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+{"text":"\\section{introduction}\n\nThe quantum information theory in the relativistic framework has\nreceived considerable attention due to its theoretical importance\nand practical application \\cite{Peres,Boschi,Pan}. Especially, more\nand more efforts have been expended on the study of quantum\nentanglement in a relativistic setting because people consider the\nentanglement to be a major resource for quantum information tasks\nsuch as quantum teleportation, quantum computation and so on\n\\cite{Bouwmeester}. With the intention of studying the entanglement\nbetween accelerated observers, the fidelity of teleportation between\ntwo parties in relative uniform acceleration was discussed by Alsing\n\\emph{et al.} \\cite{Alsing-Milburn,Alsing-McMahon-Milburn}. Xian-Hui\nGe \\emph{et al.} extended the gravitational field of the\nteleportation to the four and higher dimensional spacetimes, and\neven explicitly discussed what effects the shape of the cavity in\nwhich particles are confined has on the teleportation in a black\nhole spacetime \\cite{Ge-Shen,Ge-Kim}. In order to further\ninvestigate the observer-dependent character of the entanglement,\nFuentes-Schuller \\emph{et al.} analyzed the entanglement between two\nmodes of a non-interacting  massless scalar field when one of the\nobservers describing the state is uniformly accelerated\n\\cite{Schuller-Mann}. And then Alsing \\emph{et al.} calculated the\nentanglement between two modes of a free Dirac field described by\nrelatively accelerated parties in a flat spacetime\n\\cite{Alsing-Mann}. Their results \\cite{Schuller-Mann,Alsing-Mann}\nalso showed that the different type of field will have a\nqualitatively different effect on the degradation of entanglement\nproduced by the Unruh effect \\cite{Davies,unruh}. More recently, Ahn\n\\emph{et al.} extended the investigation to the entanglement of a\ntwo-mode squeezed state in Riemannian spacetime \\cite{Ahn-Kim}, Yi\nLing \\emph{et al.} discussed the entanglement of electromagnetic\nfield in noninertial reference frames \\cite{Ling}, and Adesso\n\\emph{et al.} investigated the distribution of entanglement between\nmodes of a free scalar field from the perspective of observers in\nuniform acceleration \\cite{Adesso}.\n\nAs a further step along this line, we will provide an analysis of\nthe entanglement for the scalar field in the spacetime of a most\ngeneral, static and asymptotically flat black hole with spherical\nsymmetry. It seems to be an interesting study to consider the\ninfluences of the Hawking effect\n\\cite{Hawking-1,Hawking-2,Hawking-3} on the quantum entangled states\nand show how the Hawking temperature will change the properties of\nthe entanglement and teleportation. Choosing a generically entangled\nstate as the initially entangled state for two observers in the flat\nregion of this black hole, we will also try to see what effects the\nuncertain entangled state will have on the degradation of\nentanglement in our scheme due to the presence of an arbitrary state\nparameter. Our scheme proposes that the two observers, Alice and\nBob, share an initially entangled state at the same initial point in\nflat Minkowski spacetime before the black hole is formed. After the\ncoincidence of Alice and Bob, Alice stays stationary at the\nasymptotically flat region, while Bob falls in toward the mass and\nthen hovers outside of it. Once Bob is safely hovering outside of\nthe object at some constant acceleration, let it collapse to form a\nblack hole. By Birkhoff's theorem \\cite{Birkhoff} this won't change\nthe metric outside of the black hole and therefore won't change\nBob's acceleration. Thus, Bob's detector registers only thermally\nexcited particles due to the Hawking effect \\cite{unruh-1,unruh-2}.\nIn order to investigate the teleportation between two modes of a\nscalar field as detected by the two observers, we assume that Alice\nand Bob each hold an optical cavity which is small and perfect for\nthe teleportation in the black hole spacetime. Just as suggested by\nRefs. \\cite{Alsing-Milburn,Alsing-McMahon-Milburn}, we further\nsuppose that each cavity supports two orthogonal modes, with the\nsame frequency, which are each excited to a single photon Fock state\nat the coincidence point for Alice and Bob. Different from the\nstandard teleportation protocol, our scheme assumes that Bob hovers\noutside of the object before it collapses, and turns on his detector\nafter the formation of the black hole. Then, Bob can check to see\nwhether any thermal photons have been excited in his local cavity\nusing the non-absorbing detector.\n\nThe organization of this paper is as follows. In Sec. 2 we discuss\nthe vacuum structure of the background spacetime and the Hawking\neffect for the scalar particles as experienced by the observer\noutside the black hole. In Sec. 3 we analyze the effects of the\nHawking temperature on the entanglement between the modes for the\ndifferent state parameter. In Sec. 4 we describe the process of the\nteleportation between Alice and Bob, and calculate the fidelity of\nteleportation. We summarize and discuss our conclusions in the last\nsection.\n\n\\section{Vacuum structure and Hawking Radiation of scalar field}\n\nIt is well known that the spherically symmetric line element of a\nstatic and asymptotically flat black hole such as Schwarzschild\nblack hole, Reissner-Nordstr\\\"{o}m black hole \\cite{Chandrasekhar},\nGarfinkle-Horowitz-Strominger dilaton black hole \\cite{Horowitz},\nCasadio-Fabbri-Mazzacurati (CFM) brane black hole \\cite{Casadio} and\nso on can be written in the form\n\\begin{eqnarray}\\label{metric}\nds^2=f(r)dt^{2}-\\frac{1}{h(r)}dr^{2}-R^{2}(r)(d\\theta^{2}+\\sin\\theta^{2}d\\varphi^{2}),\n\\end{eqnarray}\nwhere the functions $f(r)$ and $h(r)$ vanish at the event horizon\n$r=r_{+}$ of the black hole. Throughout this paper we use\n$G=c=\\hbar=\\kappa_{B}=1$. It is obvious that the surface gravity of\nthe event horizon is determined by\n$\\kappa=\\sqrt{f'(r_{+})h'(r_{+})}\/2$. Defining the tortoise\ncoordinates $r_{*}$ as $dr_{*}=dr\/\\sqrt{f(r)h(r)}$, we can rewrite\nthe metric (\\ref{metric}) as\n\\begin{eqnarray}\\label{new metric}\nds^2=f(r)(dt^{2}-dr_{*}^{2})-R^{2}(r)(d\\theta^{2}+\\sin\\theta^{2}d\\varphi^{2}).\n\\end{eqnarray}\n\nThe massless scalar field $\\psi$ satisfies the Klein-Gordon equation\n\\begin{eqnarray}\n\\label{K-G Equation}\\frac{1}{\\sqrt{-g}}\\frac{{\\partial}}{\\partial\nx^{\\mu}} \\left(\\sqrt{-g}g^{\\mu\\nu}\\frac{\\partial\\psi}{\\partial\nx^{\\nu}}\\right)=0.\n\\end{eqnarray}\nAfter expressing the normal mode solution as \\cite{unruh,D-R}\n\\begin{eqnarray}\n\\psi_{\\omega lm}=\\frac{1}{R(r)}\\chi_{\\omega\nl}(r)Y_{lm}(\\theta,\\varphi)e^{-i\\omega t},\n\\end{eqnarray}\nwe can easily get the radial equation\n\\begin{eqnarray}\\label{radial equation}\n\\frac{d^{2}\\chi_{\\omega\nl}}{dr_{*}^{2}}+[\\omega^{2}-V(r)]\\chi_{\\omega l}=0,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nV(r)=\\frac{\\sqrt{f(r)h(r)}}{R(r)}\\frac{d}{dr}\\left[\\sqrt{f(r)h(r)}\\frac{d\nR(r)}{dr}\\right]+\\frac{l(l+1)f(r)}{R^{2}(r)},\n\\end{eqnarray}\nwhere $Y_{lm}(\\theta,\\varphi)$ is a scalar spherical harmonic on the\nunit twosphere. Solving Eq. (\\ref{radial equation}) near the event\nhorizon, we obtain the incoming wave function which is analytic\neverywhere in the spacetime manifold \\cite{D-R}\n\\begin{eqnarray}\n\\psi_{in,\\omega lm}=e^{-i\\omega v}Y_{lm}(\\theta,\\varphi),\n\\end{eqnarray}\nand the outgoing wave functions for the inside and outside region of\nthe event horizon\n\\begin{eqnarray}\\label{inside mode}\n\\psi_{out,\\omega lm}(r<r_{+})=e^{i\\omega u}Y_{lm}(\\theta,\\varphi),\n\\end{eqnarray}\n\\begin{eqnarray}\\label{outside mode}\n\\psi_{out,\\omega lm}(r>r_{+})=e^{-i\\omega u}Y_{lm}(\\theta,\\varphi),\n\\end{eqnarray}\nwhere $v=t+r_{*}$ and $u=t-r_{*}$. Eqs. (\\ref{inside mode}) and\n(\\ref{outside mode}) are analytic inside and outside the event\nhorizon respectively, so they form a complete orthogonal family. In\nsecond-quantizing the field $\\Phi_{out}$ in the exterior of the\nblack hole we can expand it as follows \\cite{unruh}\n\\begin{eqnarray}\\label{First expand}\n&&\\Phi_{out}=\\sum_{lm}\\int d\\omega[b_{in,\\omega lm}\\psi_{out,\\omega\nlm}(r<r_{+})+b^{\\dag}_{in,\\omega lm}\\psi^{*}_{out,\\omega\nlm}(r<r_{+})\\nonumber\\\\\n&& \\quad \\quad \\quad  \\quad \\quad \\quad +b_{out,\\omega\nlm}\\psi_{out,\\omega lm}(r>r_{+})+b^{\\dag}_{out,\\omega\nlm}\\psi^{*}_{out,\\omega lm}(r>r_{+})],\n\\end{eqnarray}\nwhere $b_{in,\\omega lm}$ and $b^{\\dag}_{in,\\omega lm}$ are the\nannihilation and creation operators acting on the vacuum of the\ninterior region of the black hole, and $b_{out,\\omega lm}$ and\n$b^{\\dag}_{out,\\omega lm}$ are the annihilation and creation\noperators acting on the vacuum of the exterior region respectively.\nThus, the Fock vacuum state can be defined as\n\\begin{eqnarray}\\label{dilaton vacuum}\nb_{in,\\omega lm}|0\\rangle_{in}=b_{out,\\omega lm}|0\\rangle_{out}=0.\n\\end{eqnarray}\n\nIntroducing the generalized light-like Kruskal coordinates\n\\cite{D-R,Sannan,Zhao,Birrell}\n\\begin{eqnarray}\n&&U=-\\frac{1}{\\kappa}e^{-\\kappa u},\\quad V=\\frac{1}{\\kappa}e^{\\kappa\nv},\\quad {\\rm if\\quad r>r_{+}};\\nonumber\\\\\n&&U=\\frac{1}{\\kappa}e^{-\\kappa u},\\quad V=\\frac{1}{\\kappa}e^{\\kappa\nv}, \\quad {\\rm if\\quad r<r_{+}},\n\\end{eqnarray}\nand noticing that near the event horizon\n\\begin{eqnarray}\nr_{*}\\simeq\\frac{1}{\\sqrt{f'(r_{+})h'(r_{+})}}\\ln(r-r_{+}),\n\\end{eqnarray}\nwe can obtain a complete basis of the outgoing modes according to\nthe suggestion of Damour-Ruffini \\cite{D-R}\n\\begin{eqnarray}\n\\psi_{I,\\omega lm}=e^{\\frac{\\pi\\omega}{2\\kappa}}\\psi_{out,\\omega\nlm}(r>r_{+})+e^{-\\frac{\\pi\\omega}{2\\kappa}}\\psi^{*}_{out,\\omega\nlm}(r<r_{+}),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\psi_{II,\\omega\nlm}=e^{-\\frac{\\pi\\omega}{2\\kappa}}\\psi^{*}_{out,\\omega\nlm}(r>r_{+})+e^{\\frac{\\pi\\omega}{2\\kappa}}\\psi_{out,\\omega\nlm}(r<r_{+}).\n\\end{eqnarray}\nThus, we can also quantize the quantum field $\\Phi_{out}$ in terms\nof $\\psi_{I,\\omega lm}$ and $\\psi_{II,\\omega lm}$ in the Kruskal\nspacetime as\n\\begin{eqnarray}\\label{Second expand}\n\\Phi_{out}=\\sum_{lm}\\int\nd\\omega&[2&\\sinh(\\pi\\omega\/\\kappa)]^{-1\/2}[a_{out,\\omega\nlm}\\psi_{I,\\omega lm}+a^{\\dag}_{out,\\omega lm}\\psi^{*}_{I,\\omega\nlm}\\nonumber \\\\ &+&a_{in,\\omega lm}\\psi_{II,\\omega\nlm}+a^{\\dag}_{in,\\omega lm}\\psi^{*}_{II,\\omega lm}] ,\n\\end{eqnarray}\nwhere the annihilation operator $a_{out,\\omega lm}$ can be used to\ndefine the Kruskal vacuum outside the event horizon\n\\begin{eqnarray}\\label{Kruskal vacuum}\na_{out,\\omega lm}|0\\rangle_{K}=0.\n\\end{eqnarray}\n\nAccording to Eqs. (\\ref{First expand}) and (\\ref{Second expand}), we\nobtain the Bogoliubov transformations \\cite{Birrell,Barnett} for the\nparticle creation and annihilation operators in the black hole and\nKruskal spacetimes\n\\begin{eqnarray}\n&&a_{out,\\omega lm}=\\frac{b_{out,\\omega\nlm}}{\\sqrt{1-e^{-2\\pi\\omega\/\\kappa}}}-\\frac{b^{\\dag}_{in,\\omega lm}}{\\sqrt{e^{2\\pi\\omega\/\\kappa}-1}}, \\nonumber\\\\\n&&a^{\\dag}_{out,\\omega lm}=\\frac{b^{\\dag}_{out,\\omega\nlm}}{\\sqrt{1-e^{-2\\pi\\omega\/\\kappa}}}-\\frac{b_{in,\\omega\nlm}}{\\sqrt{e^{2\\pi\\omega\/\\kappa}-1}}.\n\\end{eqnarray}\nWe assume that the Kruskal vacuum $|0\\rangle_{K}$ is related to the\nvacuum of the black hole $|0\\rangle_{in}\\otimes|0\\rangle_{out}$ by\n\\begin{eqnarray}\\label{two vacuum}\n|0\\rangle_{K}=\\Upsilon(b_{in,\\omega lm},b^{\\dag}_{in,\\omega\nlm},b_{out,\\omega lm},b^{\\dag}_{out,\\omega\nlm})|0\\rangle_{in}\\otimes|0\\rangle_{out}.\n\\end{eqnarray}\nFrom $[b_{in,\\omega lm},b^{\\dag}_{in,\\omega lm}]=[b_{out,\\omega\nlm},b^{\\dag}_{out,\\omega lm}]=1$ and Eq. (\\ref{Kruskal vacuum}), we\nget \\cite{unruh,Ahn}\n\\begin{eqnarray}\n\\Upsilon\\propto \\exp({b^{\\dag}_{out,\\omega lm}b^{\\dag}_{in,\\omega\nlm}}~e^{-\\pi\\omega\/\\kappa}).\n\\end{eqnarray}\nAfter properly normalizing the state vector, we obtain the Kruskal\nvacuum which is a maximally entangled two-mode squeezed state\n\\cite{Barnett,Ahn}\n\\begin{eqnarray}\\label{Scalar-vacuum}\n|0\\rangle_{K}=\\sqrt{1-e^{-2\\pi\\omega\/\\kappa}}\n\\sum_{n=0}^{\\infty}e^{-n\\pi\\omega\/\\kappa}|n\\rangle_{in}\\otimes|n\\rangle_{out},\n\\end{eqnarray}\nand the first excited state\n\\begin{eqnarray}\\label{Scalar-excited}\n&&|1\\rangle_{K}=a^{\\dag}_{out,\\omega lm}|0\\rangle_{K}\\nonumber\\\\\n&&=(1-e^{-2\\pi\\omega\/\\kappa})\\sum_{n=0}^{\\infty}\n\\sqrt{n+1}~e^{-n\\pi\\omega\/\\kappa}|n\\rangle_{in}\\otimes|n+1\\rangle_{out},\n\\end{eqnarray}\nwhere $\\{|n\\rangle_{in}\\}$ and $\\{|n\\rangle_{out}\\}$ are the\northonormal bases for the inside and outside region of the event\nhorizon respectively. For the observer outside the black hole, he\nneeds to trace over the modes in the interior region since he has no\naccess to the information in this causally disconnected region.\nTherefore, when an outside observer travels through the Kruskal\nparticle vacuum $|0\\rangle_{K}$ of mode $\\omega$ his detector\nregisters a number of particles given by\n\\begin{eqnarray}\\label{Hawking}\n_{K}\\langle0|b^{\\dag}_{out,\\omega lm}b_{out,\\omega\nlm}|0\\rangle_{K}=\\frac{1}{e^{2\\pi\\omega\/\\kappa}-1}=\\frac{1}{e^{\\omega\/T}-1},\n\\end{eqnarray}\nwhere we have defined the Hawking temperature as\n\\cite{Kerner-Mann,Jiang-Wu-Cai}\n\\begin{eqnarray}\\label{Hawking temperature}\nT=\\frac{\\kappa}{2\\pi}=\\frac{\\sqrt{f'(r_{+})h'(r_{+})}}{4\\pi}.\n\\end{eqnarray}\nEq. (\\ref{Hawking}) is well known as the Hawking effect\n\\cite{Hawking-1,Hawking-2,Hawking-3}, which shows that the observer\nin the exterior of the black hole detects a thermal Bose-Einstein\ndistribution of particles as he traverses the Kruskal vacuum.\n\n\\section{Quantum Entanglement in background of a black hole}\n\nNow we assume that Alice has a detector which only detects mode\n$|n\\rangle_{A}$ and Bob has a detector sensitive only to mode\n$|n\\rangle_{B}$, and they share a generically entangled state at the\nsame initial point in flat Minkowski spacetime before the black hole\nis formed\n\\begin{eqnarray}\\label{initial}\n|\\Psi\\rangle=\\alpha|0\\rangle_{A}|0\\rangle_{B}\n+\\sqrt{1-\\alpha^{2}}|1\\rangle_{A}|1\\rangle_{B},\n\\end{eqnarray}\nwhere $\\alpha$ is some real number which satisfies\n$|\\alpha|\\in(0,1)$, $\\alpha$ and $\\sqrt{1-\\alpha^{2}}$ are the\nso-called ``normalized partners\". After the coincidence of Alice and\nBob, Alice remains at the asymptotically flat region but Bob freely\nfalls in toward the mass with his detector and then hovers outside\nof it before it collapses to form a black hole. Obviously, there\nwill be some thermal effects due to changes in Bob's acceleration,\nbut there will be no Hawking radiation. Note that such effects can\nbe negligible, or at least will disperse after some time. Then, once\nBob is safely hovering outside of the object at some constant\nacceleration, let it collapse to form a black hole. By Birkhoff's\ntheorem \\cite{Birkhoff} this won't change the metric outside of the\nblack hole and therefore won't change Bob's acceleration. Thus,\nBob's detector registers only thermally excited particles due to the\nHawking effect \\cite{unruh-1,unruh-2}. The states corresponding to\nmode $|n\\rangle_{B}$ must be specified in the coordinates of the\nblack hole in order to describe what Bob sees in this curved\nspacetime. Thus, using Eqs. (\\ref{Scalar-vacuum}) and\n(\\ref{Scalar-excited}), we can rewrite Eq. (\\ref{initial}) in terms\nof Minkowski modes for Alice and black hole modes for Bob. Since Bob\nis causally disconnected from the interior region of the black hole,\nwe will take the trace over the states in this region and obtain the\nmixed density matrix between Alice and Bob in exterior region\n\\begin{eqnarray}\\label{Bos-density}\n&&\\rho_{AB}=(1-e^{-\\omega\/T})\\sum_{n=0}^{\\infty}\\rho_{n}~e^{-n\\omega\/T},\n\\nonumber\\\\&& \\rho_{n}=\\alpha^{2}|0n\\rangle\\langle0n|\n+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})|1(n+1)\\rangle\\langle1(n+1)|\n\\nonumber\\\\&&\\qquad+\\alpha\\sqrt{(n+1)(1-\\alpha^{2})\n(1-e^{-\\omega\/T})}|0n\\rangle\\langle1(n+1)|\\nonumber\\\\&&\\qquad+\n\\alpha\\sqrt{(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})}\n|1(n+1)\\rangle\\langle0n|,\n\\end{eqnarray}\nwhere $|nm\\rangle=|n\\rangle_{A}|m\\rangle_{B,out}$.