\ensuremath{\\theta}^{BP}}}\n\\newcommand{\\ensuremath{\\ensuremath{\\theta}^{ML}}}{\\ensuremath{\\ensuremath{\\theta}^{ML}}}\n\\newcommand{\\ensuremath{\\ensuremath{\\theta}^{RE}}}{\\ensuremath{\\ensuremath{\\theta}^{RE}}}\n\n\\newcommand{\\ensuremath{n}}{\\ensuremath{n}}\n\\newcommand{\\ensuremath{g}}{\\ensuremath{g}}\n\\newcommand{\\ensuremath{l}}{\\ensuremath{l}}\n\n\n\\newcommand{\\ensuremath{L}}{\\ensuremath{L}}\n\\newcommand{\\ensuremath{\\wtil{\\Loglike}}}{\\ensuremath{\\wtil{\\ensuremath{L}}}}\n\n\n\n\\newcommand{\\ensuremath{\\mathcal{A}}}{\\ensuremath{{\\mathcal{C}_e}}}\n\\newcommand{\\ensuremath{\\mathcal{B}}}{\\ensuremath{{\\mathcal{C}_m}}}\n\n\n\\newcommand{\\ensuremath{\\varphi}}{\\ensuremath{\\varphi}}\n\\newcommand{\\ensuremath{\\rho}}{\\ensuremath{\\rho}}\n\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\\newcommand{\\ensuremath{\\estim{\\ensuremath{\\theta}}}}{\\ensuremath{\\estim{\\ensuremath{\\theta}}}}\n\\newcommand{\\ensuremath{\\estim{\\ensuremath{\\Theta}}}}{\\ensuremath{\\estim{\\ensuremath{\\Theta}}}}\n\n\\newcommand{\\ensuremath{\\operatorname{COR}}}{\\ensuremath{\\operatorname{COR}}}\n\n\\newcommand{tree-reweighted max-sum$\\;$}{tree-reweighted max-sum$\\;$}\n\\newcommand{tree-reweighted max-sum}{tree-reweighted max-sum}\n\n\\newcommand{tree-reweighted max-product$\\;$}{tree-reweighted max-product$\\;$}\n\\newcommand{tree-reweighted max-product}{tree-reweighted max-product}\n\n\n\n\\newcommand{\\ensuremath{\\wtil{\\ensuremath{P}}}}{\\ensuremath{\\wtil{\\ensuremath{P}}}}\n\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\\newcommand{\\ensuremath{\\mathbf{\\trmaxmarg}}}{\\ensuremath{\\mathbf{\\ensuremath{T}}}}\n\n\\newcommand{\\ensuremath{Q}}{\\ensuremath{Q}}\n\\newcommand{\\ensuremath{\\mathbf{\\newtmarg}}}{\\ensuremath{\\mathbf{\\ensuremath{Q}}}}\n\n\\newcommand{\\ensuremath{\\trmaxmarg^*}}{\\ensuremath{\\ensuremath{T}^*}}\n\\newcommand{\\ensuremath{\\trmaxmargb^*}}{\\ensuremath{\\ensuremath{\\mathbf{\\trmaxmarg}}^*}}\n\n\\newcommand{\\ensuremath{{\\mathcal{H}}}}{\\ensuremath{{\\mathcal{H}}}}\n\n\n\n\\newcommand{\\ensuremath{\\operatorname{MARG}}}{\\ensuremath{\\operatorname{MARG}}}\n\\newcommand{\\ensuremath{\\operatorname{TREE}}}{\\ensuremath{\\operatorname{TREE}}}\n\\newcommand{\\ensuremath{v}}{\\ensuremath{v}}\n\\newcommand{\\ensuremath{u}}{\\ensuremath{u}}\n\\newcommand{\\ensuremath{{\\bf{v}}}}{\\ensuremath{{\\bf{v}}}}\n\\newcommand{\\ensuremath{{\\bf{u}}}}{\\ensuremath{{\\bf{u}}}}\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\\newcommand{\\ensuremath{\\mathbf{\\cbit}}}{\\ensuremath{\\mathbf{\\ensuremath{C}}}}\n\\newcommand{\\ensuremath{p}}{\\ensuremath{p}}\n\n\n\\newcommand{\\ensuremath{\\operatorname{OPT}}}{\\ensuremath{\\operatorname{OPT}}}\n\n\n\\newcommand{0.1}{0.1}\n\\newcommand{0.9}{0.9}\n\n\\newcommand{0.1}{0.1}\n\\newcommand{0.9}{0.9}\n\n\\newcommand{0.4}{0.4}\n\\newcommand{0.6}{0.6}\n\n\\newcommand{\\ensuremath{\\epsilon}}{\\ensuremath{\\epsilon}}\n\n\n\\newcommand{\\begin{itemize}}{\\begin{itemize}}\n\\newcommand{\\end{itemize}}{\\end{itemize}}\n\\newcommand{\\begin{enumerate}}{\\begin{enumerate}}\n\\newcommand{\\end{enumerate}}{\\end{enumerate}}\n\n\\newcommand{\\begin{eqnarray}}{\\begin{eqnarray}}\n\\newcommand{\\end{eqnarray}}{\\end{eqnarray}}\n\n\\newcommand{\\hspace*{.1in}}{\\hspace*{.1in}}\n\\newcommand{\\vspace*{.1in}}{\\vspace*{.1in}}\n\\newcommand{\\vspace*{.2in}}{\\vspace*{.2in}}\n\\newcommand{\\hspace*{.2in}}{\\hspace*{.2in}}\n\n\\newcommand{\\footnotesize}{\\footnotesize}\n\n\n\\newcommand{\\myspec}[2]{\\ensuremath{\\frac{\\ensuremath{T}^*_{#1 #2}}{\\ensuremath{T}^*_{#1} \\, \\ensuremath{T}^*_{#2}}}}\n\\newcommand{\\mysing}[1]{\\ensuremath{\\ensuremath{T}^*_{#1}}}\n\n\n\\newcommand{\\myspectwo}[2]{\\ensuremath{\\ensuremath{\\psi}_{#1 #2}}}\n\n\n\\newcommand{\\ensuremath{\\estim{P}}}{\\ensuremath{\\estim{P}}}\n\\newcommand{\\ensuremath{{\\bf{B}}}}{\\ensuremath{{\\bf{B}}}}\n\\newcommand{\\ensuremath{{\\bf{F}}}}{\\ensuremath{{\\bf{F}}}}\n\\newcommand{embedded trees }{embedded trees }\n\\newcommand{ET }{ET }\n\\newcommand{ET}{ET}\n\\newcommand{\\ensuremath{l}}{\\ensuremath{l}}\n\\newcommand{\\ensuremath{L_x}}{\\ensuremath{L_x}}\n\\newcommand{\\ensuremath{\\wtil{\\ensuremath{{\\mathbf{e}}}}}}{\\ensuremath{\\wtil{\\ensuremath{{\\mathbf{e}}}}}}\n\n\\newcommand{\\ensuremath{i}}{\\ensuremath{i}}\n\\newcommand{\\ensuremath{I}}{\\ensuremath{I}}\n\\newcommand{\\ensuremath{\\omega}}{\\ensuremath{\\omega}}\n\n\\newcommand{\\ensuremath{r}}{\\ensuremath{r}}\n\n\\newcommand{\\ensuremath{{H}}}{\\ensuremath{{H}}}\n\\newcommand{\\ensuremath{H}}{\\ensuremath{H}}\n\\newcommand{\\ensuremath{\\mathbf{0}}}{\\ensuremath{\\mathbf{0}}}\n\\newcommand{\\ensuremath{\\bar{\\gamma}^0}}{\\ensuremath{\\bar{\\gamma}^0}}\n\\newcommand{\\ensuremath{r}}{\\ensuremath{r}}\n\\newcommand{\\qform}{\\ensuremath{A}} \n\\newcommand{\\qformm}{\\ensuremath{a}} \n\\newcommand{pdf$\\;\\,$}{pdf$\\;\\,$}\n\\newcommand{compatibility function$\\;$}{compatibility function$\\;$}\n\\newcommand{compatibility functions$\\;$}{compatibility functions$\\;$}\n\n\n\\newcommand{\\ensuremath{F}}{\\ensuremath{F}}\n\\newcommand{\\ensuremath{\\bar{\\gamma}}}{\\ensuremath{\\bar{\\gamma}}}\n\\newcommand{\\ensuremath{\\mathcal{U}}}{\\ensuremath{\\mathcal{U}}}\n\\newcommand{\\ensuremath{\\bar{\\eta}}}{\\ensuremath{\\bar{\\eta}}}\n\\newcommand{\\ensuremath{\\mathbf{\\bar{\\eta}}}}{\\ensuremath{\\mathbf{\\bar{\\eta}}}}\n\n\\newcommand{\\ensuremath{\\underline{\\Lambda}}}{\\ensuremath{\\underline{\\Lambda}}}\n\\newcommand{\\ensuremath{\\underline{{\\mathcal{R}}}}}{\\ensuremath{\\underline{{\\mathcal{R}}}}}\n\n\\newcommand{\\ensuremath{\\delta(x_s = j)}}{\\ensuremath{\\delta(x_s = j)}}\n\\newcommand{\\dellf}[2]{\\ensuremath{\\delta(x_{#1}={#2})}}\n\n\n\\newcommand{reparameterization }{reparameterization }\n\\newcommand{Reparameterization }{Reparameterization }\n\n\n\\newcommand{\\ensuremath{K}}{\\ensuremath{K}}\n\n\n\n\n\n\n\\newcommand{\\tilt}[2]{\\ensuremath{{#1}_{#2}}}\n\\newcommand{\\Partpl}[1]{\\ensuremath{\\ensuremath{\\Phi}_{#1}}}\n\n\\newcommand{\\begin{flushright} $\\Box$ \\end{flushright}}{\\begin{flushright} $\\Box$ \\end{flushright}}\n\\newcommand{\\ensuremath{\\mathcal{S}}}{\\ensuremath{\\mathcal{S}}}\n\\newcommand{\\ensuremath{c}}{\\ensuremath{c}}\n\n\\newcommand{\\ensuremath{L}}{\\ensuremath{L}}\n\n\\newcommand{\\ensuremath{a}}{\\ensuremath{a}}\n\\newcommand{\\ensuremath{b}}{\\ensuremath{b}}\n\\newcommand{\\ensuremath{c}}{\\ensuremath{c}}\n\n\\newcommand{\\ensuremath{\\mathcal{U}}}{\\ensuremath{\\mathcal{U}}}\n\\newcommand{\\ensuremath{S}}{\\ensuremath{S}}\n\n\\newcommand{\\ensuremath{\\operatorname{supp}}}{\\ensuremath{\\operatorname{supp}}}\n\n\n\\newcommand{\\ensuremath{{\\mathbf{e}}}}{\\ensuremath{{\\mathbf{e}}}}\n\\newcommand{\\ensuremath{{\\mathbf{t}}}}{\\ensuremath{{\\mathbf{t}}}}\n\\newcommand{\\ensuremath{{{\\mathbf{a}}}}}{\\ensuremath{{{\\mathbf{a}}}}}\n\\newcommand{\\ensuremath{{{\\mathbf{b}}}}}{\\ensuremath{{{\\mathbf{b}}}}}\n\\newcommand{\\ensuremath{{{\\mathbf{c}}}}}{\\ensuremath{{{\\mathbf{c}}}}}\n\\newcommand{\\ensuremath{{\\boldsymbol{\\ensuremath{\\phi}}}}}{\\ensuremath{{\\boldsymbol{\\ensuremath{\\phi}}}}}\n\\newcommand{\\ensuremath{{\\boldsymbol{\\ensuremath{\\psi}}}}}{\\ensuremath{{\\boldsymbol{\\ensuremath{\\psi}}}}}\n\n\\newcommand{\\ensuremath{\\ensuremath{\\Phi}}}{\\ensuremath{\\ensuremath{\\Phi}}}\n\\newcommand{\\ensuremath{\\mathcal{P}}}{\\ensuremath{\\mathcal{P}}}\n\n\\newcommand{\\ensuremath{I}}{\\ensuremath{I}}\n\\newcommand{\\ensuremath{H}}{\\ensuremath{H}}\n\n\\newcommand{\\ensuremath{\\mathcal{F}}}{\\ensuremath{\\mathcal{F}}}\n\\newcommand{\\ensuremath{\\mathcal{H}}}{\\ensuremath{\\mathcal{H}}}\n\\newcommand{\\ensuremath{\\mathbb{T}}}{\\ensuremath{\\mathbb{T}}}\n\\newcommand{\\ensuremath{\\mathbb{F}}}{\\ensuremath{\\mathbb{F}}}\n\n\\newcommand{\\ensuremath{\\mathbb{T}_+}}{\\ensuremath{\\mathbb{T}_+}}\n\\newcommand{\\ensuremath{\\mathcal{L}}}{\\ensuremath{\\mathcal{L}}}\n\\newcommand{\\ensuremath{\\mathcal{F}}}{\\ensuremath{\\mathcal{F}}}\n\n\\newcommand{\\ensuremath{\\mathbb{T}}}{\\ensuremath{\\mathbb{T}}}\n\n\\newcommand{\\ensuremath{t}}{\\ensuremath{t}}\n\n\\newcommand{\\ensuremath{\\mathbb{L}}}{\\ensuremath{\\mathbb{L}}}\n\\newcommand{\\ensuremath{\\mathbb{M}}}{\\ensuremath{\\mathbb{M}}}\n\n\n\\newcommand{\\ensuremath{\\mathbb{M}}}{\\ensuremath{\\mathbb{M}}}\n\\newcommand{\\ensuremath{\\operatorname{OPT}}}{\\ensuremath{\\operatorname{OPT}}}\n\\newcommand{\\ensuremath{\\operatorname{conv}}}{\\ensuremath{\\operatorname{conv}}}\n\n\\newcommand{\\ensuremath{F}}{\\ensuremath{F}}\n\\newcommand{\\ensuremath{J}}{\\ensuremath{J}}\n\n\\newcommand{\\ensuremath{e}}{\\ensuremath{e}}\n\n\\newcommand{\\ensuremath{G_{\\operatorname{HYP}}}}{\\ensuremath{G_{\\operatorname{HYP}}}}\n\\newcommand{\\ensuremath{\\mathcal{H}}}{\\ensuremath{\\mathcal{H}}}\n\\newcommand{\\ensuremath{h}}{\\ensuremath{h}}\n\\newcommand{\\ensuremath{\\ensuremath{E}}}{\\ensuremath{\\ensuremath{E}}}\n\\newcommand{\\ensuremath{\\mathcal{T}}}{\\ensuremath{\\mathcal{T}}}\n\\newcommand{\\ensuremath{g}}{\\ensuremath{g}}\n\\newcommand{\\ensuremath{f}}{\\ensuremath{f}}\n\n\\newcommand{\\ensuremath{\\varphi}}{\\ensuremath{\\varphi}}\n\n\n\\newcommand{\\ensuremath{{\\mathcal{T}}}}{\\ensuremath{{\\mathcal{T}}}}\n\\newcommand{\\mathfrak{T}}{\\mathfrak{T}}\n\\newcommand{\\ensuremath{{\\boldsymbol{{\\vec{\\ensuremath{\\mu}}}}}}}{\\ensuremath{{\\boldsymbol{{\\vec{\\ensuremath{\\mu}}}}}}}\n\\newcommand{\\ensuremath{\\dweivec}}{\\ensuremath{\\ensuremath{{\\boldsymbol{{\\vec{\\ensuremath{\\mu}}}}}}}}\n\n\n\\newcommand{\\ensuremath{\\beta}}{\\ensuremath{\\beta}}\n\n\\newcommand{\\ensuremath{\\boldsymbol{\\ensuremath{\\mu}_e}}}{\\ensuremath{\\boldsymbol{\\ensuremath{\\mu}_e}}}\n\\newcommand{\\ensuremath{\\boldsymbol{\\ensuremath{\\mu}_\\hyperedge}}}{\\ensuremath{\\boldsymbol{\\ensuremath{\\mu}_\\ensuremath{h}}}}\n\\newcommand{\\ensuremath{\\dwvec}}{\\ensuremath{\\ensuremath{\\boldsymbol{\\ensuremath{\\mu}_e}}}}\n\n\n\\newcommand{\\ensuremath{\\boldsymbol{\\nu}}}{\\ensuremath{\\boldsymbol{\\nu}}}\n\\newcommand{\\ensuremath{{{\\mathcal{Q}}}}}{\\ensuremath{{{\\mathcal{Q}}}}}\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\\newcommand{\\ensuremath{{\\boldsymbol{\\lambda}}}}{\\ensuremath{{\\boldsymbol{\\lambda}}}}\n\\newcommand{\\ensuremath{x^{(i)}}}{\\ensuremath{x^{(i)}}}\n\\newcommand{\\ensuremath{\\Theta}}{\\ensuremath{\\Theta}}\n\\newcommand{\\ensuremath{\\veparam^*}}{\\ensuremath{\\ensuremath{\\Theta}^*}}\n\\newcommand{\\ensuremath{\\ensuremath{\\mu}}}{\\ensuremath{\\ensuremath{\\mu}}}\n\n\n\\newcommand{\\ensuremath{\\boldsymbol{\\mu}}}{\\ensuremath{\\boldsymbol{\\mu}}}\n\n\\newcommand{\\ensuremath{{\\mathcal{L}}}}{\\ensuremath{{\\mathcal{L}}}}\n\n\\newcommand{\\ensuremath{\\mathcal{U}}}{\\ensuremath{\\mathcal{U}}}\n\n\\newcommand{\\ensuremath{\\Lambda}}{\\ensuremath{\\Lambda}}\n\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\n\\newcommand{\\ensuremath{\\ensuremath{\\theta}_{\\ensuremath{\\mathbf{A}}}}}{\\ensuremath{\\ensuremath{\\theta}_{\\ensuremath{\\mathbf{A}}}}}\n\\newcommand{\\parampl}[1]{\\ensuremath{\\ensuremath{\\theta}_{\\ensuremath{\\mathbf{A}} \\; \\cup \\:\n#1}}}\n\n\\newcommand{\\ensuremath{\\nu}}{\\ensuremath{\\nu}}\n\\newcommand{\\ensuremath{\\Sigma}}{\\ensuremath{\\Sigma}}\n\n\\newcommand{\\ensuremath{\\bar{\\ensuremath{\\theta}}}}{\\ensuremath{\\bar{\\ensuremath{\\theta}}}}\n\n\\newcommand{\\operatorname{or}}{\\operatorname{or}}\n\n\\newcommand{\\ensuremath{\\Upsilon}}{\\ensuremath{\\Upsilon}}\n\\newcommand{\\ensuremath{\\mathbb{{M}}}}{\\ensuremath{\\mathbb{{M}}}}\n\\newcommand{\\ensuremath{\\mathfrak{{P}}}}{\\ensuremath{\\mathfrak{{P}}}}\n\n\\newcommand{{\\operatorname{clique}}}{{\\operatorname{clique}}}\n\\newcommand{{\\operatorname{separator}}}{{\\operatorname{separator}}}\n\\newcommand{{\\operatorname{sep}}}{{\\operatorname{sep}}}\n\n\\newcommand{{\\operatorname{core}}}{{\\operatorname{core}}}\n\\newcommand{{\\operatorname{full}}}{{\\operatorname{full}}}\n\\newcommand{\\ensuremath{Q}}{\\ensuremath{Q}}\n\\newcommand{\\ensuremath{\\wtil{\\qdist}}}{\\ensuremath{\\wtil{\\ensuremath{Q}}}}\n\\newcommand{\\ensuremath{\\wtil{\\qdist}}}{\\ensuremath{\\wtil{\\ensuremath{Q}}}}\n\n\\newcommand{\\ensuremath{\\mathbf{\\vec{\\qdist}}}}{\\ensuremath{\\mathbf{\\vec{\\ensuremath{Q}}}}}\n\n\\newcommand{\\ensuremath{\\qdistvec^*}}{\\ensuremath{\\ensuremath{\\mathbf{\\vec{\\qdist}}}^*}}\n\n\\newcommand{\\ensuremath{\\qdist_{\\ensuremath{\\mathbf{A}}}}}{\\ensuremath{\\ensuremath{Q}_{\\ensuremath{\\mathbf{A}}}}}\n\\newcommand{\\qcorepl}[1]{\\ensuremath{\\ensuremath{Q}_{\\ensuremath{\\mathbf{A}} \\: \\cup \\; #1}}}\n\n\n\\newcommand{\\ensuremath{\\Delta}}{\\ensuremath{\\Delta}}\n\\newcommand{\\ensuremath{\\wtil{\\edset}}}{\\ensuremath{\\wtil{\\ensuremath{\\Delta}}}}\n\n\\newcommand{\\ensuremath{\\mathbf{R}}}{\\ensuremath{\\mathbf{R}}}\n\\newcommand{\\ensuremath{\\wtil{\\mathbf{R}}}}{\\ensuremath{\\wtil{\\mathbf{R}}}}\n\\newcommand{\\ensuremath{\\mathbf{A}}}{\\ensuremath{\\mathbf{A}}}\n\\newcommand{\\ensuremath{\\mathbf{B}}}{\\ensuremath{\\mathbf{B}}}\n\\newcommand{\\ensuremath{\\bullet}}{\\ensuremath{\\bullet}}\n\n\\newcommand{\\ensuremath{\\ensuremath{P}_{\\coreset}}}{\\ensuremath{\\ensuremath{P}_{\\ensuremath{\\mathbf{A}}}}}\n\\newcommand{\\pcorepl}[1]{\\ensuremath{\\ensuremath{P}_{\\ensuremath{\\mathbf{A}}\\: \\cup \\; #1}}}\n\n\n\n\\newcommand{CTD}{CTD}\n\\newcommand{CAD}{CAD}\n\\newcommand{CTD$\\;$}{CTD$\\;$}\n\\newcommand{CAD$\\;$}{CAD$\\;$}\n\n\\newcommand{\\ensuremath{P}}{\\ensuremath{P}}\n\n\n\n\\newcommand{\\ensuremath{\\{0,1\\}^\\ensuremath{N}}}{\\ensuremath{\\{0,1\\}^\\ensuremath{N}}}\n\n\\newcommand{\\ensuremath{\\mathcal{F}}}{\\ensuremath{\\mathcal{F}}}\n\\newcommand{{\\ensuremath{\\ensuremath{\\gamma}^{*}}}}{{\\ensuremath{\\ensuremath{\\gamma}^{*}}}}\n\n\\newcommand{\\ensuremath{\\mathcal{M}_e}}{\\ensuremath{\\mathcal{M}_e}}\n\\newcommand{\\ensuremath{\\mathcal{M}_m}}{\\ensuremath{\\mathcal{M}_m}}\n\n\n\\newcommand{\\ensuremath{{\\mathcal{R}}}}{\\ensuremath{{\\mathcal{R}}}}\n\n\n\\newcommand{\\ensuremath{\\Lambda}}{\\ensuremath{\\Lambda}}\n\n\n\\newcommand{\\ensuremath{F}}{\\ensuremath{F}}\n\\newcommand{\\ensuremath{\\mathcal{F}}}{\\ensuremath{\\mathcal{F}}}\n\n\\newcommand{\\ensuremath{\\ensuremath{\\Phi}_{\\infty}}}{\\ensuremath{\\ensuremath{\\Phi}_{\\infty}}}\n\n\n\\newcommand{\\ensuremath{d}}{\\ensuremath{d}}\n\n\\newcommand{\\ensuremath{\\kappa}}{\\ensuremath{\\kappa}}\n\\newcommand{\\mynorm}[1]{\\ensuremath{\\ensuremath{\\kappa}_{#1}}}\n\n\n\\newcommand{\\ensuremath{{\\mathcal{G}}}}{\\ensuremath{{\\mathcal{G}}}}\n\\newcommand{\\ensuremath{{\\mathcal{G}}}}{\\ensuremath{{\\mathcal{G}}}}\n\n\\newcommand{\\ensuremath{{\\mathcal{G}}}}{\\ensuremath{{\\mathcal{G}}}}\n\\newcommand{\\bigbethe}[2]{\\ensuremath{\\ensuremath{{\\mathcal{G}}}_{{#1}; \\, {#2}}}}\n\\newcommand{\\bigbethenon}[2]{\\ensuremath{\\wtil{\\ensuremath{{\\mathcal{G}}}}_{{#1}; \\, {#2}}}}\n\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\\newcommand{\\ensuremath{{\\mathbb{C}}}}{\\ensuremath{{\\mathbb{C}}}}\n\\newcommand{\\ensuremath{{\\mathbb{D}}}}{\\ensuremath{{\\mathbb{D}}}}\n\\newcommand{\\kullind}[3]{\\ensuremath{D^{#1}({#2}\\; \\| {#3})}}\n\n\\newcommand{\\fullcompose}[1]{\\ensuremath{({\\mathcal{R}} \\circ\\ensuremath{\\Pi})^{#1}}}\n\\newcommand{\\ensuremath{{\\mathcal{Q}}}}{\\ensuremath{{\\mathcal{Q}}}}\n\n\\newcommand{{\\ensuremath{{\\ensuremath{\\theta}^{*}}}}}{{\\ensuremath{{\\ensuremath{\\theta}^{*}}}}}\n\\newcommand{{\\ensuremath{\\ensuremath{\\eparam}^*}}}{{\\ensuremath{\\ensuremath{\\eparam}^*}}}\n\\newcommand{{\\ensuremath{{\\wtil{\\ensuremath{\\eparam}}}}}}{{\\ensuremath{{\\wtil{\\ensuremath{\\eparam}}}}}}\n\n\\newcommand{{\\ensuremath{\\boldsymbol{\\gamma}}}}{{\\ensuremath{\\boldsymbol{\\gamma}}}}\n\\newcommand{{\\ensuremath{\\boldsymbol{\\ensuremath{\\wtil{\\gamma}}}}}}{{\\ensuremath{\\boldsymbol{\\ensuremath{\\wtil{\\gamma}}}}}}\n\n\\newcommand{{\\ensuremath{\\gamma}}}{{\\ensuremath{\\gamma}}}\n\\newcommand{{\\ensuremath{\\ensuremath{\\wtil{\\gamma}}}}}{{\\ensuremath{\\ensuremath{\\wtil{\\gamma}}}}}\n\n\n\\newcommand{\\ensuremath{k}}{\\ensuremath{k}}\n\n\\newcommand{\\ensuremath{\\Theta}}{\\ensuremath{\\Theta}}\n\\newcommand{\\ensuremath{\\Legmap}}{\\ensuremath{\\ensuremath{\\Lambda}}}\n\n\\newcommand{\\breg}[3]{\\ensuremath{B_{#1}(#2; \\; #3)}}\n\\newcommand{\\bregsin}[1]{\\ensuremath{B_{#1}}}\n\n\n\\newcommand{tree-based