\n\nIt should be noted that we do not trace over the states located\ninside the event horizon for Alice, even though she now is also\ncausally disconnected from the interior region of the black hole. As\na matter of fact, the causal structure of spacetime keeps every\nobserver exterior from the black hole disconnected from its\ninterior. Why do we not trace over the degree of freedom of Alice in\nthe region inaccessible to her? We now present the reasons as\nfollows. On the one hand, we can justify what we are doing by\ntheory. For a Schwarzschild black hole, the mass of which is assumed\nto be of the order of a solar mass ($M_{bh}\\sim M_{\\odot}$), the\nmagnitude of acceleration near this black hole that Bob needs is\nabout $10^{13}m\/s^{2}$, which is much larger than that of Alice\nneeds (almost equals to zero) in the asymptotical region. Thus, we\nargue that Alice's acceleration effects can be neglected, whereas\nBob's can't. On the other hand, we can think about it from\nexperiment. Though we do not observe the black hole directly,\nimpressive progress in optical, radio and $X$-ray astronomy greatly\nbolster the evidence for supermassive black holes in the centers of\ngalaxies \\cite{Frolov-Novikov}. Thus, the Earth can be argued to be\nan asymptotical region far from black holes, as far as we know. The\nstandard quantum field theory works fine for the earthbound\nexperiments, so we have at least some circumstantial empirical\nevidence that tracing over the black hole interior can be neglected\nin asymptotical regions.\n\nIt is clear that the partial transpose criterion provides a\nsufficient condition for the existence of entanglement in this case\n\\cite{peres}: if at least one eigenvalue of the partial transpose of\nthe density matrix is negative, the density matrix is entangled; but\na state with positive partial transpose can still be entangled. It\nis well-known bound or nondistillable entanglement\n\\cite{Vidal,Plenio}. Interchanging Alice's qubits\n$(|mn\\rangle\\langle pq| \\to |pn\\rangle\\langle mq|)$, we get the\nmatrix representation of the partial transpose in the ($n$,~$n+1$)\nblock {\\small\n\\begin{eqnarray}\n(\\rho_{AB}^{T_{A}})_{n,n+1}&=&e^{-n\\omega\/T}(1-e^{-\\omega\/T})\n\\nonumber \\\\ && \\times \\left(\n\\begin{array}{ll}\n n(1-\\alpha^{2})(e^{\\omega\/T}-1)\n & \\alpha\\sqrt{(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})} \\\\ \\\\\n \\alpha\\sqrt{(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})}\n & ~~~~~~~~\\alpha^{2}e^{-\\omega\/T}\\\\\n\\end{array}\\right),\n\\end{eqnarray}}\nand its eigenvalues\n\\begin{eqnarray}\n\\lambda _{\\pm }^{n}=\\frac{e^{-n\\omega\/T}(1-e^{-\\omega\/T})}{2}\n\\left[\\zeta_{n}\\pm\\sqrt{\\zeta_{n}^{2}\n+4\\alpha^{2}(1-\\alpha^{2})(1-e^{-\\omega\/T})}~\\right],\n\\end{eqnarray}\nwhere\n$\\zeta_{n}=\\alpha^{2}e^{-\\omega\/T}+n(1-\\alpha^{2})(e^{\\omega\/T}-1)$.\nObviously the eigenvalue $\\lambda_{-}^{n}$ is always negative for\nfinite value of the Hawking temperature. Hence, this mixed state is\nalways entangled for any finite value of $T$. It should be noted\nthat in the limit $T\\rightarrow\\infty$, the negative eigenvalue will\ngo to zero. In order to discuss this further, we will use the\nlogarithmic negativity which serves as an upper bound on the\nentanglement of distillation \\cite{Vidal,Plenio}. This entanglement\nmonotone is defined as $N(\\rho_{AB})=\\log\n_{2}||\\rho_{AB}^{T_{A}}||$, where $||\\rho_{AB}^{T_{A}}||$ is the\ntrace norm of the partial transpose $\\rho_{AB}^{T_{A}}$. Thus, we\nobtain the logarithmic negativity for this case\n\\begin{eqnarray}\n&&N(\\rho_{AB})=\\log_{2}\\left[\\alpha^{2}(1-e^{-\\omega\/T})\n+\\sum_{n=0}^{\\infty}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\n\\sqrt{\\zeta_{n}^{2} +4\\alpha^{2}\n(1-\\alpha^{2})(1-e^{-\\omega\/T})}\\right].\\nonumber \\\\\n\\end{eqnarray}\nThe trajectories of the logarithmic negativity $N(\\rho_{AB})$ versus\n$T$ for different $\\alpha$ in Fig. \\ref{NegBos} just show how the\nHawking temperature $T$ would change the properties of the\nentanglement.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.8]{NegBos}\n\\caption{\\label{NegBos}The logarithmic negativity as a function of\nthe Hawking temperature $T$ with the fixed $\\omega$ for different\n$\\alpha$.}\n\\end{figure}\n\nFor the Hawking temperature of zero, corresponding to the case of a\nsupermasive or an almost extreme black hole,\n$N(\\rho_{AB})=\\log_{2}(1+2|\\alpha|\\sqrt{1-\\alpha^{2}})$. In the\nrange $0<|\\alpha|\\leq1\/\\sqrt{2}$ the larger $\\alpha$, the stronger\nthe ``initial entanglement\"; but in the range\n$1\/\\sqrt{2}\\leq|\\alpha|<1$ the larger $\\alpha$, the weaker the\n``initial entanglement\". For finite Hawking temperature, the\nmonotonous decrease of $N(\\rho_{AB})$ with increasing $T$ for five\ndifferent $\\alpha$ means that the ``initial entanglement\" is lost to\nthe thermal fields generated by the Hawking effect. This result\nagrees well with the Hawking's original argument\n\\cite{Hawking-1,Hawking-2,Hawking-3}, which says that smaller black\nholes are at a higher temperature and so radiate more violently than\nmassive black holes. Fig. \\ref{NegBos} also shows that when the\n``initial entanglement\" is stronger, we lose it more rapidly. But it\nis surprisingly found that the same ``initial entanglement\" for\n$\\alpha$ and its ``normalized partner\" $\\sqrt{1-\\alpha^{2}}$ will be\ndegraded along two different curves except for the maximally\nentangled state, i.e., $|\\alpha|=1\/\\sqrt{2}$. This phenomenon, due\nto the coupling of $\\alpha$ and the exponential functions related to\n$T$, just shows the inequivalence of the quantization for a scalar\nfield in the black hole and Kruskal spacetimes. The logarithmic\nnegativity is exactly zero for any $\\alpha$ in the limit\n$T\\rightarrow\\infty$, which indicates that the state has no longer\ndistillable entanglement for the arbitrary values of $\\alpha$ when\nthe black hole evaporates completely.\n\nIn order to estimate the total amount of correlations between Alice\nand Bob, we will analyze the mutual information which is defined as\n\\cite{RAM}\n\\begin{eqnarray}\nI(\\rho_{AB})=S(\\rho_{A})+S(\\rho_{B})-S(\\rho_{AB}),\n\\end{eqnarray}\nwhere $S(\\rho )=-\\text{Tr}(\\rho \\log_{2}\\rho)$ is the entropy of the\ndensity matrix $\\rho$. From Eq. (\\ref{Bos-density}), we can give the\nentropy of this joint state\n\\begin{eqnarray}\n&&S(\\rho _{AB})=-\\sum_{n=0}^{\\infty }\ne^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})\\right]\n\\nonumber\\\\\n&&\\qquad\\qquad~\\times\n\\log_{2}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})\\right].\n\\end{eqnarray}\nTracing over Alice's states for the density matrix $\\rho_{AB}$, we\nget Bob's density matrix in exterior region of the event horizon\n\\begin{eqnarray}\n&&\\rho_{B}=(1-e^{-\\omega\/T})\\sum_{n=0}^{\\infty}e^{-n\\omega\/T}\\left[\\alpha^{2}|n\\rangle\\langle\nn|+(n+1)(1-\\alpha^{2})(1-e^{-\\omega\/T})|n+1\\rangle\\langle\nn+1|\\right],\n\\end{eqnarray}\nand its entropy\n\\begin{eqnarray}\n&&S(\\rho_{B})=-\\sum_{n=0}^{\\infty\n}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+n(1-\\alpha^{2})(e^{\\omega\/T}-1)\\right]\n\\nonumber\\\\\n&&\\qquad\\qquad\\times\n\\log_{2}e^{-n\\omega\/T}(1-e^{-\\omega\/T})\\left[\\alpha^{2}+n(1-\\alpha^{2})(e^{\\omega\/T}-1)\\right].\n\\end{eqnarray}\nWe can also obtain Alice's density matrix by tracing over Bob's\nstates\n\\begin{eqnarray}\\label{Alice-density}\n&&\\rho_{A}=\\alpha^{2}|0\\rangle\\langle0|\n+(1-\\alpha^{2})|1\\rangle\\langle1|,\n\\end{eqnarray}\nwhose entropy can be expressed as\n\\begin{eqnarray}\\label{Alice-entropy}\n&&S(\\rho_{A})=-[\\alpha^{2}\\log_{2}\\alpha^{2}\n+(1-\\alpha^{2})\\log_{2}(1-\\alpha^{2})].\n\\end{eqnarray}\nThus, we draw the behaviors of the mutual information $I(\\rho_{AB})$\nas a function of the Hawking temperature $T$ for different values of\nthe state parameter $\\alpha$ in Fig. \\ref{MuInBos}.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.8]{MuInBos}\n\\caption{\\label{MuInBos}The mutual information as a function of the\nHawking temperature $T$ with the fixed $\\omega$ for different\n$\\alpha$.}\n\\end{figure}\n\nFig. \\ref{MuInBos} shows that for the Hawking temperature of zero,\nthe ``initially mutual information\" equals to\n\\begin{eqnarray}\\label{Boc-MuIn-Initial}\nI_{i}(\\rho _{AB})=-2[\\alpha^{2}\\log_{2}\\alpha^{2}\n+(1-\\alpha^{2})\\log_{2}(1-\\alpha^{2})].\n\\end{eqnarray}\nIn the range $0<|\\alpha|\\leq1\/\\sqrt{2}$ the larger $\\alpha$, the\nstronger $I_{i}(\\rho _{AB})$; but in the range\n$1\/\\sqrt{2}\\leq|\\alpha|<1$ the larger $\\alpha$, the weaker\n$I_{i}(\\rho _{AB})$. As the Hawking temperature increases, the\nmutual information becomes smaller. It is interesting to note that\nexcept for the maximally entangled state, the same ``initially\nmutual information\"  for $\\alpha$ and $\\sqrt{1-\\alpha^{2}}$ will be\ndegraded along two different trajectories. However, in the infinite\nHawking temperature limit $T\\rightarrow\\infty$, i.e., the black hole\nevaporates completely, the mutual information converges to the same\nvalue again\n\\begin{eqnarray}\\label{Boc-MuIn-Final}\nI_{f}(\\rho _{AB})=-[\\alpha^{2}\\log_{2}\\alpha^{2}\n+(1-\\alpha^{2})\\log_{2}(1-\\alpha^{2})],\n\\end{eqnarray}\nwhich equals to just half of $I_{i}(\\rho _{AB})$. Thus, we conclude\nthat\n\\begin{eqnarray}\n&&I_{f}(\\rho _{AB})=\\frac{1}{2}I_{i}(\\rho _{AB}),\n\\end{eqnarray}\nwhich is independent of the state parameter $\\alpha$. Obviously if\n$I_{i}(\\rho _{AB})$ is higher, it is degraded to a higher degree in\nthis limit. Since the distillable entanglement in the infinite\nHawking temperature limit is zero, we are safe to say that the total\ncorrelations consist of classical correlations plus bound\nentanglement in this limit.\n\n\\section{Quantum Teleportation in background of a black hole}\n\nIn this section we will concentrate on a particular quantum\ninformation task: quantum teleportation. We assume that Alice and\nBob each hold an optical cavity, at rest in their local frame. Each\ncavity supports two orthogonal modes (labeled by $A_{i}$ and $B_{i}$\nwith $i=1,~2$), with the same frequency, which are each excited to a\nsingle photon Fock state at the coincidence point for Alice and Bob.\nWe ignore the polarization of these modes and model the photons by\nthe massless modes of a scalar field as suggested by Refs.\n\\cite{Alsing-Milburn,Alsing-McMahon-Milburn}. Considering the\ntextbook teleportation protocol \\cite{Nielsen}, we let Alice and Bob\nshare a maximally entangled state, i.e., an entangled Bell state in\nflat Minkowski spacetime\n\\begin{eqnarray}\n|\\Psi\\rangle=\\frac{1}{\\sqrt{2}}\\left(|\\mathbf{0}\\rangle_{A}|\\mathbf{0}\\rangle_{B}\n+|\\mathbf{1}\\rangle_{A}|\\mathbf{1}\\rangle_{B}\\right),\n\\end{eqnarray}\nwhere the logical states $|\\mathbf{0}\\rangle_{A}$ and\n$|\\mathbf{1}\\rangle_{A}$ are defined in terms of the physical Fock\nstates for Alice's cavity by the dual-rail basis states\n\\cite{Alsing-Milburn,Alsing-McMahon-Milburn}\n\\begin{eqnarray}\n|\\mathbf{0}\\rangle_{A}=|1\\rangle_{A_{1}}|0\\rangle_{A_{2}},~~\n|\\mathbf{1}\\rangle_{A}=|0\\rangle_{A_{1}}|1\\rangle_{A_{2}},\n\\end{eqnarray}\nwith similar expressions for Bob's cavity. It should be noted that\n$|1\\rangle_{A_{1}}$ and $|1\\rangle_{A_{2}}$ are single photon\nexcitations of the Minkowski vacuum states in Alice's cavity. Our\nconstruction implicitly assumes that we have chosen a modal\ndecomposition of the Minkowski vacuum based on intra-cavity and\nextra-cavity modes, which is a legitimate alternative to the usual\nway of quantizing the vacuum in terms of plane wave modes\n\\cite{Dalton-Knight-1,Dalton-Knight-2}. Once the cavities are loaded\nwith a photon, we also assume each cavity is perfect and cannot emit\nthe photon.\n\nRecalling the usual teleportation protocol with the unknown state\n\\cite{Nielsen}\n\\begin{eqnarray}\\label{teleportation}\n|\\varphi\\rangle=a|\\mathbf{0}\\rangle+b|\\mathbf{1}\\rangle,\n\\end{eqnarray}\nwe assume that Alice has an additional cavity which contains this\nsingle qubit (\\ref{teleportation}) with dual-rail encoding by a\nphoton excitation of a two-mode Minkowski vacuum state. This will\nallow Alice to make a joint measurement on the two orthogonal modes\nof each cavity. For the usual teleportation protocol between two\nMinkowski observers Alice and Bob, after Alice's measurement, Bob's\nstate will be projected according to the measurement outcome. We can\ngive the final state received by Bob\n\\begin{eqnarray}\n|\\varphi_{ij}\\rangle=x_{ij}|\\mathbf{0}\\rangle+y_{ij}|\\mathbf{1}\\rangle,\n\\end{eqnarray}\nwith four possible conditional state amplitudes\n$(x_{00},y_{00})=(a,b)$, $(x_{01},y_{01})=(b,a)$,\n$(x_{10},y_{10})=(a,-b)$ and $(x_{11},y_{11})=(-b,a)$. Once\nreceiving the classical information of the result of Alice's\nmeasurement, Bob can apply a unitary transformation to verify the\nprotocol. Obviously the fidelity of the teleported state is unity in\nthis idealized situation.\n\nAlice now wishes to perform the same teleportation protocol with the\nnoninertial observer Bob. We assume that prior to their coincidence,\nAlice and Bob ensure that all photons are removed from their\ncavities. When Alice and Bob instantaneously share a maximally\nentangled state at the asymptotically flat region, we suppose that\nthe two cavities overlap and simultaneously a four photon source\nexcites a two photon state in each cavity. Then Alice remains there\nbut Bob falls in toward the mass and then hovers outside of it. Once\nBob is safely hovering outside of the object at some constant\nacceleration, let it collapse to form a black hole. Then, Bob turns\non his detector after the formation of the black hole. Bob can check\nto see whether any thermal photons have been excited in his local\ncavity using the non-absorbing detector. It should be noted that the\ncommon frequency of both Alice's and Bob's cavity is just the\nfrequency $\\omega$ of Eqs. (\\ref{Scalar-vacuum}) and\n(\\ref{Scalar-excited}) \\cite{Alsing-Milburn,Alsing-McMahon-Milburn}.\nFor Bob, the observer locates near the event horizon of a black\nhole, he needs to trace over the modes in the interior region since\nhe is causally disconnected from this region. Thus, when Alice sends\nthe result of her measurement to Bob, Bob's state can be projected\ninto\n\\begin{eqnarray}\\label{Bob teleport}\n&&\\rho_{ij}=\\sum_{k=0}^{\\infty}\\sum_{l=0}^{\\infty}~_{in}\\langle\nk,l|\\varphi_{ij}\\rangle\\langle\\varphi_{ij}|k,l\\rangle_{in}\\nonumber\\\\\n&&~~~~=(1-e^{-\\omega\/T})^{3}\\sum^{\\infty}_{n=0}\\sum^{n}_{m=0}\\{e^{-(n-1)\\omega\/T}[(n-m)|x_{ij}|^{2}+m\n|y_{ij}|^{2}]|m,n-m \\rangle_{out}\\langle m,n-m|\n\\nonumber\\\\&&\\qquad~~+[\n\\sqrt{(m+1)(n-m+1)}~x_{ij}y^{*}_{ij}e^{-n\\omega\/T}|m,n-m+1\\rangle_{out}\\langle\nm+1,n-m|+{\\rm H.c.}]\\},\\nonumber \\\\\n\\end{eqnarray}\nwhere $|m,n-m \\rangle_{out}=|m\\rangle_{B_{1}}\\otimes|n-m\n\\rangle_{B_{2}}$ is a state of $n$ total excitations in the exterior\nregion product state, with $0\\leq m\\leq n$ excitations in the\nleftmost mode. Eq. (\\ref{Bob teleport}) can be rewritten as\n\\begin{eqnarray}\n&&\\rho_{ij}=\\sum^{\\infty}_{n=0}p_{n}\\rho_{ij,n},\\quad {\\rm\nwith}\\quad p_{0}=0,\\quad\np_{n}=(1-e^{-\\omega\/T})^{3}e^{-(n-1)\\omega\/T}\\quad{\\rm for}\\quad\nn\\geq1.\n\\end{eqnarray}\nSince what we concern about is to which extent\n$|\\varphi_{ij}\\rangle$ might deviate from unitarity, so upon\nreceiving the result $(i,j)$ of Alice's measurement, Bob can apply\nthe rotation operators (a unitary transformation in his local frame)\nrestricted to the one-excitation sector of his state spanned by\n$\\{|\\mathbf{0}\\rangle_{out},|\\mathbf{1}\\rangle_{out}\\}=\\{|0,1\\rangle_{out},|1,0\\rangle_{out}\\}$\nto turn this portion of his density matrix into the exterior region\nanalogue of the state in Eq. (\\ref{teleportation})\n\\cite{Alsing-Milburn,Alsing-McMahon-Milburn,Ge-Shen,Ge-Kim}\n\\begin{eqnarray}\n|\\varphi\\rangle_{out}=a|\\mathbf{0}\\rangle_{out}+b|\\mathbf{1}\\rangle_{out}.\n\\end{eqnarray}\nThus, we can obtain the fidelity of Bob's final state with\n$|\\varphi\\rangle_{out}$\n\\begin{eqnarray}\nF\\equiv{}~_{out}\\langle\\varphi|\\rho_{ij}|\\varphi_{ij}\\rangle_{out}\n=(1-e^{-\\omega\/T})^{3}.\n\\end{eqnarray}\n\nFrom Fig. \\ref{FBosw}, we can see that the fidelity of teleportation\ndepends on the Hawking temperature $T$. It has been found that the\nfidelity decreases as the Hawking temperature increases, which just\nindicates the entanglement degradation obtained in previous section\nbecause the state fidelity in conventional teleportation protocol is\nrelated to the entanglement.\n\n\\begin{figure}[ht]\n\\includegraphics[scale=0.8]{FBosw}\n\\caption{\\label{FBosw}The fidelity of teleportation as a function of\nthe Hawking temperature $T$ with the fixed $\\omega$ for a maximally\nentangled state.}\n\\end{figure}\n\n\\section{summary}\n\nWe have analytically discussed the effect of the Hawking temperature\non the entanglement between two modes of a scalar field as detected\nby Alice who stays stationary at an asymptotically flat region and\nBob who locates near the event horizon in the background of a most\ngeneral, static and asymptotically flat black hole with spherical\nsymmetry. It is shown that the entanglement is degraded by the\nHawking effect with increasing Hawking temperature. It is found that\nthe stronger the ``initial entanglement\" which corresponds to the\nHawking temperature of zero, i.e., the case of a supermasive or an\nalmost extreme black hole, the faster it loses. It is found that the\nsame ``initial entanglement\" for the state parameter $\\alpha$ and\nits ``normalized partners\" $\\sqrt{1-\\alpha^{2}}$ will be degraded\nalong two different trajectories as the Hawking temperature\nincreases except for the maximally entangled state\n$\\alpha=1\/\\sqrt{2}$, which just shows the inequivalence of the\nquantization for a scalar field in the black hole and Kruskal\nspacetimes. In the infinite Hawking temperature limit\n$T\\rightarrow\\infty$, corresponding to the case of the black hole\nevaporating completely, the state has no longer distillable\nentanglement for the arbitrary values of $\\alpha$. Further analysis\nshows that the mutual information is degraded to a nonvanishing\nminimum value which is dependent of $\\alpha$ with increasing Hawking\ntemperature. However, it is interesting to note that the mutual\ninformation in the infinite Hawking temperature limit equals to just\nhalf of the ``initially mutual information\", which is independent of\n$\\alpha$. We have also investigated the scheme of teleportation in\nthis black hole spacetime. It has been demonstrated that the\nfidelity of teleportation decreases as the Hawking temperature\nincreases, which just indicates the entanglement degradation because\nthe state fidelity in conventional teleportation protocol is related\nto the entanglement.\n\n\\begin{acknowledgments}\nThe authors would like to thank the anonymous referee for his\/her\ninsightful and constructive criticisms and suggestions, which\nallowed us to improve the manuscript significantly. This work was\nsupported by the National Natural Science Foundation of China under\nGrant No. 10675045; the FANEDD under Grant No. 200317; and the Hunan\nProvincial Natural Science Foundation of China under Grant No.\n08JJ3010.\n\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThere is no established complete theory of 4-dimensional quantum gravity as yet, and several approaches are being pursued to achieve this goal; among these, spin foam models \\cite{review, alexreview} represent a new promising one in that they seem to be a point of convergence of quite a few lines of research, including loop quantum gravity and simplicial approaches. Among the various models proposed, the Barrett-Crane model \\cite{bc} is the one on which most of the work has focused so far; the basic idea of this model is to describe and quantize gravity, in a simplicial setting, as a constrained topological field theory, where the topological field theory one starts with is the so-called BF theory. The result is a purely combinatorial and algebraic version of a sum-over-histories formulation of the theory in which spacetime is replaced by a combinatorial 2-complex and spacetime geometry is encoded in algebraic data from the representation theory of the Lorentz group. These data are of two types: representation parameters labelling (unitary irreducible) simple, i.e. class I, representations of the group (in the principal series), and vectors in the homogeneous space ${\\cal H}^3 = \\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$; both have a geometric interpretation within the simplicial spacetime one reconstructs from the spin foam (see \\cite{us} and references therein): the first are interpreted as areas of the triangles they label, while the second are thought of as normals to the tetrahedra of the simplicial complex. In this simplicial context, the spacetime behind spin foam models is made out of fundamental building blocks, \\lq\\lq atoms of a quantum spacetime\\rq\\rq, being the 4-simplices of the triangulation, or equivalently, the vertices of the spin foam, and the spin foam model describe how these building blocks interact, how they are coupled, with this coupling determining the details of the overall spacetime geometry. In the specific case of the Barrett-Crane model the coupling between 4-simplices is limited to the first set of algebraic data, i.e. to the representation labels only, in that triangles have the same area in all the 4-simplices they are shared by; on the other hand, there two normal variables for each tetrahedron in the manifold, one for each 4-simplex sharing it, and these two variables are completely un-correlated, i.e. un-coupled. This is not necessarily a problem and in fact it has a nice geometric interpretation (see \\cite{us} and references therein), but it represents a sort of \\lq\\lq ultra-locality\\rq\\rq of the model that one may want to relax (on the ultra-locality of the Barrett-Crane model seen from the point of view of group field theories, see \\cite{generalised}). One reason for this is that one expects this coupling to be affected by and somehow flow under renormalisation of the quantum amplitudes, and one would like to be able to tune the model to different degrees of locality, so to allow for propagating degrees of freedom; in fact it is not clear whether or not these are to be expected in a ultra-local model.\nTherefore, one would like to introduce an additional coupling between 4-simplices with respect to the Barrett-Crane model, and possibly in a tunable way, e.g. introducing an extra parameter in the theory that can be set, when wanted, to a value that gives back the un-coupled Barrett-Crane model, and on whose value one can play to obtain the various levels of locality. This is exactly what we achieve in this paper. The goal goes however beyond the construction of a coupled model: one would like to study the renormalisation of the coupling, as a step towards the difficult task of understanding the renormalisation of spin foam models in full generality, an issue made tricky by the absence of any background structure, as it should be the case for any sensible candidate to a theory of quantum gravity. We believe the model presented here may be of help regarding this more ambitious goal.\nIn this article we work in the framework and language of group field theory \\cite{review, alexreview,laurentgft}. Group field theories are known to be behind any spin foam model we know so far \\cite{carlomike} and to give the most complete formulation of them, in that they allow to remove the dependence on any specific discretization of spacetime obtaining thus a fully background independent model. Moreover, group field theories allow for an easy control over the degrees of freedom and the variables being dealt with in that it provides the basic ingredients of the spin foam model without being tied to any choice of underlying 2-complex, because the provide directly the basic building blocks of it, in the form of propagators and vertex amplitudes forming it as a Feynman diagram of the theory; it is then easier to work in the group field theory context when looking for generalisations of the current models because one sees clearly the structures being generalised even before one sees how the resulting spin foam amplitudes end up being modified. The use of the group field theory formalism descends however also, independently on the specific result one wants to achieve, from the general point of view that group field theories represent not only the most complete formulation of spin foam models, as said, but the most {\\it fundamental} definition of them, i.e. they are {\\it the} theory we are talking about when discussing spin foam models, whose content and significance goes far beyond that of spin foam models alone. This point of view is not that outrageous given that indeed spin foam models arise as Feynman diagrams amplitudes for the corresponding group field theory, and we know that there is much  more in a quantum field theory than its perturbative expansion in Feynman diagrams. However, at present there are too many aspects of group field theories that are not been studied and too many properties of spin foam models whose group field theory origin is not understood for being able to consider this point of view as {\\it the correct one} and not just a possible one. We content ourselves in this paper to show one instance in which working directly at the group field theory level proves advantageous over working at the level of its Feynman amplitudes. Moreover, the results we will present in the end of this work, concerning a new type of group field theory suggest that the group field theory formalism may contain more than usually suspected also with respect to the issue of renormalisation of spin foam models. \n\nThe paper is organised as follows: we start by showing how the {\\it usual} Barrett-Crane model can be obtained by a group field theory in for a field with {\\it five} arguments living in the group manifold, with the extra (with respect to the usual formulations) fifth variable playing the role, in the resulting amplitudes, of the normal to the tetrahedra of the triangulation dual to the Feynman 2-complex, but with the corresponding degrees of freedom not being propagated from vertex to vertex (from 4-simplex to 4-simplex); then in section \\ref{sec:genmodel} we construct a generalisation of this model that provides a non-trivial coupling of these variables across different 4-simplices; we write down the model in full generality, i.e. regardless of any specific choice of coupling function, but we then exhibit a particularly interesting example, with the coupling function being the heat kernel on the group, and thus carrying a tunable coupling parameter, and leading to a model interpolating between the Barrett-Crane model and a strongly coupled flat model, as we are going to discuss; in section \\ref{sec:partBC} we discuss how the mathematical structures involved in this generalised model are relevant also for the insertion of particles in the Barrett-Crane model; finally, in section \\ref{sec:gengft} we propose a further generalisation of the group field theory formalism in which the coupling parameter of the coupled model is promoted to a variable on the same footing as group variables, and in section \\ref{sec: renorm} we discuss at length the issue of renormalisation of spin foam model from the novel perspective provided by the results of this work.             \n\n\\section{Generalized Barrett-Crane model and the coupling of spacetime\natoms} \\label{sec:genBC}\n\\subsection{Cosmetics on the Barrett-Crane model} \\label{sec:cosm}\nBefore we show how it is possible to generalise the Barrett-Crane model to a new model that presents a non-trivial coupling between 4-simplices, we want to first show how the {\\it usual} Barrett-Crane model can be derived from a group field theory action defined for a field living on {\\it five} copies of the Lorentz group, instead of four, with the fifth group element having the interpretation of the normal to the tetrahedron of which the field represents the (second) quantization.\n\nWe start with a complex field $\\varphi(g_i,G)$\\footnote{Here and in the following we use a simplified notation labelling by a single group element $g_i$, indexed by $i$, the four group elements $g_1,g_2,g_3,g_4$ variables of the field} on $\\mathrm{SO}(3,1)^{\\times 5}$; this is our dynamical object\\footnote{The calculations and results of this paper are valid for both $\\mathrm{SO}(3,1)$ and $\\mathrm{SO}(4)$ in the same way, with the non-compactness of the Lorentz group causing only a higher degree of formal complications but no difference in substance; therefore in what follows we always refer explicitly only to $\\mathrm{SO}(3,1)$ because it is physically more interesting, but we use the notation that is customary in the $\\mathrm{SO}(4)$ case instead because it is easier to follow and less cumbersome.}. We define its projection on $(\\mathrm{SO}(3,1)\/\\mathrm{SO}(3))^{\\times 4}\\times \\mathrm{SO}(3,1)$ and expand it into Fourier modes:\n$$\nP_h\\varphi(g_i,G)=\\sum_{J_i A_i}\\varphi^{J_i}_{A_i,B_i}(G)D^{J_i}_{A_iB_i}(g_i)W^{J_i}_{B_i},\n$$\nwhere $\\varphi^{J_i}_{A_i}(G)$ are the Fourier modes of the field and $W^{J_i}_{A_i}$ the $\\mathrm{SO}(3)$-invariant vector in the $J_i$ (simple) representation.\nThe gauge invariant field is defined as:\n\\begin{eqnarray}\n\\phi(g_i,G)\\equiv P_gP_h\\varphi(g_i,G)\n=\\int d\\Lambda\\, P_h\\varphi(g_i\\Lambda,G\\Lambda) \\\\ = \\int d\\Lambda\\, \\sum_{J_i A_i}\\varphi^{J_i}_{A_i}(G\\Lambda)D^{J_i}_{A_iB_i}(g_i\\Lambda)W^{J_i}_{B_i} = \\label{eq:field1} \\\\\n=\\int d\\Lambda\\, \\sum_{J_i A_i}\\varphi^{J_i}_{A_i}(\\Lambda)D^{J_i}_{A_iB_i}(g_iG^{-1}\\Lambda)W^{J_i}_{B_i}.\n\\end{eqnarray}\nThe group variable $\\Lambda$ is then naturally projected down to the hyperboloid ${\\cal H}=\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$.\nThe field is a function of five variables, either five group elements or five representations of the Lorentz group, in the complete mode expansion, so it represents a generalisation of the field on which the Barrett-Crane model is based; this is also clear if one thinks that it decomposes into combinations of 5-valent intertwiners between simple representations. In a sense it can be said that it \\lq\\lq unlocks\\rq\\rq the structure of a quantum tetrahedron, by relaxing the closure constraint and allowing the four simple representations corresponding to the 4 triangles of a tetrahedron to be mapped to a non-trivial one. However, whether and how this is realised depends on what action one defines for this extended field, i.e. on whether or not one chooses to trivialise the dependence on the extra variables. Moreover, we see that for each given value of the $G$ variables, the extra fifth variables of the field are given by the $\\Lambda$s entering the amplitudes as normals to the tetrahedra, as anticipated. The presence of these variables, however, allows for $\\Lambda$s to be coupled in different 4-simplices, as we will see.  \n\nIn fact, using this definition of the field, one can define the action:\n\\begin{eqnarray} \nS[\\varphi]&=& \\frac{1}{2}\\prod_i\\int dg_i \\int dG\\int dG'\\phi(g_i,G)\\phi(g_i,G') \\\\\n&+&\\frac{\\lambda}{5!} \\prod_{i=1}^{10} \\int dg_i\\,\\int dG_A ..dG_E\n\\phi(g_1,g_2,g_3,g_4,G_A)\\phi(g_4,g_5,g_6,g_7,G_B)\\phi(g_7,..,G_C)\n\\phi(g_9,..,G_D)\\phi(g_{10},..,G_E), \\nonumber\n\\end{eqnarray}\nwith capital letters labeling the five group elements associated to the five tetrahedra of a 4-simplex.\nThis is a trivial extension of the group field theory action from which the usual Barrett-Crane model is derived, in the formulation of \\cite{DP-F-K-R}, in the sense that the above action reproduces {\\it exactly} the same amplitudes of the usual Barrett-Crane model, as it is easy to verify. For example, we can show how this is realised for the propagator of the theory. The kinetic term is decomposed in modes by standard methods, starting from expression \\Ref{eq:field1} as:\n\\begin{equation}\n\\prod_i\\int dg_i \\int dG  dG' \\phi(g_i,G)\\phi(g_i,G')= \\int d\\Lambda d\\Lambda' \\int dG dG' \\varphi^{J_i,J}_{A_i,m,n} \\varphi^{J_i,J'}_{A_i,k,l} \\prod_i D^{J_i}_{00}(\\Lambda \\Lambda'^{-1})\\, D^{J}_{mn}(G\\Lambda)\\,D^{J'}_{kl}(G'\\Lambda'),\n\\end{equation}\nwith implicit summation over all indices $J_i, A_i, J,J',m,n,k,l$. It is easy to see that, upon integration of the $G$ variables, the dependence on the variables $\\Lambda$ (those imposing the gauge invariance of the fields) is reduced and the fifth representations $J$,$J'$ are forced to be the singlet, and the usual kinetic term \\cite{DP-F-K-R} is obtained:\n\\begin{equation}\n\\prod_i\\int dg_i \\int dG \\int dG' \\phi(g_i,G)\\phi(g_i,G')= \\sum_{J_i,A_i} \\varphi^{J_i}_{A_i}\\varphi^{J_i}_{A_i}\\int d\\Lambda d\\Lambda' \\prod_i D^{J_i}_{00}(\\Lambda \\Lambda'^{-1}). \n\\end{equation}\nThe same can be verified for the interaction term, that gives the usual Barrett-Crane amplitude for a 4-simplex. In the end the Barrett-Crane model is obtained also in this 5-arguments formalism, with the (once) fifth variable entering in the model as the normal to the tetrahedra in each of the 4-simplices to which they belong \\cite{us}, but being completely decoupled in the different 4-simplices. Note that, had we imposed the projection $P_h$ only in the interaction term and not in the kinetic one, we would have obtained back the Barrett-Crane model in the version given in \\cite{P-R}; so the same relation between the various versions of the BC model is present in this five argument extension as in the usual four argument formulation of the group field theory. In the following we will discuss in parallel what happens when one does or does not impose the simplicity constraints in the kinetic term.\nThe usefulness of this extension of the model lies in the fact that, the dependence on the normal variables having been made explicit in the field theory, it is possible to modify the field theory itself in a very simple way, and thus to introduce a coupling between these variables. This is what we are now going to show.\n\n\\subsection{A generalised model with a non-trivial coupling between 4-simplices} \\label{sec:genmodel}\nWe modify the kinetic term, the one producing the propagator of the theory and thus the edge amplitudes of the spin foam model, in the above extended action by inserting an arbitrary coupling for the new extra variables $G$:\n\\begin{equation}\nS_{\\rm kin}[\\varphi]\\equiv\n\\int dg_i \\int dG \\, dG'\\,\nf(G,G')\\phi(g_i,G)\\phi(g_i,G').\n\\end{equation}\nThus the function $f(G,G')$ encodes in configuration space a non-trivial coupling between the fifth variables of the field, that, as we have seen in the previous section, enter in each 4-simplex (vertex) amplitude as the normals to the tetrahedra in the reference frame associated to each of them. It represents a non-trivial amplitude for the curvature, in the sense that it does not weight equally all the possible values of them, as it does the trivial choice $f(G,G')=1$. The coupling $f$ is naturally of the form $f(G,G')\\equiv F(G'G^{-1})$ where $F$ is a function over $\\mathrm{SO}(3,1)$.\nMoreover, we would like to have a Lorentz invariant coupling such that the correlations be invariant under $\\Lambda,\\Lambda'\\rightarrow g\\Lambda,g\\Lambda'$ for all $g\\in\\mathrm{SO}(3,1)$. This amounts to assuming the coupling function $F$ to be invariant under conjugation: $F(G)=F(g^{-1}Gg), \\,\\forall g$.\n\nWe want now to see what expression this coupling takes in momentum (representation) space; this is necessary when the simplicity constraints are imposed, i.e. in trying to construct a \\lq\\lq coupled extension\\rq\\rq of the version of the Barrett-Crane model presented in \\cite{DP-F-K-R}, because when this is done taking the inverse of the kinetic operator in configuration space is particularly cumbersome, and much simpler in momentum space. Expanding into Fourier modes, we obtain:\n\\begin{equation}\nS_{\\rm kin}[\\varphi]=\n\\int d\\Lambda\\, d\\Lambda'\\, \\sum_{J_i,A_i}\\frac{1}{\\Delta_{J_i}}\n\\varphi^{J_i}_{A_i}(\\Lambda)\\varphi^{J_i}_{A_i}(\\Lambda')\n\\,\\left[\n\\int dG\\,dG'\\, f(G,G') \\prod_{i=1}^4 K^{J_i}(\\Lambda'^{-1}G'G^{-1}\\Lambda)\n\\right],\n\\end{equation}\nwhere $K^{J}(g)=\\bar{W}^J_AD^J_{AB}(g)W^J_B=D^{J}_{00}(g)$ is the homogeneous kernel, the Hadamard propagator of 'mass' $J$ (more precisely, $m^2=J(J+1)$) on the hyperboloid ${\\cal H}$ \\cite{feynman}.\nThen the quadratic coupling in the action is given by a kind of Fourier transform of $F$ where the modes are the \\lq\\lq eye diagram evaluations\\rq\\rq:\n\\begin{equation}\n{\\cal C}^{J_i}(\\Lambda,\\Lambda')\\equiv\n\\int dG\\, F(G) \\prod_{i=1}^4 K^{J_i}(\\Lambda'^{-1}G\\Lambda).\n\\label{transform}\n\\end{equation}\nBecause $F$ is invariant under conjugation, ${\\cal C}^{J_i}$ is simply\na function of $\\Lambda'^{-1}\\Lambda$, so we will denote it more concisely\n${\\cal C}^{J_i}(\\Lambda,\\Lambda')\\equiv {\\cal\n  C}^{J_i}(\\Lambda'^{-1}\\Lambda)$, with:\n$$\n{\\cal C}^{J_i}(g)=\\int dG\\, F(G) \\prod_{i=1}^4 K^{J_i}(Gg).\n$$\n${\\cal C}^{J_i}$ is thus a zonal function, i.e. invariant under both\nthe left and right actions of $\\mathrm{SO}(3)$. Then, under the assumption\nthat it is a $L^2$ function over the hyperboloid, it can be decomposed\ninto Fourier modes:\n$$\n{\\cal C}^{J_i}(g)=\\sum_{J}\\Delta_J{\\cal C}^{J_i}_J K_J(g),\n$$\nwhere ${\\cal C}^{J_i}_J$ are its Fourier components.\n\nLet's point out that the modes of the transform \\Ref{transform} that\nwe introduced above $\\prod_{i=1}^4 K^{J_i}(\\Lambda'^{-1}G\\Lambda)$\nwere already discussed in \\cite{us}. Mathematically, it is the\nevaluation of the eye diagram labeled by the four representations\n$J_i$ on the group element $G$. It can be interpreted as the\nprobability amplitude of the quantum tetrahedron, with triangles\ndefined by the four representations $J_i$, carrying a curvature\ndefined by the parallel transport $G$ between the two 4-simplices\nsharing that tetrahedron.\n\nThe propagator of the theory will be the inverse kernel\n$P(\\Lambda,\\Lambda')$ such that:\n$$\n\\int_{{\\cal H}} d\\Lambda'\\,\n{\\cal P}^{J_i}(\\Lambda,\\Lambda'){\\cal\n  C}^{J_i}(\\Lambda',\\Lambda'')=\\delta_{{\\cal\n    H}}(\\Lambda\\Lambda''^{-1}),\n$$\nor equivalently\n\\begin{equation}\n\\int_{\\mathrm{SO}(3,1)} dg\\,\n{\\cal P}^{J_i}(Gg^{-1}){\\cal C}^{J_i}(g)=\\delta(G).\n\\end{equation}\nThen considering that $\\delta(G)=\\sum_J\\Delta_JK^J(G)$, it is straightforward to check that:\n$$\n{\\cal P}^{J_i}(g)=\\sum_{J}\\Delta_J\\frac{1}{{\\cal C}^{J_i}_J} K_J(g).\n$$\n\nIt is easy to see what changes in the above results if we do not impose the simplicity constraints in the kinetic term of our generalised group field theory action; of course there will still be a non-trivial coupling for the fifth variables of the field, resulting from a non-trivial coupling insertion $f(G,G')$ in configuration space, and the corresponding propagator in configuration space will be given by a product of delta functions for the first four arguments of the field, just as in \\cite{P-R} times a propagator for the fifth pair of variable given simply by $f^{-1}(G,G')$. The four deltas and the extra propagator will be intertwined by a global action of $\\mathrm{SO}(3,1)$; in momentum space this will lead for the kinetic term to an analogue of the operator $\\mathcal{C}^{J_{i}}$, given by:\n\\begin{equation}\n{\\cal C}_{B_i C_i}^{J_i}(g)=\\int dG\\, F(G) \\prod_{i=1}^4 D_{B_i C_i}^{J_i}(Gg).\n\\end{equation}\nwith extra indices $B_i,C_i$ (appearing also in the field modes, resulting from the fact that we do not project anymore on the invariant vectors in the relevant representation spaces. The rest of the construction proceeds analogously, leading to a propagator that carries also these extra indices (of course all the variables are in this case in the group and not in the homogeneous space).   \n\nWhat we have done up to now is completely general, in that it holds for any choice of coupling function $f$, and it gives a generalisation of the Barrett-Crane model with a non-trivial coupling between 4-simplices with respect to the normal vectors variables. \nHowever, we can do more than this: we can exhibit a specific choice of the coupling function satisfying all the properties required, thus providing us with a specific model with many interesting properties, that we are going to highlight. Moreover, this model leads naturally to a further generalisation of the Barrett-Crane model, to be discussed below, that can be the starting point for a new approach to the issue of renormalisation of spin foam models.\n  \nConsider the case in which the simplicity projections $P_h$ are not imposed in the kinetic term, as we assume from now on. The specific model we mentioned arise from choosing the (truncated) heat kernel function as the coupling function $f(G,G')$ in configuration space:\n\\begin{equation}\nf(G,G')=F(g)={\\cal K}^{-\\beta,L}(g)=\\sum_{J\\le L} \\Delta_J e^{\\beta C_J} K^J(g),\n\\end{equation}\nwhere $C_J$ is the quadratic Casimir of the Lorentz group; then the propagator is the inverse heat kernel ${\\cal K}^{\\beta,L}$, i.e. the (truncated) heat kernel function for the opposite value of the coupling $\\beta$, as it is easy to verify.\nConsidering (formally) the heat kernel ${\\cal K}^{-\\beta}(g)$ (untruncated) as coupling\nfunction $f(G,G')$, two limits are particularly interesting for the corresponding propagator ${\\cal K}^{\\beta}(g)$:\n\\begin{itemize}\n\\item $\\beta\\rightarrow \\infty$: the coupling becomes trivial, i.e.\n\\begin{equation}\n{\\cal K}^{\\beta}(g)\\rightarrow Id\n\\qquad \\Rightarrow \\qquad\nC^{J_i}(\\Lambda, \\Lambda')=1,\n\\end{equation}\nand we lose the\n  dependence on $\\Lambda$ and $\\Lambda'$. We recover the usual expression for the Barrett-Crane model in the expansion in Feynman graphs; in this limit the 4-simplices are decoupled as far as the $\\Lambda$ variables are concerned, meaning that the connection, that maps the normal vector to the tetrahedron as seen from the reference frame associated to one 4-simplex to that corresponding to the reference frame associated to the other 4-simplex, is completely arbitrary, and the only coupling is given by the representations assigned to the triangles of the tetrahedron, which are the same in both 4-simplices that share it.\n\\item $\\beta\\rightarrow 0$: the coupling becomes rigid, i.e. ${\\cal K}^{\\beta}(g)=\\delta(g)$ so we are in\n  a strong coupling limit. The propagator constraints\n  $G=G'$, where $G$ and $G'$ are the fifth variables in the two fields corresponding to the same tetrahedron and interacting in the two vertices, being propagated by ${\\cal K}^\\beta(g)$. We call this model the flat model since it\n  amounts to imposing a trivial parallel transport between\n  4-simplices, i.e. constraining the (boost part of the) connection to be flat. \n  Here we stress that we constrain the fifth group variables $G, G'$ to be equal and not the normals $\\Lambda, \\Lambda'$, so the coupling is simply the eye diagram evaluation:\n$$\nC^{J_i}(\\Lambda, \\Lambda')=\\prod_{i=1}^4K^{J_i}(\\Lambda'^{-1}\\Lambda).\n$$\nit would be interesting to understand the explicit relation between this limit of the model and BF theories. On the one hand it would seem that this limit indeed corresponds to a BF theory, since we constrain the curvature to be flat. On the other hand, it is not the BF theory based on the Lorentz group (the Crane-Yetter model), however it could well be a BF theory on the homogeneous space $\\mathrm{SL}(2,\\mathbb{C})\/SU(2)$.\n\\end{itemize}\nBoth these limits can be easily verified both in configuration space and in momentum space, by a straightforward calculation (in this context without simplicity projectors).\n\nNow let us have a look at the interaction term of the group field\ntheory, which defines the vertex amplitude of a spin foam or\nequivalently the 4-simplex amplitude. Let us consider the following\nintegral:\n\\begin{equation}\nI(G_A,..,G_E)\\equiv\n\\int \\prod_{i=1}^{10} dg_i\\,\n\\phi(g_1,g_2,g_3,g_4,G_A)\\phi(g_4,g_5,g_6,g_7,G_B)\\phi(g_7,g_3,g_8,g_9,G_C)\n\\phi(g_9,..,G_D)\\phi(g_{10},..,G_E),\n\\end{equation}\nwhere $i=1,..10$ refers to the triangles of the 4-simplex and  the letters\n$A,..,E$ to its tetrahedra.\nThen it is straightforward to check that:\n\\begin{equation}\nI(G_A,..,G_E)=\n\\int d\\Lambda_A\\, ..\\, d\\Lambda_E\\,\n\\sum_{J_i, A_i} \\frac{1}{\\Delta_{J_i}}\n\\varphi^{J_1..J_4}_{A_1..A_4}(\\Lambda_A)..\\varphi^{J_{10}..J_1}_{A_{10}..A_1}(\\Lambda_E)\n\\,\\prod_{i=1}^{10} K^{J_i}(\\Lambda_{t(i)}^{-1}G_{t(i)}G_{s(i)}^{-1}\\Lambda_{s(i)}),\n\\end{equation}\nwhere we label $s(i)$ and $t(i)$ the source and target vertices of the edge $i$ of the 4-simplex spin network \\cite{F-K}, or equivalently, the two tetrahedra sharing the triangle $i$ of the 4-simplex.\n\n\nIf we take as interaction term $\\int dG_A\\, ..\\,dG_E I(G_A,..,G_E)$,\nthen we erase all information on the normals $\\Lambda_A,..\\Lambda_E$\nand we recover the usual Barrett-Crane evaluation of a 4-simplex, as we have discussed in the previous section. As\nwe would like to keep a non-trivial dependence  of the $\\Lambda$'s, we\nsimply choose:\n\\begin{equation}\nS_{\\rm int}[\\varphi]=I(G_A={\\rm Id},..,G_E={\\rm Id}).\n\\end{equation}\n\nThis way, we keep an extended dependence on the variables $\\Lambda$, that can be coupled by the propagator in each Feynman graph, leading to a spin foam model which differs from the Barrett-Crane model. Note that choosing ${\\rm Id}$ is not essential. Fixing the $G_v$ variables to another value would amount to changing the origin -the reference point- of the hyperboloid for each variable $\\Lambda_v$.\n\nFinally, our group field theory for the field $\\varphi$ is defined by the action\n$S_{\\rm GFT}[\\varphi]\\equiv S_{\\rm kin}[\\varphi]+S_{\\rm int}[\\varphi]$, i.e. by:\n\\begin{eqnarray}\nS_{\\rm GFT}[\\varphi]&\\equiv& \\int dg_i \\int dG \\, dG'\\,\nf(G,G')\\phi(g_i,G)\\phi(g_i,G')  \\nonumber\\\\\n&&+\\frac{\\lambda}{5!}  \\int \\prod_{i=1}^{10} dg_i\\,\n\\phi(g_1,g_2,g_3,g_4,Id)\\phi(g_4,g_5,g_6,g_7,Id)\\phi(g_7,g_3,g_8,g_9,Id)\n\\phi(g_9,..,Id)\\phi(g_{10},..,Id),\n\\end{eqnarray}\nin configuration space, or\n\\begin{eqnarray}\nS_{\\rm GFT}[\\varphi]&\\equiv& \\int d\\Lambda d\\Lambda' \\sum_{J_i,A_i}\\frac{1}{\\Delta_{J_i}}\n\\varphi^{J_i}_{A_i B_i}(\\Lambda)\\varphi^{J_i}_{A_i C_i}(\\Lambda'){\\cal C}_{B_i C_i}^{J_i}(\\Lambda,\\Lambda') \\nonumber\\\\ \n&&+\\frac{\\lambda}{5!