reparameterization$\\,$}{tree-based reparameterization$\\,$}\n\\newcommand{tree-based reparameterization}{tree-based reparameterization}\n\\newcommand{\\algshorts}{TRP$\\;$} \n\\newcommand{TRP}{TRP}\n\n\\newcommand{\\gmps}{GMP$\\;$} \n\\newcommand{GMP}{GMP}\n\n\n\\newcommand{\\ensuremath{i}}{\\ensuremath{i}}\n\\newcommand{\\ensuremath{i}}{\\ensuremath{i}}\n\\newcommand{\\ensuremath{l}}{\\ensuremath{l}}\n\n\\newcommand{\\ensuremath{\\ensuremath{\\theta}^\\dagger}}{\\ensuremath{\\ensuremath{\\theta}^\\dagger}}\n\\newcommand{\\ensuremath{\\ensuremath{T}^\\dagger}}{\\ensuremath{\\ensuremath{T}^\\dagger}}\n\\newcommand{\\ensuremath{\\bar{\\ensuremath{\\theta}}}}{\\ensuremath{\\bar{\\ensuremath{\\theta}}}}\n\\newcommand{\\ensuremath{{\\wtil{\\ensuremath{\\theta}}}}}{\\ensuremath{{\\wtil{\\ensuremath{\\theta}}}}}\n\\newcommand{\\ensuremath{\\wtil{\\gamma}}}{\\ensuremath{\\wtil{\\gamma}}}\n\n\n\\newcommand{\\ensuremath{\\alpha}}{\\ensuremath{\\alpha}}\n\n\n\\newcommand{\\apmarg}[2]{\\ensuremath{T^*_{#1; #2}}}\n\\newcommand{\\trmarg}[2]{\\ensuremath{P_{#1; #2}}}\n\n\\newcommand{\\ensuremath{\\Lambda}}{\\ensuremath{\\Lambda}}\n\\newcommand{\\ensuremath{\\Psi}}{\\ensuremath{\\Psi}}\n\n\\newcommand{\\ensuremath{\\estim{\\ensuremath{T}}}}{\\ensuremath{\\estim{\\ensuremath{T}}}}\n\\newcommand{\\ensuremath{\\ensuremath{T}^*}}{\\ensuremath{\\ensuremath{T}^*}}\n\\newcommand{\\ensuremath{\\scr{F}}}{\\ensuremath{\\scr{F}}}\n\n\\newcommand{\\ensuremath{{\\scr{M}}}}{\\ensuremath{{\\scr{M}}}}\n\n\\DeclareMathOperator{\\cl}{cl}\n\\DeclareMathOperator{\\is}{is}\n\\DeclareMathOperator{\\markov}{Markov}\n\\DeclareMathOperator{\\wrt}{w.r.t}\n\\DeclareMathOperator{\\suchthat}{s.t.}\n\n\n\\newcommand{M\\\"{o}bius $\\,$}{M\\\"{o}bius $\\,$}\n\n\n\n\n\n\n\n\n\n\n\n\n\\newcommand{\\ensuremath{{\\bf{A}}}}{\\ensuremath{{\\bf{A}}}}\n\n\n\\newcommand{\\ensuremath{{{x}}^F}}{\\ensuremath{{{x}}^F}}\n\\newcommand{\\ensuremath{{{x}}^S}}{\\ensuremath{{{x}}^S}}\n\\newcommand{\\ensuremath{{{y}}^+}}{\\ensuremath{{{y}}^+}}\n\n\\newcommand{\\ensuremath{E}}{\\ensuremath{E}}\n\n\\newcommand{\\ensuremath{\\operatorname{Err}}}{\\ensuremath{\\operatorname{Err}}}\n\n\n\\newcommand{\\ensuremath{\\operatorname{Diff}}}{\\ensuremath{\\operatorname{Diff}}}\n\n\n\\newcommand{\\mathbf{d}}{\\mathbf{d}}\n\n\\newcommand{\\ensuremath{S}}{\\ensuremath{S}}\n\\newcommand{{\\ensuremath{m}}}{{\\ensuremath{m}}}\n\\newcommand{\\ensuremath{N}}{\\ensuremath{N}}\n\\newcommand{\\ensuremath{|\\ensuremath{E}|}}{\\ensuremath{|\\ensuremath{E}|}}\n\\newcommand{\\ensuremath{L}}{\\ensuremath{L}}\n\n\\newcommand{\\ensuremath{i}}{\\ensuremath{i}}\n\\newcommand{\\ensuremath{{\\mathcal{I}}}}{\\ensuremath{{\\mathcal{I}}}}\n\n\\newcommand{\\ensuremath{{\\mathcal{M}}}}{\\ensuremath{{\\mathcal{M}}}}\n\n\\newcommand{\\coeff}[4]{\\ensuremath{C\\big(#1, \\,#2; \\; #3, \\,#4\\big)}}\n\\newcommand{\\bcoeff}[5]{\\ensuremath{D\\big (#1, \\,#2; \\; #3, \\,#4; \\, #5\\big)}}\n\n\\newcommand{\\usetree}[2]{\\ensuremath{\\estim{#1}^{#2}}}\n\\newcommand{\\usecut}[3]{\\ensuremath{\\wtil{#1}^{#2;#3}}}\n\n\\newcommand{\\ensuremath{M}}{\\ensuremath{M}}\n\\newcommand{\\ensuremath{\\mathbf{M}}}{\\ensuremath{\\mathbf{M}}}\n\n\\newcommand{\\ensuremath{\\zeta}}{\\ensuremath{\\zeta}}\n\\newcommand{\\ensuremath{\\boldsymbol{\\zeta}}}{\\ensuremath{\\boldsymbol{\\zeta}}}\n\n\\newcommand{\\linf}[1]{\\ensuremath{\\|{#1}\\|_\\infty}}\n\n\\newcommand{\\ensuremath{\\mathbf{M}}}{\\ensuremath{\\mathbf{M}}}\n\\newcommand{\\ensuremath{\\Gamma}}{\\ensuremath{\\Gamma}}\n\n\\newcommand{\\ensuremath{{\\mathcal{A}}}}{\\ensuremath{{\\mathcal{A}}}}\n\\newcommand{\\ensuremath{{\\mathcal{C}}}}{\\ensuremath{{\\mathcal{C}}}}\n\\newcommand{\\ensuremath{\\psi}}{\\ensuremath{\\psi}}\n\\newcommand{\\ensuremath{\\phi}}{\\ensuremath{\\phi}}\n\\newcommand{\\ensuremath{\\wtil{\\clipot}}}{\\ensuremath{\\wtil{\\ensuremath{\\phi}}}}\n\\newcommand{\\ensuremath{\\varphi}}{\\ensuremath{\\varphi}}\n\\newcommand{\\ensuremath{\\Delta}}{\\ensuremath{\\Delta}}\n\\newcommand{\\ensuremath{{\\mathcal{P}}}}{\\ensuremath{{\\mathcal{P}}}}\n\\newcommand{\\ensuremath{{\\mathcal{P'}}}}{\\ensuremath{{\\mathcal{P'}}}}\n\n\\newcommand{\\ensuremath{\\eta}}{\\ensuremath{\\eta}}\n\\newcommand{\\ensuremath{{\\wtil{\\dualvar}}}}{\\ensuremath{{\\wtil{\\ensuremath{\\eta}}}}}\n\\newcommand{\\ensuremath{\\bar{\\dualvar}}}{\\ensuremath{\\bar{\\ensuremath{\\eta}}}}\n\n\\newcommand{\\ensuremath{{\\boldsymbol{\\eta}}}}{\\ensuremath{{\\boldsymbol{\\eta}}}}\n\\newcommand{\\ensuremath{\\Psi}}{\\ensuremath{\\Psi}}\n\\newcommand{\\ensuremath{\\dualvar^{\\ensuremath{\\scr{T}}}}}{\\ensuremath{\\ensuremath{\\eta}^{\\ensuremath{\\scr{T}}}}}\n\n\n\\newcommand{\\ensuremath{\\mathcal{P}}}{\\ensuremath{\\mathcal{P}}}\n\\newcommand{\\ensuremath{\\mathcal{M}}}{\\ensuremath{\\mathcal{M}}}\n\n\\newcommand{\\ensuremath{\\theta}}{\\ensuremath{\\theta}}\n\\newcommand{\\ensuremath{\\Theta}}{\\ensuremath{\\Theta}}\n\\newcommand{\\ensuremath{\\Theta}}{\\ensuremath{\\Theta}}\n\\newcommand{\\ensuremath{\\eparam}}{\\ensuremath{\\ensuremath{\\theta}}}\n\n\\newcommand{\\ensuremath{\\gamma}}{\\ensuremath{\\gamma}}\n\\newcommand{\\ensuremath{F}}{\\ensuremath{F}}\n\\newcommand{\\ensuremath{z}}{\\ensuremath{z}}\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\\newcommand{\\ensuremath{\\bar{\\nu}}}{\\ensuremath{\\bar{\\nu}}}\n\n\\newcommand{\\ensuremath{{\\scr{M}}}}{\\ensuremath{{\\scr{M}}}}\n\\newcommand{\\ensuremath{{\\scr{X}}}}{\\ensuremath{{\\scr{X}}}}\n\\newcommand{\\ensuremath{{\\scr{Y}}}}{\\ensuremath{{\\scr{Y}}}}\n\\newcommand{\\ensuremath{{\\scr{\\estim{X}}}}}{\\ensuremath{{\\scr{\\estim{X}}}}}\n\n\\newcommand{\\ensuremath{\\statesp^{\\nodenum}}}{\\ensuremath{\\ensuremath{{\\scr{X}}}^{\\ensuremath{N}}}}\n\\newcommand{\\ensuremath{\\ystatesp^{\\nodenum}}}{\\ensuremath{\\ensuremath{{\\scr{Y}}}^{\\ensuremath{N}}}}\n\n\\newcommand{\\ensuremath{\\qstatesp^{\\nodenum}}}{\\ensuremath{\\ensuremath{{\\scr{\\estim{X}}}}^{\\ensuremath{N}}}}\n\n\\newcommand{\\ensuremath{\\scr{T}}}{\\ensuremath{\\scr{T}}}\n\\newcommand{\\ensuremath{e}}{\\ensuremath{e}}\n\\newcommand{\\ensuremath{G}}{\\ensuremath{G}}\n\\newcommand{\\ensuremath{g}}{\\ensuremath{g}}\n\\newcommand{\\ensuremath{\\nu}}{\\ensuremath{\\nu}}\n\\newcommand{\\ensuremath{\\Phi}}{\\ensuremath{\\Phi}}\n\\newcommand{\\ensuremath{\\Psi}}{\\ensuremath{\\Psi}}\n\n\n\\newcommand{\\ensuremath{U}}{\\ensuremath{U}}\n\\newcommand{\\mathbf{U}}{\\mathbf{U}}\n\\newcommand{\\ensuremath{\\Pi}}{\\ensuremath{\\Pi}}\n\\newcommand{{\\mathcal{R}}}{{\\mathcal{R}}}\n\\newcommand{{\\mathcal{I}}}{{\\mathcal{I}}}\n\n\\newcommand{\\ensuremath{P}}{\\ensuremath{P}}\n\\newcommand{\\mathbf{P}}{\\mathbf{P}}\n\n\\newcommand{\\ensuremath{\\mathbf{U}}}{\\ensuremath{\\mathbf{U}}}\n\\newcommand{\\mathbf{V}}{\\mathbf{V}}\n\n\\newcommand{\\ensuremath{{U}}}{\\ensuremath{{U}}}\n\\newcommand{{V}}{{V}}\n\n\n\\newcommand{{\\mathcal{T}}}{{\\mathcal{T}}}\n\\newcommand{{\\mathcal{T}}}{{\\mathcal{T}}}\n\\newcommand{{\\mathcal{C}}}{{\\mathcal{C}}}\n\\newcommand{{\\mathcal{F}}}{{\\mathcal{F}}}\n\n\\newcommand{\\eig}[1]{\\ensuremath{\\lambda_{#1}}}\n\\newcommand{\\ensuremath{\\eig{\\max}}}{\\ensuremath{\\eig{\\max}}}\n\\newcommand{\\ensuremath{\\eig{\\min}}}{\\ensuremath{\\eig{\\min}}}\n\\newcommand{\\eigset}[1]{\\ensuremath{\\sigma\\left({#1}\\right)}}\n\\newcommand{\\eigrad}[1]{\\ensuremath{\\rho\\left({#1}\\right)}}\n\\newcommand{\\eigval}[1]{\\ensuremath{\\lambda_{#1}}}\n\\newcommand{\\eigvalset}[1]{\\ensuremath{\\left\\{\\eigval{i}\\left(#1\\right)\\right\\}}}\n\\newcommand{\\singval}[1]{\\ensuremath{\\sigma_{#1}}}\n\\newcommand{\\singvalset}[1]{\\ensuremath{\\left\\{\\singval{i}\\left(#1\\right)\\right\\}}}\n\\newcommand{\\ensuremath{B}}{\\ensuremath{B}}\n\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\n\\newcommand{\\ensuremath{T}}{\\ensuremath{T}}\n\n\n\\newcommand{\\ensuremath{P}}{\\ensuremath{P}}\n\\newcommand{\\ensuremath{\\exafun}}{\\ensuremath{\\ensuremath{P}}}\n\n\n\\newcommand{\\ensuremath{\\exafun}}{\\ensuremath{\\ensuremath{P}}}\n\n\n\\newcommand{\\ensuremath{\\scr{I}}}{\\ensuremath{\\scr{I}}}\n\n\n\\newcommand{\\condprob}[3]{\\ensuremath{p_{#1}(#2 | #3)}}\n\n\\newcommand{\\sinprob}[2]{\\ensuremath{p_{#1}(#2)}}\n\n\\newcommand{\\pcond}[2]{\\ensuremath{p(#1 \\, | \\, #2 )}}\n\n\n\n\\newcommand{\\ensuremath{P}}{\\ensuremath{P}}\n\\newcommand{\\ensuremath{R}}{\\ensuremath{R}}\n\n\n\\newcommand{\\ensuremath{A}}{\\ensuremath{A}}\n\n\\newcommand{\\ensuremath{{\\scr{A}}}}{\\ensuremath{{\\scr{A}}}}\n\\newcommand{\\ensuremath{{\\scr{B}}}}{\\ensuremath{{\\scr{B}}}}\n\\newcommand{\\ensuremath{{{\\scr{C}}}}}{\\ensuremath{{{\\scr{C}}}}}\n\n\\newcommand{\\ensuremath{A}}{\\ensuremath{A}}\n\\newcommand{\\ensuremath{B}}{\\ensuremath{B}}\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\n\n\n\n\\newcommand{\\ensuremath{\\scr{I}}}{\\ensuremath{\\scr{I}}}\n\\newcommand{\\ensuremath{\\scr{A}}}{\\ensuremath{\\scr{A}}}\n\n\\newcommand{\\ensuremath{\\scr{J}}}{\\ensuremath{\\scr{J}}}\n\\newcommand{\\ensuremath{\\scr{S}}}{\\ensuremath{\\scr{S}}}\n\n\\newcommand{\\ensuremath{{\\mathcal{D}}}}{\\ensuremath{{\\mathcal{D}}}}\n\n\n\n\\newcommand{\\ensuremath{i}}{\\ensuremath{i}}\n\n\n\n\\newcommand{\\ensuremath{\\mu}}{\\ensuremath{\\mu}}\n\n\n\\newcommand{ET }{ET }\n\n\\newcommand{\\pi}{\\pi}\n\\newcommand{\\ensuremath{\\mathcal{A}}}{\\ensuremath{\\mathcal{A}}}\n\n\\newcommand{{\\mathcal{O}}}{{\\mathcal{O}}}\n\\newcommand{\\phi}{\\phi}\n\n\n\n\n\\newcommand{{\\mathcal{C}}}{{\\mathcal{C}}}\n\\newcommand{\\mathbf{C}}{\\mathbf{C}}\n\n\n\\newcommand{{\\mathcal{U}}}{{\\mathcal{U}}}\n\\newcommand{{\\mathbf{S}}}{{\\mathbf{S}}}\n\\newcommand{\\mathbf{\\wtil{S}}}{\\mathbf{\\wtil{S}}}\n\n\\newcommand{\\ensuremath{\\cliqueset_{\\operatorname{sep}}}}{\\ensuremath{\\mathbf{C}_{\\operatorname{sep}}}}\n\n\\newcommand{\\Cut}[1]{\\ensuremath{K_{#1}}}\n\\newcommand{\\Kut}[1]{\\ensuremath{K_{#1}}}\n\\newcommand{\\ensuremath{\\Jestt}}{\\ensuremath{\\Jestt}}\n\\newcommand{\\Jestt}[1]{\\ensuremath{\\widehat{J}_{#1}}}\n\\newcommand{\\ensuremath{\\widehat{J}_{\\operatorname{orig}}}}{\\ensuremath{\\widehat{J}_{\\operatorname{orig}}}}\n\\newcommand{\\ensuremath{\\estim{x}}}{\\ensuremath{\\estim{x}}}\n\\newcommand{\\ensuremath{\\estv_{\\operatorname{MAP}}}}{\\ensuremath{\\ensuremath{\\estim{x}}_{\\operatorname{MAP}}}}\n\\newcommand{\\ensuremath{\\estim{x}}}{\\ensuremath{\\estim{x}}}\n\n\\newcommand{\\ensuremath{\\mathbf{E}}}{\\ensuremath{\\mathbf{E}}}\n\\newcommand{\\ensuremath{\\mathbf{D}}}{\\ensuremath{\\mathbf{D}}}\n\\newcommand{\\ensuremath{\\mathbf{E}_{r}}}{\\ensuremath{\\mathbf{E}_{r}}}\n\n\\newcommand{\\ensuremath{\\Pi}}{\\ensuremath{\\Pi}}\n\n\\newcommand{\\ensuremath{\\theta}}{\\ensuremath{\\theta}}\n\\newcommand{\\ensuremath{\\Theta}}{\\ensuremath{\\Theta}}\n\n\\newcommand{\\ensuremath{\\estim{P}_{full}}}{\\ensuremath{\\estim{P}_{full}}}\n\n\n\\newcommand{\\inprod}[2]{\\ensuremath{\\langle #1 , \\, #2 \\rangle}}\n\n\\newcommand{\\inprodd}[3]{\\ensuremath{\\langle #1 , \\, #2 \\rangle_{#3}}}\n\n\\newcommand{\\covv}[3]{\\ensuremath{\\Sigma_{#3}\\{#1, \\, #2\\}}}\n\n\n\\newcommand{\\kull}[2]{\\ensuremath{D(#1\\; \\| \\; #2)}}\n\n\\newcommand{\\kullinf}[2]{\\ensuremath{D(#1\\, \\| \\, #2)}}\n\n\n\\newcommand{\\normspec}[2]{\\ensuremath{\\|#1\\|_{#2}}}\n\n\n\\newcommand{{\\cal A}}{{\\cal A}}\n\n\\newcommand{\\ensuremath{{y}}}{\\ensuremath{{y}}}\n\\newcommand{\\ensuremath{{c}}}{\\ensuremath{{c}}}\n\\newcommand{\\ensuremath{{v}}}{\\ensuremath{{v}}}\n\n\\newcommand{\\ensuremath{c}}{\\ensuremath{c}}\n\\newcommand{\\ensuremath{{\\bf{c}}}}{\\ensuremath{{\\bf{c}}}}\n\\newcommand{\\ensuremath{\\mathcal{I}}}{\\ensuremath{\\mathcal{I}}}\n\n\\newcommand{\\ensuremath{\\mathbf{1}}}{\\ensuremath{\\mathbf{1}}}\n\\newcommand{\\ensuremath{\\mathbf{0}}}{\\ensuremath{\\mathbf{0}}}\n\\newcommand{\\ensuremath{\\mathcal{L}}}{\\ensuremath{\\mathcal{L}}}\n\n\n\\newcommand{\\chooser}[2]{\\ensuremath{\\big(\\frac{#1}{#2}\\big)}}\n\n\n\n\n\n\\newcommand{\\ensuremath{G}}{\\ensuremath{G}}\n\\newcommand{\\ensuremath{H}}{\\ensuremath{H}}\n\\newcommand{\\ensuremath{V}}{\\ensuremath{V}}\n\\newcommand{\\ensuremath{E}}{\\ensuremath{E}}\n\\newcommand{\\ensuremath{{F}}}{\\ensuremath{{F}}}\n\n\n\\newcommand{\\ensuremath{P}}{\\ensuremath{P}}\n\n\\newcommand{\\ensuremath{\\wtil{\\graph}}}{\\ensuremath{\\wtil{\\ensuremath{G}}}}\n\\newcommand{\\ensuremath{\\wtil{\\graph}}}{\\ensuremath{\\wtil{\\ensuremath{G}}}}\n\\newcommand{\\ensuremath{\\mathbf{\\wtil{C}}}}{\\ensuremath{\\mathbf{\\wtil{C}}}}\n\n\\newcommand{\\ensuremath{r}}{\\ensuremath{r}}\n\\newcommand{\\ensuremath{c}}{\\ensuremath{c}}\n\\newcommand{\\ensuremath{n}}{\\ensuremath{n}}\n\\newcommand{\\ensuremath{v}}{\\ensuremath{v}}\n\n\n\n\n\\newcommand{\\ensuremath{\\mathcal{T}}}{\\ensuremath{\\mathcal{T}}}\n\n\n\\newcommand{\\ensuremath{\\omega}}{\\ensuremath{\\omega}}\n\\newcommand{\\ensuremath{w}}{\\ensuremath{w}}\n\n\\newcommand{\\ensuremath{h}}{\\ensuremath{h}}\n\n\\newcommand{\\ensuremath{Z}}{\\ensuremath{Z}}\n\n\\newcommand{\\ensuremath{\\backslash}}{\\ensuremath{\\backslash}}\n\\newcommand{\\ensuremath{K}}{\\ensuremath{K}}\n\n\\newcommand{\\ensuremath{\\bar{\\gamma}}}{\\ensuremath{\\bar{\\gamma}}}\n\\newcommand{\\childa}[1]{\\ensuremath{\\alpha_{#1}}}\n\\newcommand{\\ensuremath{\\bar{\\alpha_1}}}{\\ensuremath{\\bar{\\alpha_1}}}\n\\newcommand{\\ensuremath{\\bar{\\alpha_2}}}{\\ensuremath{\\bar{\\alpha_2}}}\n\\newcommand{\\ensuremath{\\bar{\\alpha_4}}}{\\ensuremath{\\bar{\\alpha_4}}}\n\\newcommand{\\ancest}[2]{\\ensuremath{{#1} \\wedge {#2}}}\n\\newcommand{\\ensuremath{\\operatorname{Ch}}}{\\ensuremath{\\operatorname{Ch}}}\n\n\\newcommand{\\ensuremath{\\operatorname{pa}}}{\\ensuremath{\\operatorname{pa}}}\n\\newcommand{\\ensuremath{\\Delta}}{\\ensuremath{\\Delta}}\n\n\\newcommand{\\ensuremath{\\operatorname{parent}}}{\\ensuremath{\\operatorname{parent}}}\n\\newcommand{\\ensuremath{\\operatorname{is}}}{\\ensuremath{\\operatorname{is}}}\n\\newcommand{\\ensuremath{\\operatorname{of}}}{\\ensuremath{\\operatorname{of}}}\n\n\\newcommand{\\future}[1]{\\ensuremath{{\\bf{y}}^{+}_{#1}}}\n\\newcommand{\\past}[1]{\\ensuremath{{\\bf{y}}^{-}_{#1}}}\n\\newcommand{\\lamplus}[2]{\\ensuremath{\\lambda^{+}_{#1}(#2({#1}))}}\n\\newcommand{\\lamhatplus}[3]{\\ensuremath{\\widehat{\\lambda}^{+}_{#1}(#2({#3}))}}\n\\newcommand{\\aplus}[2]{\\ensuremath{{#1}^{+}_{#2}}}\n\\newcommand{\\aplushat}[2]{\\ensuremath{\\widehat{#1}^{+}_{#2}}}\n\n\\newcommand{compplus}{compplus}\n\\newcommand{comphat}{comphat}\n\n\n\n\\newcommand{\\ensuremath{{\\bf{y}}}}{\\ensuremath{{\\bf{y}}}}\n\\newcommand{\\probd}[2]{\\ensuremath{p(#1 \\, | \\, #2 )}}\n\n\n\n\\newcommand{\\ensuremath{x}}{\\ensuremath{x}}\n\\newcommand{\\ensuremath{y}}{\\ensuremath{y}}\n\n\n\n\\newcommand{\\invn}[1]{\\ensuremath{{#1}^{-1}}}\n\\newcommand{\\ensuremath{\\mathcal{K}}}{\\ensuremath{\\mathcal{K}}}\n\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\\newcommand{\\ensuremath{\\beta}}{\\ensuremath{\\beta}}\n\n\\newcommand{\\ensuremath{\\log_2}}{\\ensuremath{\\log_2}}\n\n\\newcommand{\\lambda_{max}}{\\lambda_{max}}\n\\newcommand{\\lambda_{min}}{\\lambda_{min}}\n\\newcommand{\\earb}[1]{\\lambda_{#1}}\n\n\\newcommand{\\ensuremath{\\stackrel}{\\mathcal{P}}{\\longrightarrow}}{\\ensuremath{\\stackrel}{\\mathcal{P}}{\\longrightarrow}}\n\\newcommand{\\ensuremath{\\stackrel}{\\mathitalic{d}}{\\longrightarrow}}{\\ensuremath{\\stackrel}{\\mathitalic{d}}{\\longrightarrow}}\n\n\\newcommand{\\pderiv}[2]{\\ensuremath{\\frac{\\partial{#1}}{\\partial{#2}}}}\n\\newcommand{\\pderivtwo}[3]{\\ensuremath{\\frac{\\partial^2{#1}}{\\partial{#2}\n\\, \\partial{#3}}}}\n\\newcommand{\\pderivtwod}[2]{\\ensuremath{\\frac{\\partial^2{#1}}{\\partial{#2}^2}}}\n\n\\newcommand{\\pdif}[2]{\\ensuremath{\\frac{\\partial{#1}}{\\partial{#2}}}}\n\\newcommand{\\pdifftwo}[3]{\\ensuremath{\\frac{\\partial^2{#1}}{\\partial{#2}\n\\, \\partial{#3}}}}\n\\newcommand{\\pdiff}[2]{\\ensuremath{\\frac{\\partial^2{#1}}{\\partial{#2}^2}}}\n\\newcommand{\\pdifn}[3]{\\ensuremath{\\frac{\\partial^{#1}{#2}}{\\partial{#3}^{#1}}}}\n\n\n\\newcommand{\\ensuremath{\\mathcal{L}}}{\\ensuremath{\\mathcal{L}}}\n\n\\newcommand{\\ensuremath{\\mathit{Ra}}}{\\ensuremath{\\mathit{Ra}}}\n\n\n\\newcommand{\\ensuremath{(0,1)^{\\df(\\meparam)}}}{\\ensuremath{(0,1)^{\\ensuremath{d}(\\ensuremath{\\eparam})}}}\n\\newcommand{\\ensuremath{(0,1)^{\\df(\\eparam)}}}{\\ensuremath{(0,1)^{\\ensuremath{d}(\\ensuremath{\\theta})}}}\n\n\\newcommand{\\ensuremath{{\\mathbb{E}}}}{\\ensuremath{{\\mathbb{E}}}}\n\n\n\\newcommand{\\mcite}[2]{\\citetext{#1  \\citealp{#2}}}\n\\newcommand{\\incite}[2]{\\citealp{#1,\\citetext{#2}}}\n\n\n\\newcommand{\\begin{equation}}{\\begin{equation}}\n\\newcommand{\\end{equation}}{\\end{equation}}\n\n\\newcommand{\\pnorm}[2]{\\|\\, #1 \\,\\|_{#2}}\n\n\\newcommand{\\fastgraph}[2]{\\includegraphics[scale=#1]{#2}}\n\n\\newcommand{\\widgraph}[2]{\\includegraphics[keepaspectratio,width=#1]{#2}}\n\n\\newcommand{\\epsgraph}[2]{\\epsfig{file=#2,width=#1}}\n\n\\newcommand{\\scr}[1]{\\ensuremath{\\mathcal{#1}}}\n\n\\newcommand{\\int_{0}^{\\infty}}{\\int_{0}^{\\infty}}\n\\newcommand{\\ensuremath{\\overset{d}{=}}}{\\ensuremath{\\overset{d}{=}}}\n\\newcommand{\\ensuremath{\\overset{\\cdot}{=}}}{\\ensuremath{\\overset{\\cdot}{=}}}\n\\newcommand{\\ensuremath{\\triangleq}}{\\ensuremath{\\triangleq}}\n\\newcommand{\\ensuremath{:  =}}{\\ensuremath{:  =}}\n\\newcommand{\\ensuremath{= :}}{\\ensuremath{= :}}\n\n\\newcommand{\\ensuremath{d}}{\\ensuremath{d}}\n\\newcommand{\\ensuremath{S}}{\\ensuremath{S}}\n\n\\newcommand{\\ensuremath{d_H}}{\\ensuremath{d_H}}\n\\newcommand{\\ensuremath{\\operatorname{APP}}}{\\ensuremath{\\operatorname{APP}}}\n\n\\newcommand{x}{\\ensuremath{{{\\mathbf{x}}}}}\n\\newcommand{\\ensuremath{{{\\mathbf{X}}}}}{\\ensuremath{{{\\mathbf{X}}}}}\n\\newcommand{\\ensuremath{{{\\mathbf{\\estim{x}}}}}}{\\ensuremath{{{\\mathbf{\\estim{x}}}}}}\n\\newcommand{\\ensuremath{{\\mathbf{X}}}}{\\ensuremath{{\\mathbf{X}}}}\n\\newcommand{\\ensuremath{{\\mathbf{Y}}}}{\\ensuremath{{\\mathbf{Y}}}}\n\n\\newcommand{\\empav}[1]{\\bar{#1}}\n\n\\newcommand{\\ensuremath{x^*}}{\\ensuremath{x^*}}\n\n\\newcommand{\\ensuremath{x}}{\\ensuremath{{\\bf{x}}}}\n\n\\newcommand{{\\bf{u}}}{{\\bf{u}}}\n\\newcommand{\\ensuremath{Z}}{\\ensuremath{Z}}\n\\newcommand{{\\bf{y}}}{{\\bf{y}}}\n\\newcommand{{\\bf{u}}}{{\\bf{u}}}\n\\newcommand{{\\bf{z}}}{{\\bf{z}}}\n\\newcommand{{\\bf{w}}}{{\\bf{w}}}\n\\newcommand{{\\bf{v}}}{{\\bf{v}}}\n\n\\newcommand{{\\bf{Y}}}{{\\bf{Y}}}\n\\newcommand{{\\bf{U}}}{{\\bf{U}}}\n\\newcommand{{\\bf{Z}}}{{\\bf{Z}}}\n\\newcommand{{\\bf{W}}}{{\\bf{W}}}\n\\newcommand{{\\bf{V}}}{{\\bf{V}}}\n\n\n\\newcommand{{{x}}}{{{x}}}\n\\newcommand{{{y}}}{{{y}}}\n\\newcommand{{{z}}}{{{z}}}\n\\newcommand{{{w}}}{{{w}}}\n\\newcommand{{{u}}}{{{u}}}\n\\newcommand{{{v}}}{{{v}}}\n\n\\newcommand{{{X}}}{{{X}}}\n\\newcommand{{{Y}}}{{{Y}}}\n\\newcommand{{{Z}}}{{{Z}}}\n\\newcommand{{{W}}}{{{W}}}\n\\newcommand{{{U}}}{{{U}}}\n\\newcommand{{{V}}}{{{V}}}\n\n\\newcommand{\\vect}[1]{{\\bf{#1}}}\n\n\n\\newcommand{\\ensuremath{J}}{\\ensuremath{J}}\n\n\\newcommand{\\ensuremath{u}}{\\ensuremath{u}}\n\\newcommand{\\ensuremath{z}}{\\ensuremath{z}}\n\\newcommand{\\ensuremath{z}}{\\ensuremath{z}}\n\n\n\\newcommand{\\ensuremath{\\left\\{2^m}{\\ensuremath{\\left\\{2^m}\n\\psi^{m}_n(2^m t - n)\\right\\}}\n\\newcommand{\\ensuremath{x^{m}_n}}{\\ensuremath{x^{m}_n}}\n\\newcommand{\\ensuremath{\\psi(2^m t - n)}}{\\ensuremath{\\psi(2^m t - n)}}\n\n\\newcommand{\\updmn}[1]{\\ensuremath{{#1}^{m}_{n}}}\n\\newcommand{\\updn}[3]{\\ensuremath{{#1}^{#2}_{#3}}}\n\n\\newcommand{\\ensuremath{w}}{\\ensuremath{w}}\n\\newcommand{\\ensuremath{W}}{\\ensuremath{W}}\n\n\\newcommand{\\ensuremath{\\zeta}}{\\ensuremath{\\zeta}}\n\\newcommand{\\ensuremath{v}}{\\ensuremath{v}}\n\n\\newcommand{\\ensuremath{A}}{\\ensuremath{A}}\n\\newcommand{\\ensuremath{B}}{\\ensuremath{B}}\n\\newcommand{\\ensuremath{C}}{\\ensuremath{C}}\n\\newcommand{\\ensuremath{D}}{\\ensuremath{D}}\n\n\\newcommand{\\wavdet}[2]{\\ensuremath{x^{#1}_{#2}}}\n\n\n\\newcommand{\\begin{center}}{\\begin{center}}\n\\newcommand{\\end{center}}{\\end{center}}\n\n\\newcommand{\\begin{itemize}}{\\begin{itemize}}\n\\newcommand{\\end{itemize}}{\\end{itemize}}\n\n\\newcommand{\\begin{enumerate}}{\\begin{enumerate}}\n\\newcommand{\\end{enumerate}}{\\end{enumerate}}\n\n\n\\newcommand{\\begin{slide}}{\\begin{slide}}\n\\newcommand{\\begin{slide*}}{\\begin{slide*}}\n\n\\newcommand{\\comsld}[2]{\\begin{slide}[#1,#2]}\n\\newcommand{\\comspord}[2]{\\begin{slide*}[#1,#2]}\n\n\\newcommand{\\end{slide}}{\\end{slide}}\n\\newcommand{\\end{slide*}}{\\end{slide*}}\n\n\n\\newcommand{\\ensuremath{\\msca  \\gvec}}{\\ensuremath{\\ensuremath{z}  {\\bf{u}}}}\n\\newcommand{\\ensuremath{\\msca  \\gsca}}{\\ensuremath{\\ensuremath{z}  \\ensuremath{u}}}\n\n\\newcommand{\\ensuremath{\\sqrt{\\ssca}  \\gvec}}{\\ensuremath{\\sqrt{\\ensuremath{z}}  {\\bf{u}}}}\n\\newcommand{\\ensuremath{\\sqrt{\\ssca}  \\gsca}}{\\ensuremath{\\sqrt{\\ensuremath{z}}  \\ensuremath{u}}}\n\n\\newcommand{\\mbox{$\\xvec \\, \\edist \\, \\atgvec$}}{\\mbox{$x \\, \\ensuremath{\\overset{d}{=}} \\, \\ensuremath{\\msca  \\gvec}$}}\n\\newcommand{\\mbox{$\\xsca \\, \\edist \\, \\atgsca$}}{\\mbox{${{x}} \\, \\ensuremath{\\overset{d}{=}} \\, \\ensuremath{\\msca  \\gsca}$}}\n\n\\newcommand{\\mbox{$\\xvec \\, \\edist \\, \\satgvec$}}{\\mbox{$x \\, \\ensuremath{\\overset{d}{=}} \\, \\ensuremath{\\sqrt{\\ssca}  \\gvec}$}}\n\\newcommand{\\mbox{$\\xsca \\, \\edist \\, \\satgsca$}}{\\mbox{${{x}} \\, \\ensuremath{\\overset{d}{=}} \\, \\ensuremath{\\sqrt{\\ssca}  \\gsca}$}}\n\n\n\\newcommand{\\normal}[1]{\\ensuremath{\\mathcal{N}(0,#1)}}\n\\newcommand{\\normall}[2]{\\ensuremath{\\mathcal{N}(#1,#2)}}\n\n\\newcommand{\\opt}[1]{\\ensuremath{{#1}^{*}}}\n\\newcommand{\\estim}[1]{\\ensuremath{\\widehat{#1}}}\n\\newcommand{\\bls}[1]{\\ensuremath{\\widehat{#1}_{BLS}}}\n\\newcommand{\\map}[1]{\\ensuremath{\\widehat{#1}_{MAP}}}\n\\newcommand{\\wtil}[1]{\\ensuremath{\\widetilde{#1}}}\n\n\\newcommand{\\inv}[1]{\\ensuremath{\\big(#1\\big)^{-1}}}\n\\renewcommand{\\t}[1]{\\ensuremath{[#1]^{{\\scriptscriptstyle\\top}}}}\n\n\\newcommand{\\ensuremath{f}}{\\ensuremath{f}}\n\n\\newcommand{r}{r}\n\\newcommand{\\ensuremath{{\\mathbb{R}}}}{\\ensuremath{{\\mathbb{R}}}}\n\\newcommand{\\ensuremath{{\\mathbb{R^{+}}}}}{\\ensuremath{{\\mathbb{R^{+}}}}}\n\\newcommand{\\ensuremath{{\\mathbb{N}}}}{\\ensuremath{{\\mathbb{N}}}}\n\\newcommand{\\ensuremath{{\\mathbb{Z}}}}{\\ensuremath{{\\mathbb{Z}}}}\n\\newcommand{{\\mathbf Z}}{{\\mathbf Z}}\n\n\\newcommand{\\ensuremath{\\int_0^{\\infty}}}{\\ensuremath{\\int_0^{\\infty}}}\n\\newcommand{\\ensuremath{\\int_{-\\infty}^{\\infty}}}{\\ensuremath{\\int_{-\\infty}^{\\infty}}}\n\\newcommand{\\intar}[2]{\\ensuremath{\\int_{#1}^{#2}}}\n\n\\newcommand{\\ensuremath{{\\mathcal{L}}}}{\\ensuremath{{\\mathcal{L}}}}\n\n\\newcommand{\\student}[3]{\\ensuremath{ (1+ \\frac{#1^2}{2 #2^2})^{-#3}}}\n\\newcommand{\\gamd}[1]{\\ensuremath{Z({#1})}}\n\\newcommand{\\symgam}[3]{\\ensuremath{\\left({#3}\/({#3} + {#1}^2)\\right)^{#2}}}\n\\newcommand{\\gausss}[2]{\\ensuremath{\\frac{1}{\\sqrt{2\\pi} #2} \\exp{(-\\frac{#1^2}{2 #2^2})}}}\n\\newcommand{\\gaussm}[3]{\\ensuremath{\\frac{1}{\\sqrt{2\\pi}\n|#2|^{\\frac{#3}{2}}} \\exp{\\big(-\\frac{1}{2} #1'#2^{-1} #1 \\big)}}}\n\n\\newcommand{\\dif}[2]{\\ensuremath{\\frac{\\partial #1}{\\partial #2}}}\n\\newcommand{\\diff}[2]{\\ensuremath{\\frac{\\partial^2 #1}{\\partial #2^2}}}\n\\newcommand{\\dift}[3]{\\ensuremath{\\frac{\\partial^2 #1}{\\partial #2 \\partial #3}}}\n\n\\newcommand{\\odif}[2]{\\ensuremath{\\frac{d #1}{d #2}}}\n\\newcommand{\\odiff}[2]{\\ensuremath{\\frac{d^2 #1}{d #2^2}}}\n\\newcommand{\\odift}[3]{\\ensuremath{\\frac{d #1}{d #2 d #3}}}\n\n\\newcommand{H-ss}{H-ss}\n\\newcommand{\\beta}{\\beta}\n\\newcommand{\\pcon}[2]{p(#1 | #2)}\n\\newcommand{\\scaleform}[3]{\\frac{1}{#2} {#3}(\\frac{#1}{#2})}\n\n\n\\newcommand{\\ensuremath{\\lambda}}{\\ensuremath{\\lambda}}\n\\newcommand{\\ensuremath{\\alpha}}{\\ensuremath{\\alpha}}\n\n\\newcommand{\\charfunc}[2]{\\ensuremath{\\phi_{#1}{(#2)}}}\n\\newcommand{\\charf}[1]{\\ensuremath{\\phi_{#1}}}\n\n\\newcommand{\\laptran}[2]{\\ensuremath{\\psi_{#1}{(#2)}}}\n\n\\newcommand{\\gaussd}[2]{\\ensuremath{\\frac{1}{\\sqrt{2 \\pi #1}}\n\\exp{(-\\frac{#2^2}{2 #1})}}}\n\n\n\n\n\n\\newcommand{\\ensuremath{^{\\frac{1}{2}}}}{\\ensuremath{^{\\frac{1}{2}}}}\n\\newcommand{\\ensuremath{w}}{\\ensuremath{w}}\n\\newcommand{\\ensuremath{x}}{\\ensuremath{x}}\n\\newcommand{\\ensuremath{u}}{\\ensuremath{u}}\n\n\\newcommand{\\ensuremath{\\; \\:}}{\\ensuremath{\\; \\:}}\n\n\\newcommand{\\iden}[1]{\\ensuremath{I_{#1}}}\n\\newcommand{\\zeros}[1]{\\ensuremath{0_{#1}}}\n\n\n\\newcommand{{I}t\\^{o}\\hspace*{3pt}}{{I}t\\^{o}\\hspace*{3pt}}\n\\newcommand{{C}sisz\\'{a}r\\hspace*{3pt}}{{C}sisz\\'{a}r\\hspace*{3pt}}\n\\newcommand{{C}sisz\\'{a}r}{{C}sisz\\'{a}r}\n\n\n\n\n\\DeclareMathOperator{\\modd}{mod}\n\\DeclareMathOperator{\\probab}{Pr}\n\\DeclareMathOperator{\\diag}{diag}\n\\DeclareMathOperator{\\var}{var}\n\\DeclareMathOperator{\\stable}{stable}\n\\DeclareMathOperator{\\Sigma}{cov}\n\\DeclareMathOperator{\\argmax}{argmax}\n\\DeclareMathOperator{\\trace}{trace}\n\\DeclareMathOperator{\\abs}{abs}\n\\DeclareMathOperator{\\floor}{floor}\n\\DeclareMathOperator{\\spanner}{span}\n\\DeclareMathOperator{\\vol}{vol}\n\\DeclareMathOperator{\\child}{child}\n\\DeclareMathOperator{\\parent}{parent}\n\\DeclareMathOperator{\\ensuremath{\\scr{T}}}{tree}\n\\DeclareMathOperator{\\loopy}{loop}\n\\DeclareMathOperator{\\textrm{sign}}{sign}\n\\DeclareMathOperator{\\for}{for}\n\\DeclareMathOperator{\\all}{all}\n\\DeclareMathOperator{\\some}{some}\n\\DeclareMathOperator{\\cum}{cum}\n\\DeclareMathOperator{\\toeplitz}{toeplitz}\n\n\\renewcommand{\\Re}{\\mathbb{R}}\n\\newcommand{\\tr}[2]{\\left<#1,#2\\right>}\n\\def\\textrm{logdet}{\\textrm{logdet}}\n\\def\\textrm{sign}{\\textrm{sign}}\n\\def\\textrm{Cmax}{\\textrm{Cmax}}\n\n\\def\\tilde{\\Theta}^{*}{\\tilde{\\Theta}^{*}}\n\\def{\\tilde{\\Theta}^{*}}^{-1}{{\\tilde{\\Theta}^{*}}^{-1}}\n\\def\\tilde{\\Delta}{\\tilde{\\Delta}}\n\\def\\widetilde{W}{\\widetilde{W}}\n\\def\\widetilde{Z}{\\widetilde{Z}}\n\\def\\widetilde{R}{\\widetilde{R}}\n\\newcommand{\\matnorm}[2]{| \\! | \\! | #1 | \\! | \\! |_{{#2}}}\n\\newcommand{\\vecnorm}[2]{\\| #1 \\|_{{#2}}}\n\\renewcommand{\\sp}[2]{{#1}^{(#2)}}\n\\def\\Theta^{*}{\\Theta^{*}}\n\\def\\Theta^{*}{\\Theta^{*}}\n\\def\\Theta^{*^{-1}}{\\Theta^{*^{-1}}}\n\\newcommand\\mvect[1]{\\overline{#1}}\n\\newcommand{\\ensuremath{E}}{\\ensuremath{E}}\n\\newcommand{{\\ensuremath{{\\Eset^{c}}}}}{{\\ensuremath{{\\ensuremath{E}^{c}}}}}\n\n\\def\\Theta{\\Theta}\n\\def\\Sigma{\\Sigma}\n\n\\def\\theta^{*}_{\\min}{\\theta^{*}_{\\min}}\n\\def\\frac{1}{\\lambda_n}{\\frac{1}{\\lambda_n}}\n\\newcommand\\Op[1]{\\mathcal{O}_{P}\\left(#1\\right)}\n\\newcommand\\Oa[1]{\\mathcal{O}\\left(#1\\right)}\n\\newcommand\\Cl[1]{\\textrm{Cl}(#1)}\n\\renewcommand\\Vec{\\textrm{Vec}}\n\\newcommand\\ThetaWitness{\\widetilde{\\Theta}}\n\n\n\n\\defC_1{C_1}\n\\defC_2{C_2}\n\\defC{C}\n\\defC_1{C_1}\n\\defC_2{C_2}\n\\defC_3{C_3}\n\\defC_4{C_4}\n\\defL{L}\n\\defK{K}\n\\defc_{*}{c_{*}}\n\n\\defK{K}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}