} \\int d\\Lambda_A ..\\Lambda_E \\sum_{J_i,A_i} \\varphi^{J_1..J_4}_{A_1..A_4}(\\Lambda_A)...\\varphi^{J_{10}..J_1}_{A_{10}..A_1}(\\Lambda_E) \\prod_i D^{J_i}_{00}(\\Lambda_{s(i)}\\Lambda_{t(i)}^{-1}) \\\\ &=& \\sum_{J_i,A_i}\n\\varphi^{J_i,J}_{A_i,B_i,k,l}\\varphi^{J_i,J'}_{A_i,C_i,m,n}\\left(\\int d\\Lambda d\\Lambda' D^{J}_{kl}(\\Lambda)D^{J'}_{mn}(\\Lambda')\\frac{1}{\\Delta_{J_i}} {\\cal C}_{B_i C_i}^{J_i}(\\Lambda,\\Lambda')\\right) \\nonumber\\\\ &&+\\frac{\\lambda}{5!}  \\sum_{J_i,A_i} \\varphi^{J_1..J_4,J_A}_{A_1..A_4,m_A,n_A}...\\varphi^{J_{10}..J_1,J_E}_{A_{10}..A_1,m_E,n_E} \\left(\\int d\\Lambda_A ..\\Lambda_E D^{J_A}_{m_A n_A}(\\Lambda_A)...D^{J_E}_{m_E n_E}(\\Lambda_E) \\prod_i D^{J_i}_{00}(\\Lambda_{s(i)}\\Lambda_{t(i)}^{-1})\\right)\\nonumber,\n\\end{eqnarray}\nin momentum space, from which it is immediate to read out the vertex amplitude for our Feynman diagrams encoding the interaction of our field $\\varphi$, given in configuration space by:\n\\begin{equation}\n\\mathcal{I}_v (g_{\\mid v}, \\tilde{g}_{f\\mid v}, \\Lambda_{e \\mid v}) = \\prod_{f\\mid v} \\delta(g_{f\\mid v}\\Lambda_{e1}^{-1}\\Lambda_{e2}\\tilde{g}_{f\\mid v}^{-1}) \\label{eq:vertex}\n\\end{equation}\n\nwhile the propagator is given by the expression: \n\n\\begin{equation}\n{\\cal P}(g_i,G; \\tilde{g}_i, G') = \\int d\\Lambda\\int d\\Lambda' \\prod_i \\delta(g_i \\Lambda^{-1},\\tilde{g}_i\\Lambda'^{-1}) \\, f^{-1}(G\\Lambda^{-1},G'\\Lambda'^{-1})\n\\end{equation} \nin configuration space, and by:\n\\begin{equation}\n{\\cal P}(J_i,A_i, B_i,J,mn; \\tilde{J}_i,\\tilde{A}_i, C_i,J',kl)=\\prod_i\\delta_{J_i,\\tilde{J}_i}\\prod_i \\delta_{A_i,\\tilde{A}_i} \\int dGdG' D^{J}_{mn}(G)D^{J'}_{kl}(G') {\\cal P}^{J_i}_{B_i C_i}(G,G')\n\\end{equation}\nin momentum space, with ${\\cal P}^{J_i}$ having been defined above. This is for the general case; for the specific choice of coupling function proposed above, i.e. the heat kernel on the group, the analogous expressions are easily written down: the interaction term is unaltered, while the propagator takes the form:\n\\begin{equation}\n{\\cal P}^\\beta(g_i,G; \\tilde{g}_i, G') = \\int d\\Lambda\\int d\\Lambda' \\prod_i \\delta(g_i \\Lambda^{-1},\\tilde{g}_i\\Lambda'^{-1}) \\, {\\cal K}^{\\beta}(G\\Lambda^{-1},G'\\Lambda'^{-1})\n\\end{equation}\nthat reduces as we had discussed to the usual Barrett-Crane propagator for $\\beta\\rightarrow \\infty$ and gives instead a BF-type product of five deltas if the limit $\\beta\\rightarrow 0$ is taken instead. \n\nAnalogously, one can impose the simplicity projectors in the kinetic term and work fully in the homogeneous space; the construction is not altered substantially, but checking the two limiting properties in $\\beta$ for the resulting Feynman diagrams is much less straightforward, especially in configuration space.   \n\nThe partition function is then defined in terms of its Feynman expansion, as usual, and it is given by:\n\\begin{equation}\nZ(\\lambda,\\beta) = \\int \\mathcal{D}\\varphi\\; e^{- S[\\varphi]} = \\sum_{\\Gamma} \\frac{\\lambda^N}{sym(\\Gamma)} \\,Z_{\\Gamma}[\\beta]\n\\end{equation}\nwhere $N$ is the number of vertices in the graph and $sym$ its order of automorphisms, and with the amplitude for each graph being given by the appropriate convolution of vertex amplitudes and propagators as:\n\n\\begin{equation}\nZ_{\\Gamma}[\\beta]= \\left(\\prod_e \\prod_{f\\mid e} \\int dg_{f\\mid e} \\int d\\tilde{g}_{f\\mid e}\\right) \\left( \\prod_e \\int dG_e \\int dG'_e\\right) \\prod_e {\\cal P}^{\\beta}(g_{f\\mid e},\\tilde{g}_{f\\mid e}, G_e,G'_e) \\prod_v \\mathcal{I}_v(g_{f\\mid v}, \\tilde{g}_{f\\mid v}, G_{e\\mid v})\n\\end{equation}\n \nIt is now easier to understand the structure of the resulting models; in particular, we see how indeed, in the $\\beta\\rightarrow 0$ limit of the coupled model, the fifth variables of the fields, having the interpretation of normals to the tetrahedra in the reference frames defined by each 4-simplex, are forced to be mapped trivially from one 4-simplex to the next. It is indeed a flat model, with a rigid coupling between 4-simplices and similar to a BF theory in that allows {\\it only} such a flat configuration at least for what concerns the extra variables. It is not however a GFT formulation of a true BF theory for a 5-dimensional spacetime, and differs of course also from a true BF theory in 4-dimensions, in that: 1) the first four variables of the field are projected down to the homogeneous space $\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$ while maintaining the gauge invariance under the full Lorentz group; 2) the fifth variables in the interaction term, i.e. in each 4-simplex, are completely decoupled.\n\n\nIt is important to stress that not only the new model gives a simple way of interpolating between the Barrett-Crane model and the flat model by moving in parameter space, i.e. by following a renormalisation group flow, but most importantly perhaps it allows for perturbation analysis around these limiting configurations, i.e. it permits to study the physics of the model near the \\lq decoupled phase', i.e. the ordinary Barrett-Crane model, and near the \\lq strongly coupled phase' instead. Interestingly, one would expect to see propagating \\lq gravitons', i.e. propagating perturbations in the curvature, in the almost flat case, thus in the strongly coupled phase, and \\lq confined gravitons' in the decoupled phase, when perturbing around the usual Barrett-Crane configurations. In the almost flat case it would correspond to a definition of a gravity theory in terms of a perturbative expansion around a flat connection configuration; in both cases it would amount to a fully covariant perturbation expansion, analogous to the one proposed in \\cite{laurent-artem}, and indeed the exact relationship with the model in \\cite{laurent-artem} should be analysed and we leave it for further study. In a renormalisation group interpretation, according to which the $\\beta$-dependent model would possess two physically very different phases, the Barrett-Crane or decoupled phase and the strongly coupled BF-type one, this perturbation expansion would then be an expansion around the two possible fixed points of the renormalisation group flow. \nMoreover, we are going to show in the following that the $\\beta$-dependent model bases on the heat kernel coupling lend itself to a further generalisation which could be the basis for further progress concerning the renormalisation of spin foam models. But before discussing this further generalised model, we would like to show how the five argument formalism for the field theory described above and the associated mathematical structures arise naturally when particles are inserted in the spin foam model.\n\n\n\\subsection{About particle insertions in the Barrett-Crane model} \\label{sec:partBC}\nIn flat spacetime, a basis of one-particle states is given by momentum\neigenstates $\\psi_{p,\\sigma}$, where $p$ is the four-momentum and\n$\\sigma$ the other degrees of freedom - the spin. A Lorentz\ntransformation $U(\\Lambda)$ will map a state of momentum $p$ to a\nstate of momentum $\\Lambda p$. Then we can choose a reference momentum\n$k^\\mu$ and define the states $\\psi_{p,\\sigma}$ from the action of Lorentz\nboosts on the reference state:\n\n$$\n\\psi_{p,\\sigma}=N(p)U(s(p))\\psi_{k,\\sigma},\n$$\nwhere $N(p)$ is a normalization factor and $s(p)$ is a Lorentz\ntransformation mapping $k$ to $p$.\nThe usual normalization is $N(p)=\\sqrt{k_0\/p_0}$ so that the scalar product\nis normalized\n$\\langle\\psi_{p,\\sigma},\\psi_{p,\\sigma}\\rangle=\n\\delta_{\\sigma\\sigma'}\\delta^{(2)}(\\vec{p}-\\vec{p'})$.\nFor massive particles, we usually take $k=(1,0,0,0)$ and the little\ngroup of Lorentz transformations $W$ leaving $k$ invariant is the\nrotation group $\\mathrm{SU}(2)$. Then the action of a Lorentz transformation\nreads:\n$$\nU(\\Lambda)\\psi_{p,\\sigma}=N(p)U(s(\\Lambda p))U(W(\\Lambda,p))\\psi_{k,\\sigma},\n$$\nwith $W(\\Lambda,p)=s(\\Lambda p)^{-1}\\Lambda s(p)$ living in the little group.\nFrom this, we see that it is enough to postulate the action of the little group:\n$$\nU(W)\\psi_{k,\\sigma}=\\sum_{\\sigma'}D_{\\sigma',\\sigma}(W)\\psi_{k,\\sigma'},\n$$\nin order to get the action of an arbitrary Lorentz transformations:\n$$\nU(\\Lambda)\\psi_{p,\\sigma}=\n\\left(\\frac{N(p)}{N(\\Lambda p)}\\right)\n\\sum_{\\sigma'}D_{\\sigma',\\sigma}(W(\\Lambda,p))\\psi_{\\Lambda p,\\sigma'}.\n$$\nThe {\\it spin} $s$ is the choice of a irreducible representation of the little group $\\mathrm{SU}(2)$,\nwhich defines the coefficient $D_{\\sigma',\\sigma}$, and $\\sigma$ is the angular momentum. Then we have induced a representation of the Lorentz group from a $\\mathrm{SU}(2)$ representation.\n\n\\medskip\n\nTo include particles in the Barrett-Crane model, we first look at the\nparticle insertion on a 4-simplex. The one-4-simplex amplitude is\nsimply the evaluation of its boundary spin network. Thus we need to\ninclude the particles in a spin network wave function. Inserting\nparticles at all vertices of the spin network graph, the wave function\nis then a function of $E$ holonomies $U_e\\in\\mathrm{SL}(2,\\mathbb{C})$, describing the\nparallel transport between vertices and carrying the gravitational\ndegrees of freedom, and $V$ group elements\n$\\Lambda_v\\in\\mathrm{SL}(2,\\mathbb{C})\/\\mathrm{SU}(2)$ in the hyperboloid ${\\cal H}$, defining the\nstate of the particles, where $E$ and $V$ are respectively the number\nof edges and vertices of the graph. Considering the transformations of\nholonomies and momenta under gauge transformations, we are looking at\nthe following gauge invariant functions:\n\\begin{equation}\n\\psi(U_1,..,U_E,\\Lambda_1,..,\\Lambda_V)=\n\\psi(g_{s(1)}^{-1}U_1g_{t(1)},..,g_{s(E)}^{-1}U_Eg_{t(E)},g_1\\Lambda_1,..,g_V\\Lambda_V),\n\\qquad\n\\forall g_v\\in(\\mathrm{SL}(2,\\mathbb{C}))^{\\otimes V}.\n\\end{equation}\nSuch wave functions have already been considered in the context of\n(covariant) loop quantum gravity, and in \\cite{proj} a basis for\nthe Hilbert space of such wave functions ($L^2$ invariant functions\nfor the Haar measure on the Lorentz group) is defined in terms of {\\it projected spin\n  networks}. Roughly, these are spin networks labeled with $\\mathrm{SL}(2,\\mathbb{C})$\nrepresentations on every edge, and with $\\mathrm{SU}(2)$ representations on\nevery edge and $\\mathrm{SU}(2)$ intertwiners at every vertex. In the case that\nwe choose the $\\mathrm{SU}(2)$ to be all the trivial scalar representation, we\nfind back the simple spin networks of the standard Barrett-Crane model\n\\cite{proj,us}:\n$$\n\\psi_{J_e}(U_1,..,U_E,\\Lambda_1,..,\\Lambda_V)=\\prod_e K^{J_e}(\\Lambda_{s(e)}^{-1}U_e\\Lambda_{t(e)}),\n$$\nwhere we note $s(e)$ and $t(e)$ the source and target vertices of the edge $e$.\nTo each vertex of the simple spin network is associated the\n$\\Lambda_v$ dependent $\\mathrm{SU}(2)$ intertwiner between the $\\mathrm{SL}(2,\\mathbb{C})$\nrepresentations $R^{J_i}$:\n\\begin{equation}\n\\bigotimes_i \\langle J_i,\\Lambda_v,j=m=0|\\,:\\bigotimes_i R^{J_i}\\,\\rightarrow\\,\\mathbb{C},\n\\label{simple}\n\\end{equation}\nwhere $i$ runs over all the edges meeting at the vertex\n$v$. $|J_i,\\Lambda,j=m=0\\rangle=D^{J_i}(\\Lambda)W_{J_i}$ is the vector in\n$R_{J_i}$ which is invariant under the $\\mathrm{SU}(2)$ subgroup of\n$\\mathrm{SL}(2,\\mathbb{C})$ leaving invariant the direction $\\Lambda\\in{\\cal H}$.\n\nWe recognize in the above structure the basis of states in which we decompose the 4-valent field of the usual group field theory formulation of the Barrett-Crane model, or, equivalently, the basis states arising in the trivial 5-valent extension of the same group field theory, discussed at the beginning. The field is indeed decomposed in 4-valent simple intertwiners of the Lorentz group, i.e. carrying simple representations of the Lorentz group on every link, and trivial $SU(2)$ representations on the same links, and a $\\Lambda$ dependent $SU(2)$ intertwiner at each vertex, with the various $\\Lambda$'s being integrated over at the end when evaluating the spin network amplitude. \n\nThere is another way of building a basis of the space of such\ninvariant wave functions, which makes the link with the particle\ninsertions clearer. In fact the construction is very close to the way spinning particles are coupled in 3d quantum gravity (see \\cite{pr1,pr3,3d} for the construction of coupled spin foam amplitudes, and \\cite{particle, dj} for the group field theory formalism). To each $n$-valent\nvertex of the spin network graph, we associate a $(n+1)$ valent intertwiner ${\\cal I}$\nwhich intertwines the $n$ edges meeting at the vertex and an extra\nedge describing the particle insertion. The particle is then\ncharacterized by a choice of a Lorentz representation $J$ and a vector\n${\\cal V}^J$ in the corresponding Hilbert space $R^J$. At each vertex, the\nintertwiner ${\\cal I}_v$ is a $\\mathrm{SL}(2,\\mathbb{C})$ invariant tensor in $\\otimes_i\nR^{J_i}\\,\\otimes R^J$. The wave functions is then the evaluation of\nthe holonomies on the spin network with the following tensors at each\nvertex:\n\\begin{equation}\n{\\cal I}_v(\\,\\cdot\\, ,D^J(\\Lambda){\\cal V}^J)\n\\,:\\bigotimes_i R^{J_i}\\,\\rightarrow\\,\\mathbb{C}.\n\\label{particle}\n\\end{equation}\nTo describe a particle of spin $s$, we simply choose the vector\n${\\cal V}^J$ to lie in the subspace of $R^{J}$ corresponding to the\n$\\mathrm{SU}(2)$ representation of spin $j\\equiv s$. Therefore, to describe a\nspinless particle, we choose the $\\mathrm{SU}(2)$ invariant vector\n${\\cal V}^J\\equiv W^J$.\n\nWe recognize in this structure the basis of states arising in the mode expansion of our 5-valent field, in the non-trivial extension of the group field theory presented above.\n\n\nThese two basis, projected spin networks and particle insertions,\ncorrespond thus to two choices of Fourier modes for the decomposition of\nthe $\\phi(g_i,G)$ in our group field theory context. It is easy to see\nthat the simple intertwiner \\Ref{simple} corresponds to the sum over\nall possible particle insertions \\Ref{particle} for fixed\nrepresentations $J_i$, i.e. sum over all possible $J$ and\n$(n+1)$-valent intertwiners.\nIndeed:\n\\begin{equation}\n\\bigotimes_i |J_i,\\Lambda, j=m=0\\rangle\n=\\prod_i D^{J_i}_{A_iB_i}(\\Lambda)W^{J_i}_{B_i}\n=\\sum_J\\sum_{{\\cal I}}{\\cal I}^{J_1..J_n,J}_{A_1..A_n,A}D^J_{AB}(\\Lambda)W^J_B,\n\\end{equation}\nwhere we are summing over an orthonormal basis of $(n+1)$ valent intertwiners between the representations $J_1,..,J_n,J$.\nNevertheless, more generally, if we want to include a particle of some\ngiven fixed spin $s$, we will have to go beyond simple spin networks\nand allow arbitrary $\\mathrm{SU}(2)$ representations in the projected spin\nnetwork basis. Here we will not write in details the explicit change\nof basis between the two choices of modes. \nInstead we would like to\nstress again that the generalized group field theory introduced in the\nprevious section with an extra group variable for each field (or\nquantized tetrahedron) would allow to compute transition amplitudes\nfor gravity plus particles in the spin foam formalism.\nThis also means that, on the one hand, as we anticipated, this kind of mathematical structures would probably play a role in any group field theoretic formulation of 4d quantum gravity coupled to point particles, and, on the other hand, that the insertion of point particle in the Barrett-Crane model, or a similar one, would provide the model with a non-trivial coupling between spacetime atoms, i.e. between 4-simplices, possibly of the type we have presented here. \n\n\\section{A new type of group field theory: turning the coupling into a variable} \\label{sec:gengft}\nWe have just seen that the choice of the heat kernel\n${\\cal K}^{\\beta}(g,g')$ for the propagator term, relating the two normals\ncorresponding to the same tetrahedron in two neighboring 4-simplices,\ngenerates a model that interpolates nicely between the usual\nBarrett-Crane model and a strongly coupled model where the only\ngeometric configuration allowed is flat space, with all the normals\nbeing identified. This very same choice leads naturally to a\nfurther generalization of the group field theory and of spin foam\nmodels, that we believe has even more interesting properties and may\nbe a good starting point for further investigations.\n\nConsider any given spin foam (i.e. Feynman graph) amplitude for the\nabove model, with the heat kernel with parameter $\\beta$ on each dual\nedge connecting the two group variables corresponding to the fifth\nvariables of the field, i.e. to the tetrahedron normals. Now\nre-interpret the $\\beta$ as a {\\it variable} of the model, as opposed\nto a freely specified {\\it parameter}. This implies that there is an\nintegral over it, with range from zero to infinity in the definition\nof the amplitudes; for the same reason, the two limits mentioned for\nthis interpolating model correspond now to two possible approximations\none can make for this integral, considering only the region of\nintegration near zero or the asymptotic region at infinity; this last\none describes perturbations around the Barrett-Crane configurations,\nwhile the first corresponds to a strong coupling region. \n\nEven more interesting is the group field theory derivation of this\nextended model, with $\\beta$ treated as a variable. First of all\nnotice that, if $\\beta$ has to be considered as a variable, the\ndefinition of the field has to be extended to a function \n$$ \\phi(g_1,...,g_4;G,\\beta) : SO(3,1)^5\\times \\mathbb{R} \\rightarrow\n\\mathbb{C}, $$ with the global invariance under $SO(3,1)$\ntransformations and the invariance under $SO(3)$ transformations of\nthe first four arguments to be then imposed in the action. \n\nNow, considering just the Feynman amplitudes in configuration space, i.e. not performing any mode expansion, we have now a propagator of the form:\n\\begin{equation}\n{\\cal P}(g_i,g'_i,G,G',\\beta)=\\left(\\prod_i\\delta(g_i,g'_i)\\right) {\\cal K}(G,G';\\beta),\n\\end{equation}\nwhere we switched to a different notation for the heat kernel to\nhighlight the fact that now $\\beta$ has the interpretation of a \\lq\\lq\nEuclidean time \\rq\\rq for a particle living on the homogeneous space\nto which $G$, and $G'$ belong, and the heat kernel function indeed\npropagates it from the point $G'$ at time 0 to the point $G$ at (Euclidean) time\n$\\beta$. We assume that ${\\cal K}(G,G';\\beta)$ is zero for $\\beta<0$, in order to have a well-defined mode expansion for the propagator, so we effectively work with ${\\cal K}^+(G,G';\\beta)=\\theta(\\beta){\\cal K}(G,G';\\beta)$, where $\\theta$ is the Heaviside function. \n\nIt is easy to identify the kinetic term from which this\npropagator comes. Indeed the heat kernel with \\lq\\lq time\\rq\\rq variable $\\beta$ is\nsimply the propagator for the heat equation \n\\begin{equation}\n\\left( -\\frac{\\partial}{\\partial \\beta} + \\nabla \\right) \\psi(g,\\beta) = 0,\n\\end{equation}\nwith $\\nabla$ being the Laplace-Beltrami operator on the group manifold.\nTherefore the appropriate kinetic term of the group field theory action is identified with:\n\n\\begin{equation}\nS_{kin}[\\phi] = \\prod_i \\int dg_i \\int dG\\int d\\beta \\;\\phi(g_1,...,g_4;G,\\beta)\\left[\\left( -\\partial_\\beta + \\nabla_G \\right) \\phi\\right](g_1,...,g_4;G,\\beta).\n\\end{equation}\nand the kinetic operator with:\n\\begin{equation}\nK(g_i,G,\\beta; \\tilde{g}_i,G',\\beta')= \\left( \\prod_i \\delta(g_i,\\tilde{g}_i) \\right) \\left( -\\partial_\\beta + \\nabla_G\\right)\\delta(G,G')\\delta(\\beta,\\beta').\n\\end{equation}\n\nIn the following we will actually use a symmetrized version of the propagator above, so to erase any asymmetry between the two possible sign of the $\\beta$ variable entering in it, so we will work with a propagator of the form:\n\\begin{equation}\n{\\cal P}_H(g_i,g'_i,G,G',\\beta)=\\left(\\prod_i\\delta(g_i,g'_i)\\right) \\left[ \\theta(\\beta){\\cal K}(G,G';\\beta) + \\theta(-\\beta) {\\cal K}(G,G',-\\beta)\\right].\n\\end{equation}\nThis has an interpretation in terms of different transition amplitudes one can define for quantum gravity in a group field theory context \\cite{generalised}.\n\nWe can now consider the interaction term for our generalized group field theory. A first choice is to define the dependence on the extra variable $\\beta$ in each field to be trivialized, i.e. a different $\\beta$ enters in the various fields in the interaction term, but no relation is assumed between them; the interaction is then of the form:\n\n\\begin{eqnarray}\n\\lefteqn{S_{int}^{(1)}[\\phi]=\\int d\\beta_A ...d\\beta_E\\,\\int d\\Lambda_A ...d\\Lambda_E \\int \\prod_{i=1}^{10} dg_i\\prod_i \\int dh_i d\\tilde{h}_i\\,\n\\phi(g_1 h_1\\Lambda_A,g_2 h_2\\Lambda_A,g_3 h_3\\Lambda_A,g_4 h_4\\Lambda_A,\\Lambda_A, \\beta_A)} \\nonumber \\\\ &&\\phi(g_4 \\tilde{h}_4 \\Lambda_B,g_5 h_5\\Lambda_B,g_6 h_6 \\Lambda_B,g_7 h_7 \\Lambda_B,\\Lambda_B, \\beta_B)\\; \\phi(g_7\\tilde{h}_7\\Lambda_C,g_3 \\tilde{h}_3\\Lambda_C,g_8 h_8 \\Lambda_C,g_9 h_9 \\Lambda_C,\\Lambda_C, \\beta_C) \\nonumber\n\\\\ &&\\phi(g_9 \\tilde{h}_9 \\Lambda_D,g_6\\tilde{h}_6\\Lambda_D,g_2\\tilde{h}_2\\Lambda_D,g_{10}h_{10}\\Lambda_D,\\Lambda_D, \\beta_D)\\; \\phi(g_{10}\\tilde{h}_{10}\\Lambda_E,g_8\\tilde{h}_8\\Lambda_E,g_5\\tilde{h}_5\\Lambda_E,g_1\\tilde{h}_1\\Lambda_E,\\Lambda_E,\\beta_E).\n\\end{eqnarray}\n\nThis is an admissible choice for the interaction term; however, the resulting dependent on the $\\beta$ variables is somehow unsatisfactory: there are two $\\beta$ variables associated with each tetrahedron in the dual formulation of the associated Feynman graphs, i.e. two $\\beta$ variables for each edge of the Feynman graph; the amplitudes depend only on the difference between them, but this dependence is \\lq\\lq localised\\rq\\rq along the edge of the Feynman graph only, i.e. the variables appearing there do not appear anywhere else in the amplitude, so the integral can actually be performed separately in each edge, leading to a modified but $\\beta$-independent propagator. Explicitly, this modified $\\beta$-independent propagator of this model is given by:\n\\begin{equation}\n{\\cal P}_H(g_i,\\tilde{g}_i,G,G')= \\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\int d\\beta \\int d\\tilde{\\beta}\\;\\; {\\cal P}_H(G,G';\\beta-\\tilde{\\beta}), \n\\end{equation}\nwhere $\\beta$ and $\\tilde{\\beta}$ are the two \\lq\\lq Euclidean time\\rq\\rq variables associated to the give edge. The integral can be performed easily if one uses the mode expansion of the heat kernel, assuming that one can exchange the sum over representation and the integral over the $\\beta$ variables, leading to:\n\\begin{eqnarray}\n\\lefteqn{{\\cal P}_H(g_i,\\tilde{g}_i,G,G')=\\,2\\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\int_0^\\infty d(\\beta -\\tilde{\\beta}) \\;\\;{\\cal K}(G,G';\\beta-\\tilde{\\beta})=} \\nonumber \\\\ &=& 2\\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\int_0^\\infty d(\\beta -\\tilde{\\beta}) \\sum_J \\Delta_J e^{-(\\beta - \\tilde{\\beta}) C_J} \\chi^{J}(G G'^{-1}) = 2\\prod_i \\delta(g_i,\\tilde{g}_i)\\,\\sum_J \\frac{\\Delta_J}{C_J}\\, \\chi^{J}(G G'^{-1}),\n\\end{eqnarray}  \nwhere $C_J$ is the non-zero quadratic Casimir for simple representations of the Lorentz group\n(with an appropriate analytic continuation needed to regularise the integral), to which the sum refers, $\\chi^J$ is the character of the representation, and we have discarded an infinite factor coming from the integral over $\\beta + \\tilde{\\beta}$ and that is clearly pure gauge, the amplitude being completely independent of such a variable.\n\nOne can still consider the different regimes of integration over $\\beta - \\tilde{\\beta}$, i.e. the regime in which the difference is large and the model then approximates the usual Barrett-Crane model, and the one in which the difference is very small, and the model gives a BF-type behavior; however, such a localized dependence on the variables and the triviality of the amplitudes with respect to this dependence, to the point that the $\\beta$ variables can be quite simply eliminated from the amplitudes by integrating them out, make this choice of interaction term somewhat unpleasant, and we are led to look for a different and less trivial choice.\n\nIf we want a relationship between the various $\\beta$ variables in the fields in the same vertex term, then the simplest choice is to choose them to be all equal; the interaction term in the action would then be:\n\n\\begin{eqnarray}\n\\lefteqn{S_{int}^{(2)}[\\phi]=\\int d\\beta \\; \\int d\\Lambda_A ...d\\Lambda_E \\int \\prod_{i=1}^{10} dg_i\\prod_i \\int dh_i d\\tilde{h}_i\\,\n\\phi(g_1 h_1\\Lambda_A,g_2 h_2\\Lambda_A,g_3 h_3\\Lambda_A,g_4 h_4\\Lambda_A,\\Lambda_A, \\beta)} \\nonumber \\\\ &&\\phi(g_4 \\tilde{h}_4 \\Lambda_B,g_5 h_5\\Lambda_B,g_6 h_6 \\Lambda_B,g_7 h_7 \\Lambda_B,\\Lambda_B, \\beta)\\; \\phi(g_7\\tilde{h}_7\\Lambda_C,g_3 \\tilde{h}_3\\Lambda_C,g_8 h_8 \\Lambda_C,g_9 h_9 \\Lambda_C,\\Lambda_C,\\beta) \\nonumber\n\\\\ &&\\phi(g_9 \\tilde{h}_9 \\Lambda_D,g_6\\tilde{h}_6\\Lambda_D,g_2\\tilde{h}_2\\Lambda_D,g_{10}h_{10}\\Lambda_D,\\Lambda_D, \\beta)\\; \\phi(g_{10}\\tilde{h}_{10}\\Lambda_E,g_8\\tilde{h}_8\\Lambda_E,g_5\\tilde{h}_5\\Lambda_E,g_1\\tilde{h}_1\\Lambda_E,\\Lambda_E,\\beta)\n\\end{eqnarray}     \n\nThis choice is more satisfactory for several reasons; first of all it avoids the problems mentioned above for $S_{int}^{(1)}$, in that the dependence on $\\beta$ is much less trivial and not localized along each edge; second, with the interpretation of $\\beta$ as a interaction coupling parameter in each spin foam amplitude, this choice corresponds to having a different coupling parameter in each spacetime building block, i.e. 4-simplex, with the amplitude for the interaction between these building blocks depending only on the difference between the parameters associated to them; third, if one keeps in mind instead the interpretation of the $\\beta$ variable as a kind of \\lq\\lq Euclidean time\\rq\\rq, then the choice of $S_{int}^{(2)}$ made above is actually the most sensible one, as it corresponds to an interaction between the five fields corresponding to the five tetrahedra in 4-simplex that is {\\it local} in this time variable.     \nNote that also in this case we have a different coupling of spacetime atoms, i.e. 4-simplices, for each edge of the spin foam, but with a non-trivial relation among those referring to the same 4-simplex.\nThe vertex amplitude in configuration space corresponding to the interaction term $S_{int}^{(2)}[\\phi]$ is therefore still given by \\Ref{eq:vertex}:\n\\begin{equation}\n\\mathcal{I}_v(g_f,\\tilde{g}_f, \\Lambda_e, \\beta)= \\prod_{f\\mid v} \\delta(g_{f\\mid v}\\Lambda_{e1}^{-1}\\Lambda_{e2}\\tilde{g}_{f\\mid v}^{-1}),\n\\end{equation}\nwhere the dependence on $\\beta$ is limited to the fact that the same $\\beta$ should appear in any mode expansion of the fields with respect to this variable, that we do not perform explicitly, and in that the same $\\beta$ has to appear in any propagator connecting this vertex to any other one in the Feynman expansion. This is given of course by:\n\n\\begin{equation}\nZ(\\lambda) = \\int \\mathcal{D}\\phi\\; e^{- S[\\phi]} = \\int \\mathcal{D}\\phi\\; e^{- S_{kin}[\\phi] - \\frac{\\lambda}{5!} S_{int}^{(2)}[\\phi]} = \\sum_{\\Gamma} \\frac{\\lambda^N}{sym(\\Gamma)} \\,Z_{\\Gamma}, \\label{eq:Zvar}\n\\end{equation}\nwith:\n\\begin{equation}\nZ_{\\Gamma}= \\left(\\prod_e \\prod_{f\\mid e} \\int dg_{f\\mid e} \\int d\\tilde{g}_{f\\mid e}\\right) \\left( \\prod_e \\int dG_e \\int dG'_e\\right) \\prod_v \\int d\\beta_v \\prod_e {\\cal P}_H(g_{f\\mid e},\\tilde{g}_{f\\mid e}, G_e,G'_e, \\beta_{e}, \\tilde{\\beta}_e) \\prod_v \\mathcal{I}_v(g_{f\\mid v}, \\tilde{g}_{f\\mid v}, G_{e\\mid v}, \\beta_v).\n\\end{equation}\n\nWe think this new type of group field theory model possesses many interesting features, and the resulting spin foam amplitudes should be analysed in more detail, but we leave this analysis for future work; now we prefer to simply highlight some of its general aspects, its possible interpretation and uses. Note for example that in this new model the coupling relative to different edges of the spin foam, i.e. too different tetrahedra, may be different so that it is possible to study models that are almost flat, but not quite, and with small {\\it different} curvatures in different regions of space.\n\nFirst of all the model presented above can be seen at the same time as an extension and as a special case of the new type of group field theories studied and to be presented in \\cite{generalised}; there a \\lq\\lq proper time\\rq\\rq parametrisation of group field theories is studied in order to achieve a generalized formulation of them from which both orientation dependent and orientation independent spin foam models \\cite{us,feynman} can be obtained; the field is defined to be of the general form:\n\\begin{equation}\n\\varphi(g_1, s_1; g_2, s_2; g_3, s_3; g_4, s_4) : \\left( \\mathrm{SO}(3,1) \\times \\mathbb{R}\\right)^{\\times 4}\n\\end{equation}  \nand on this field both a simplicity projector onto $\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$ for the four arguments and a global \\lq\\lq gauge invariance\\rq\\rq projector, under $\\mathrm{SO}(3,1)\\times \\mathbb{R}$ transformations acting simultaneously (diagonally) on the four arguments, are applied at the level of the action; with respect to a model based on this type of field, the one we have outlined above represents a restriction, because the extension of the domain of the first four arguments of the field from $\\mathrm{SO}(3,1)$ to $\\mathrm{SO}(3,1)\\times \\mathbb{R}$, with an extra \\lq\\lq proper time\\rq\\rq variable, is dropped; but it is also a generalization, in the same sense as the model presented in section \\Ref{sec:genBC} is a generalization of the Barrett-Crane model, because it extends the field to a five argument one, again by basically turning the variables enforcing the gauge invariance of the four argument field into true extra variables of it, and extending appropriately the gauge invariance to be imposed.   \nIndeed we expect a direct generalisation along these lines of the type of models studied in \\cite{generalised} to be straightforward and to lead to a model of the same type as the one defined by the partition function \\Ref{eq:Zvar}.  The main difference is however that what appears in the model defined by \\Ref{eq:Zvar} is a kind of \\lq\\lq Euclidean time\\rq\\rq, not a Minkowskian one, so that the corresponding differential operator appearing in the action is of heat- or diffusion-type, and not of Schr\\\"oedinger type; this is needed for maintaining the property that the model reduces to the Barrett-Crane one in the large $\\beta-\\tilde{\\beta}$ limit. \n\nIt is interesting at this point to discuss briefly the classical equations of motion for the field $\\phi$  coming from the action $S[\\phi] = S_{kin}[\\phi] + S_{int}^{(2)}[\\phi]$. These are:\n\\begin{eqnarray}\n\\lefteqn{\\left(-\\partial_\\beta + \\nabla_G\\right)\\phi(g_1,g_2,g_3,g_4,G,\\beta) +} \\nonumber \\\\  &+&\\frac{\\lambda}{5!} \\int dg_5...dg_{10}\\;\\varphi(g_1,g_5,g_6,g_7,G,\\beta)\\phi(g_2,g_5,g_8,g_9,G,\\beta)\\varphi(g_3,g_6,g_8,g_{10},G,\\beta)\\varphi(g_4,g_7,g_9,g_{10},G,\\beta)= 0,\n\\end{eqnarray} \nwhich is indeed the form of a diffusion or heat transfer equation in Euclidean time $\\beta$, with an additional time dependent \\lq\\lq driving potential\\rq\\rq, for what concerns the fifth extra arguments; the dynamics defined by this equation for the other arguments of the field is basically the usual one, modulo the modified gauge invariance requirement.  However, the physical interpretation of such an equation is not straightforward from a quantum gravity perspective, mainly because, while the field $\\phi$ itself has the (quantum) geometric interpretation of a (second) quantized tetrahedron, the geometric interpretation of $\\beta$ is unclear at the present stage. On the other hand, recalling the interpretation of $\\beta$ as a coupling parameter, the above equation can be interpreted as a kind of renormalisation group equation for our coupled model and it is quite striking, in our opinion, that what is a renormalisation group equation from the point of view of the spin foam amplitudes as presented in the previous section is just an equation of motion from the point of view of the generalised group field theory described above. Indeed, this new type of group field theory can be a useful starting point, we think, for tackling the issue of renormalisation of spin foam models, or, better, of the group field theories generating them as Feynman graphs and amplitudes. Therefore we conclude now by discussing how this issue can be tackled from the starting point provided in this paper. \n\n\n\\section{Outlook: renormalisation of spin foam models} \\label{sec: renorm}\n\nThe final goal is to understand the semi-classical behavior of the spin foam quantum gravity, recover the standard effective physics on flat or curved space-time and compute the corrections due to the quantum fluctuations of the gravitational field. This is a long-term program, and a first step in this direction would be to understand the coarse-graining and renormalisation of the spin foam models such as the Barrett-Crane model. We could expect a semi-classical space-time to be built out of a large number of elementary Planck-size blocks (the 4-simplices) and the macroscopic effective spin foam model to have a structure similar to that of the microscopic theory, but with a slightly different form of the amplitudes and a different value of the couplings: the Barrett-Crane model would then be only be a particular model along the renormalisation flow. We propose to use our generalized Barrett-Crane with arbitrary couplings between 4-simplices as an ansatz to study explicitly this renormalisation flow.\n \n\\subsection{About the renormalisation of the coupling}\n\nThe Barrett-Crane amplitude for a 4-simplex is a function of the 10 representation labels $\\rho_f\\in\\mathbb{R}_+$ living on its faces and of 5 variables $\\Lambda_T\\in\\mathrm{SO}(3,1)\/\\mathrm{SO}(3)$ attached to each tetrahedron. We note it ${\\cal A}(\\rho_f,\\Lambda_T)$. The $\\rho_f$'s have the interpretation of the area of the corresponding triangle while the $\\Lambda_T$'s are thought the (time-like) normal to the corresponding tetrahedron. Then the Barrett-Crane $\\{10\\rho\\}$ symbol can be considered as a (quite peculiar) quantum amplitude for a single quantum 4-simplex described in terms of 1st order Regge calculus\\cite{us,barrett,barrettruth}. A full spin foam is constructed by gluing 4-simplices together. As explained above, the representation label attached to a triangle will be the same for all 4-simplices who share that triangle. However, the normal variables $\\Lambda_T$ are attached to the corresponding tetrahedron in the reference frame of a given 4-simplex: they are not shared between 4-simplices and we will not denote them $\\Lambda_{T,\\sigma}$ referring to both the tetrahedron $T$ and the 4-simplex $\\sigma$ to which it belongs. Then the Barrett-Crane spin foam amplitude is the product of the 4-simplex amplitudes summed over all assignments $\\{\\rho_f,\\Lambda_{T,\\sigma}\\}$.\n\nLet us consider a particular tetrahedron $T_0$ in a fixed spin foam. It belongs to two 4-simplices, $\\sigma_1$ and $\\sigma_2$. The two normal variables for $T_0$, $\\Lambda_{T_0,\\sigma_1}$ and $\\Lambda_{T_0,\\sigma_2}$, are  a priori independent. They are indeed both integrated out independently to get the Barrett-Crane amplitude. Nevertheless, if we compute the spin foam amplitude summing and integrating over all representation labels $\\rho_f$ and all other normal variables $\\Lambda_T$, $T\\ne T_0$, we will obtain a probability amplitude on  the space of these two variables, $\\Lambda_1$ and $\\Lambda_2$ in short; this amplitude will not be factorised in general, which implies that they will not be treated as fully independent by the model after this sort of \\lq\\lq renormalisation procedure\\rq\\rq involving the mentioned summation over representations and normal variables. Therefore, with the aim of understanding the renormalisation of the Barrett-Crane model, it is natural to work in a generalized formulation allowing for generic coupling between the two normals associated to each tetrahedron. This should be intended as an effective way for encoding the coupling between these variables that would result from renormalisation, before the actual renormalisation procedure is carried out. \nAnother way to state the same point is the following. The Lorentz transformation mapping $\\Lambda_{T_0,\\sigma_1}$ to $\\Lambda_{T_0,\\sigma_2}$ can naturally be interpreted as the parallel transport between the two reference frames associated to the two 4-simplices $\\sigma_1$ and $\\sigma_2$. In the \"fundamental\" model, this parallel transport is arbitrary, and therefore $\\Lambda_1$ and $\\Lambda_2$ are integrated over independently. Nevertheless, under renormalisation of that fundamental model, we expect that the effective probability amplitude for this parallel transport evolves to a non-trivial correlation ${\\cal C}(\\Lambda_1,\\Lambda_2)$ between $\\Lambda_1$ and $\\Lambda_2$.\n\n\\medskip\n\nFrom this perspective, it would be interesting to look for the existence of a fixed point of the renormalisation of the generalized coupled Barrett-Crane model, which we described above. We think that the group field theory provides us with the right framework to address this issue. A first result would be to check that the uncoupled spin foam model is not a fixed point (or at least not a stable fixed point) and then whether or not the rigidly coupled model is a fixed point. This rigid coupling, or strong coupling limit, ${\\cal C}(\\Lambda_1,\\Lambda_2)\\sim \\delta (\\Lambda_1^{-1}\\Lambda_2)$ would corresponds to a flat model, as we said: the 4-simplices would be glued with a trivial parallel transport, so that the spin foam should be describing a flat space-time \\cite{us}.\n\nThe recent study \\cite{wade} of the Lorentzian Barrett-Crane partition function in terms of the normal variables $\\Lambda$'s should be particularly relevant to this project: the author looks at the properties of the amplitudes obtained by integrating first the representation labels $\\rho_f$ and uses them to present a new proof of the finiteness theorems for the Barrett-Crane amplitudes. It would actually be rather natural to include arbitrary couplings between the $\\Lambda$'s in such a reformulation of the model.\n\n\nWe also proposed a more specific group field theory for a restricted class of coupling given by the heat kernel. It should be simpler to study the coarse-graining properties of this model for which the couplings are described by a single real parameter. This model is close to a very interesting model proposed and studied by Oeckl \\cite{robert}.\nHe introduces a new model replacing the $\\delta$-functions constituting the edge and vertex amplitudes (in configuration space) in the spin foam quantization of the topological BF theory and the various Barrett-Crane-like models by heat kernels, and discusses the stability properties of the weak and strong coupling limits under coarse-graining moves generalizing the standard Pachner moves, i.e. under renormalisation. Our proposal is in the same spirit. However, while Oeckl's model corresponds to using the usual definition of the field and just changing the coupling between its arguments, the areas of triangles in momentum space, we first introduce a generalisation of the same field to five arguments, and then use explicitly the $\\Lambda$ variables, representing the normal to the tetrahedra.\nWe feel that using the $\\Lambda$ allows us to introduce the coupling between 4-simplices in a transparent way, easier to interpret physically. Moreover, Oeckl's model interpolates between the Barrett-Crane model and BF theory, while as we have discussed our $\\beta\\rightarrow 0$ limit does not give a true quantum BF theory but a similar theory in which the only configuration allowed for the (boost part of the) connection is the flat one. We believe it would be very interesting to analyse in more detail the similaritites and differences between these two modifications of the Barrett-Crane model, and in particular check whether the respective fixed points in the renormalisation flow in $\\beta$ correspond to the same type of theory. \n\nThe \\lq\\lq parametrised \\rq\\rq group field theory we have introduced last, in which the heat kernel coupling is treated as a variable of the theory, suggests (actually, forces) a different way of dealing with the same issues. In fact, as we said, the renormalisation flow of the coupling is encoded there in the {\\it classical} equations of motion coming from the group field theory action; therefore, if we are to study the presence and nature of fixed points, we should check whether particular regimes for the coupling are favored by the classical equations of motion.\n\n\\medskip\n\nWe conclude this discussion by a speculation that the study of the renormalisation flow of the spin foam model could \nshed light on the issue of the diffeomorphism invariance of the spin foam amplitudes: diffeomorphisms may appear as renormalisation group transformations in the group field theory formulation since they seem to be related to Pachner moves in the state sum formulation of spin foams and to the Ward identities in the group field theory formulation. This was suggested in \\cite{laurentgft,instantons}, following the results of \\cite{laurentdiffeo}. Indeed Ward identities in quantum field theory relate the amplitudes of different Feynman diagrams. In the group field theory, they would then relate the amplitudes of different spin foams. It is likely that looking more precisely to this point would lead to a deeper understand of both the renormalisation of spin foams and the action of diffeomorphisms in this discrete quantum setting.  In the 2d spin foam case, the group field theory reduces to matrix models, for which the explicit link between the Ward identities of the path integral and the conformal symmetry is well-understood (see for example the review \\cite{matrix}), so methods and ideas from matrix models can be of help in analysing this same issue for group field theories. Finally, it would be interesting to compare the resulting Ward identities relating different spin foam amplitudes to the background independent coarse graining scheme proposed by Markopoulou \\cite{fotini}, using the Hopf algebra structure of 2-complexes in a way very similar to the Connes-Kreimer approach to the renormalisation of quantum field theory \\cite{connes}.\n\n\n\\subsection{A numerical simulation project}\n\nThe coarse-graining of the coupling can well be studied numerically. First, we would need to look at the single 4-simplex amplitude ${\\cal A}(\\rho_f,\\Lambda_T)$. It would be interesting to look at the probability amplitude of the $\\Lambda$ variables. We could restrict the study to regular 4-simplices where all the representation labels are identical, $\\rho_f =\\rho$. Fixing $\\rho$ and integrating over three of the $\\Lambda$ variables, we would get a probability amplitude on the remaining two $\\Lambda$ variables, which would describe the correlations between two tetrahedra within the considered 4-simplex. Actually, due to the Lorentz gauge invariance of the 4-simplex, we would obtain a simple distribution on the angle between the two normals. A first issue is whether this distribution is peaked or not at a given angle. A second issue is whether this peak moves or not when rescaling the representation label $\\rho$: this would show that the coupling would definitely be sensitive to the size of the 4-simplices (measured in Planck units). \n\nStill looking at a single 4-simplices, we could consider almost identical assignments of representation labels $\\rho_f$. More precisely, singling out two tetrahedra, and keeping fixed the 7 representation labels attached to these tetrahedra, we could look at the evolution of the coupling between these tetrahedra when we change the 3 remaining representation labels. This would show how the geometry of a tetrahedron does not solely depend on the representation labels assigned to it but on the representation labels assigned to the 4-simplex to which it belongs: the intrinsic geometry of a tetrahedron is determined by the space-time geometry of the 4-simplex.\n\nLet us point out that we could carry on this analysis unfolding the Barrett-Crane intertwiner at each tetrahedron in terms of $\\mathrm{SO}(3,1)$ simple representations $\\rho_{int}$ instead of the normal variable $\\Lambda$. $\\rho_{int}$ would be interpreted as describing the internal space geometry of the tetrahedron while the $\\Lambda$ seem to describe its embedding in the surrounding spacetime. A first computation would be to study the correlations between the internal representation label $\\rho_{int}$ of two tetrahedra within the same 4-simplex and see whether the maximal probability corresponds to small\/large-small\/large configurations. One might then hope that such correlations propagate within the spinfoam helped by the coupling between 4-simplices.\n\nThe next step would be to consider two 4-simplices or more and the coupling between two tetrahedra belonging to different 4-simplices. The coupling between the 4-simplices would then play a role, and therefore the new coupled model we have proposed in this paper can be of direct use. We could study how long-range correlations between spacetime points (tetrahedra) would depend on the coupling appearing between two glued 4-simplices. In this framework, we could either keep the representation labels fixed and look at the correlations induced solely by the coupling imposed by hand, or we could sum over internal representation labels (attached to the internal faces i.e not on the spin foam boundary) and we would look at the full effective correlations. \n\n\n\\section{Conclusions}\nLet us summarize what we have achieved in this work. Our main result has been to construct a group field theory formalism that generalizes the Barrett-Crane model to allow a non-trivial coupling between the two normals to the tetrahedra in the two 4-simplices that share them; the way we have done it is simple: we have shown first how the usual Barrett-Crane model can be seen as the result of a quantization of a field theory on five copies of a group manifold, with no coupling among the fifth arguments of the field, and then we have modified this field theory to introduce a coupling. We have exhibited a specific choice of coupling function and therefore constructed a specific coupled model; this model interpolates nicely between the usual Barrett-Crane model, with the parallel transport between 4-simplices being completely randomized, and a flat or BF-like model, in which the only allowed connection is the flat one, according to the value that a new additional parameter of the theory, not present in the usual Barrett-Crane model, takes. The tuning of the coupling parameter therefore corresponds to a tuning of the degree of \\lq\\lq locality\\rq\\rq of the resulting spin foam amplitudes. This new coupled model is amenable for further study of correlations between simplices, both analytical and numerical, that can shed light on the presence or absence of local propagating degrees of freedom in spin foams, and can play an important role in studying renormalisation of the spin foam amplitudes under coarse graining. Moreover, it can be the basis for developing fully covariant perturbation expansions in the coupling parameter that do not make any use of background structures and therefore remain fully background independent from the quantum spacetime point of view. We have also discussed how the mathematical structures on which the coupled model is based are going to be relevant for studies of particle and matter insertions in the Barrett-Crane or similar spin foam models. Finally, we have gone further in this process of generalization, by showing how a new type of group field theory can be constructed in which the coupling parameter is promoted to a variable of the theory, on the same footing as group variables, leading to the presence of derivatives in the action and therefore to a different type of classical equations of motion for the field. We concluded by discussing the issue of renormalisation of spin foam models from the new perspective that the results of this work suggest, and how the new coupled model can be a useful new starting point for the application of renormalisation group techniques, and therefore for understanding the semi-classical properties of spin foam models.  \n\n\n\n\n\n\n\\section*{Acknowledgements}\nWe would like to thank, for hospitality and beverages, Caffe' Nero in Cambridge and Cafe' 1842 in Waterloo, where part of this work was done. D.O. thanks also the Perimeter Institute for hospitality during the early stages of this project, and the Kalahari Meerkat Project for hospitality in the Kuruman River Reserve, South Africa, during its completion.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{0pt}{0.4em plus 0.2em minus 0.2em}{0.4em plus 0.2em minus 0.2em}\n\n\\title{Policy Evaluation with Latent Confounders via Optimal Balance}\n\n\n\\author{%\n  Andrew~Bennett\\thanks{Alphabetical order.} \\\\\n  Cornell University \\\\\n  \\texttt{awb222@cornell.edu} \n \n  \\and\n  Nathan~Kallus\\samethanks\\\\\n  Cornell University\\\\\n  \\texttt{kallus@cornell.edu} \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n}\n\\date{}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nEvaluating novel contextual bandit policies using logged data is crucial in applications where exploration is costly, such as medicine. But it usually relies on the assumption of no unobserved confounders, which is bound to fail in practice. We study the question of policy evaluation when we instead have proxies for the latent confounders and develop an importance weighting method that avoids fitting a latent outcome regression model. We show that unlike the unconfounded case no single set of weights can give unbiased evaluation for all outcome models, yet we propose a new algorithm that can still provably guarantee consistency by instead minimizing an adversarial balance objective. We further develop tractable algorithms for optimizing this objective and demonstrate empirically the power of our method when confounders are latent.\n\\end{abstract}\n\n\\input{src\/intro.tex}\n\\input{src\/problem.tex}\n\\input{src\/theory.tex}\n\\input{src\/algorithms.tex}\n\\input{src\/experiments.tex}\n\\input{src\/conclusion.tex}\n\n\n\\section{Introduction}\n\\label{sec:intro}\n\nPersonalized intervention policies are of increasing importance in education \\citep{mandel2014offline}, healthcare \\citep{bertsimas2017personalized}, and public policy \\citep{kube2019allocating}.\nIn many of these domains exploration is costly or otherwise prohibitive, and so it is crucial to evaluate new policies using existing observational data.\nUsually, this relies on an assumption of no unobserved confounding (aka unconfoundedness or ignorability): that conditioned on observables, interventions are independent of idiosyncrasies that affect outcomes, so that counterfactuals can be reliably and correctly predicted. In particular, this enables the use of inverse propensity score (IPS) estimators of policy value \\citep{beygelzimer2009offset,li2011unbiased,kallus2018policy} that eschew the need to actually fit outcome prediction models and doubly robust estimators that work even if such models are misspecified \\citep{dudik2011doubly}.\n\nIn practice, however, it may be unlikely that we observe confounders exactly. Nonetheless, if we observe very many features they may serve as good proxies for the true confounders, which can enable an alternative route to identification \\citep{louizos2017causal,kallus2018causal}. \nIn particular, noisy observations of true confounders can serve as valid proxies.\nFor example, if intelligence is latent but affects both selection and outcome, we can instead use many noisy observations of intelligence such as school grades, IQ test, etc. Similarly, many medical measurements taken together can serve as proxies for underlying healthfulness.\n\n\n\n\n\nIn this paper, we study the problem of policy evaluation from observational data where we observe proxies instead of true confounders and we develop new weighting estimators based on optimizing balance in the latent confounders. Unlike the unconfounded setting where IPS weights ensure balance regardless of outcome model, we show that in this new setting there cannot exist any single of weights that ensure such unbiasedness regardless of outcome model. Instead, we develop an adversarial objective that bounds the conditional mean square error (CMSE) of any weighted estimator and, by appealing to game theoretic and empirical process arguments, we show that this objective can actually be driven to zero by a single set of weights. We therefore propose a novel policy evaluation method that minimizes this objective, thus provably ensuring consistent estimation in the face of latent confounders. We develop tractable algorithms for this optimization problem. Finally, we provide empirical evidence demonstrating our method's consistent evaluation compared to standard evaluation methods and its improved performance compared to using fitted latent outcome models.\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe presented theory for how to do optimal balancing for policy evaluation when we only have proxies for the true confounders, given an already identified model for the confounders, treatment, and proxies, but not for the outcomes. We provided an adversarial objective for selecting optimal weights given some class of mean outcome functions to be balanced, and proved that under mild conditions these optimal weights result in consistent policy evaluation. In addition, we presented a tractable algorithm for minimizing this objective when our function class is an RKHS, and we conducted a series of experiments to demonstrate that our method can achieve consistent evaluation even under sufficient levels of confounding where standard approaches fail.\n\nFor future work we note that the adversarial objective and theory presented here is fairly general, and could be used to develop new algorithms for balancing different function classes such as neural networks. An alternative direction would be to study how best to apply this methodology when an identified model is not already given.\n\n\\section{Weight-Balancing Objectives}\n\\label{sec:method}\n\n\\subsection{Infeasibility of IPS-Style Unbiased Weighting}\n\nIf we had unconfoundedness given $X$ (\\ie, $Y(t)\\indep T\\mid X$), the IPS weights $\\pi_T(X)\/\\eta_T(X)$ are immediately gotten as the solution to making every term in the weighted sum \\cref{eq:weighted} unbiased:\n\\begin{equation}\n\\label{eq:ips}\n\\mathbb{E}[W(X, T) \\delta_{T_i t} Y(t)] = \\mathbb{E}[\\pi_t(X) Y(t)].\n\\end{equation}%\nNotably the IPS weights do not depend on the outcome function.\nHowever, without unconfoundedness given $X$ and given only \\cref{asm:ignorability,asm:overlap,asm:variance}, this approach fails.\n\\begin{theorem}\n\\label{thm:ips}\nIf $W(x,t)$ satisfies \\cref{eq:ips} then for any $t\\in\\{1,\\dots,m\\}$\n\\begin{equation}\\label{eq:generalips} W(X,t) = \\pi_t(X) \\frac{\\sum_{t' \\in \\mathcal{\\tau}} \\eta_{t'}(X) \\nu_t(X,t') + \\Omega_t(X)}{\\eta_t(X) \\nu_t(X,t)}, \\end{equation}\nfor some $\\Omega_t(x)$ such that $\\E\\Omega_t(X)=0\\,\\forall t$.\n\\end{theorem}\nThe proof of \\cref{thm:ips} is given in \\cref{apx:ips}. \n\nNote that \\emph{if} we had unconfoundedness given $X$ then $\\nu_t(x,t')=\\nu_t(x,t)=\\rho_t(x)$ so that\nchoosing $\\Omega_t(X) = 0$ would recover the standard IPS weights. However, in our setting we generally have $\\nu_t(x,t')\\neq\\nu_t(x,t)$, and so \\cref{thm:ips} tells us that we cannot do unbiased IPS-style weighted evaluation without knowing the mean outcome functions $\\nu_t(x,t')$. In particular, there exists no single weight function that is simultaneously unbiased for all outcome functions.\n\n\nOn the other hand, \\cref{thm:ips} tells us that there do exist some weights that give unbiased and consistent policy evaluation via \\cref{eq:weighted} or \\cref{eq:dr}: we just may not be able to calculate them. \nThe existence of such weights motivates our subsequent approach, which seeks weights that mimic these weights for a wide class of possible outcome functions.\n\n\n\\subsection{Adversarial Error Objective}\n\nOver all weights that are functions of $X_{1:n},T_{1:n}$, the \\emph{optimal} choice of weights for estimating $\\tau^{\\pi}$ via \\cref{eq:weighted} would minimize the (unknown) conditional MSE (CMSE):\n\\begin{equation}\\label{eq:cmsetau}\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ].\\end{equation} \nIn particular, the weights in \\cref{eq:generalips} achieve $O_p(1\/n)$ control on this CMSE for many outcome functions, as long as the denominator is well behaved, which can be seen by applying concentration inequalities to \\cref{eq:cmsetau}. However, as discussed above the outcome function is unknown and these weights are therefore practically infeasible.\nOur aim is to find weights with similar near-optimal behavior but that do not depend on the particular unknown outcome function. To do this, we will find an upper bound for \\cref{eq:cmsetau} that we can actually compute.\n\nLet $f_{it} = W_i \\delta_{T_i t} - \\pi_t(X_i)$ and\n$$\nJ(W,\\mu) =  \\left( \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m f_{it} \\nu_t(X_i,T_i) \\right)^2 + \\frac{2\\sigma^2}{n^2} \\|W\\|_2^2,\n$$\nwhere we embedded the dependence on $\\mu$ inside $\\nu_t(x,t')=\\Eb{\\mu_t(Z)\\mid X=x,T=t'}$.\n\n\\begin{theorem}\n\\label{thm:jbound}\n$\\displaystyle\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]\n\\leq 2J(W,\\mu)+O_p(1\/n)\n$.\n\\end{theorem}\nWe prove this in \\cref{apx:jbound}.\n\nTherefore, if we find weights that obtain $O_p(1\/n)$ control on $J(W,\\mu)$, we can ensure that we also have $O_p(1\/n)$ control on $\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]$. Combined with the following result, which follows from \\citep[Lemma~31]{kallus2016generalized}, this would give root-n consistent estimation.\n\\begin{lemma}\n\\label{lem:control}\nIf $\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]=O_p(1\/n)$ then\n$\\hat{\\tau}^{\\pi}_W=\\tau^{\\pi}+O_p(1\/\\sqrt{n})$.\n\\end{lemma}\n\n\n\n\n\n\n\n\nIt remains to find weights that control $J(W,\\mu)$. The key obstacle for this is that $\\mu$ is unknown. Instead, we show how we can obtain weights that control $J(W,\\mu)$ over a whole class of given functions $\\mu$.\n\nSuppose we are given a set $\\mathcal{F}$ of functions mapping $\\mathcal{Z}$ to $\\mathbb{R}^m$, where each $\\mu \\in \\mathcal{F}$ corresponds to a vector of mean outcome functions $\\mu = (\\mu_1, \\ldots, \\mu_m)$. Then, motivated by \\cref{thm:jbound,lem:control}, we define our adversarial optimization problem as\n\n\\begin{equation}\n\\label{eq:optim}\n\\mathcal{W}^* = \\argmin_{W \\in \\mathcal{W}} \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu).\n\\end{equation}\n\nOne question the reader might ask at this point is why not solve the above optimization problem by ignoring the hidden confounders and directly balancing the conditional mean outcome functions $\\nu_t(x,t')$. The problem is that this would be impossible to do over any kind of generic flexible function space, since we have no data corresponding to terms in the form $\\nu_t(x,t')$ when $t \\neq t'$, so this is akin to an overlap problem. Conversely, if we were to ignore the conditioning on $t$ and balance against functions of the form $\\nu_t(x) = \\nu_t(x,t)$ this would be inadequate, as we couldn't hope for such a space to cover the true $\\mu$ since we don't assume ignorability given $Z$.\n\nIn light of these limitations, we can view what we are doing in optimizing \\cref{eq:optim} using an identified model $\\varphi(z;x,t)$ as implicitly balancing some controlled space of functions $\\nu_t(x,t')$ that do not have this overlap issue between different $t$ values. The following lemma makes this explicit, as it implies that that the terms $\\nu_t(x,t')$ are all mutually bounded by each other for fixed $x$ and $t$:\n\n\\begin{lemma}\n\\label{lem:gbound}\nAssuming $\\|\\mu_t\\|_\\infty\\leq b$, under \\cref{asm:overlap}, for all $x \\in \\mathcal{X}$, and $t,t',t'' \\in \\{1,\\ldots,m\\}$ we have\n\\[ |\\nu_t(x,t'')| \\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\sqrt{8b\n\\Eb{e^{-2}_t(Z) \\mid X=x, T=t'} |\\nu_t(x,t')|}. \\]\n\\end{lemma}\n\nWe prove this in \\cref{apx:gbound}. \n\n\n\\subsection{Consistency of Adversarially Optimal Estimator}\n\nNow we analyze the consistency of our weighted estimator based on \\cref{eq:optim}. Given \\cref{lem:control}, all we need to justify to prove consistency is that $\\mu \\in \\mathcal{F}$ and that $\\inf_W \\sup_{\\mu \\in \\mathcal{F}} J(W,\\mu) \\in O_p(\\frac{1}{n})$. Define $\\mathcal{F}_t$ as the space of all functions for treatment level $t$ allowed by $\\mathcal{F}$. That is $\\mathcal{F}_t = \\{\\mu_t : \\exists (\\mu_1', \\ldots, \\mu_m') \\in \\mathcal{F} \\ \\text{with} \\ \\mu_t' = \\mu_t\\}$. We will use the following assumptions about $\\mathcal{F}$ to prove control of $J$:\n\n\\begin{assumption}[Normed]\n\\label{asm:norm}\nFor each $t \\in \\{1,\\ldots,m\\}$ there exists a norm $\\|\\cdot\\|_t$ on $\\op{span}(\\mathcal{F}_t)$, and there exists a norm $\\|\\cdot\\|$ on $\\op{span}(\\mathcal{F})$ which is defined given some $\\mathbb{R}^m$ norm as $\\|\\mu\\|~=~\\|(\\|\\mu_1\\|_1,\\ldots,\\|\\mu_m\\|_m)\\|$.\n\\end{assumption}\n\n\\begin{assumption}[Absolutely Star Shaped]\n\\label{asm:symmetry}\nFor every $\\mu \\in \\mathcal{F}$ and $\\abs{\\lambda} \\leq 1$, we have $\\lambda \\mu \\in \\mathcal{F}$.\n\\end{assumption}\n\n\\begin{assumption}[Convex Compact]\n\\label{asm:compact}\n$\\mathcal{F}$ is convex and compact \n\\end{assumption}\n\n\\begin{assumption}[Square Integrable]\n\\label{asm:ltwo}\nFor each $t \\in \\{1,\\ldots,m\\}$ the space $\\mathcal F_t$ is a subset of $\\ltwo(\\mathcal Z)$, and its norm dominates the $\\ltwo$ norm (i.e., $\\inf_{\\mu_t \\in \\mathcal F_t} \\|\\mu_t\\| \/ \\|\\mu_t\\|_{\\ltwo} > 0$).\n\\end{assumption}\n\n\\begin{assumption}[Nondegeneracy]\n\\label{asm:nondegen}\nDefine $\\mathcal B(\\gamma) = \\{\\mu \\in \\op{span}(\\mathcal F) : \\| \\mu \\| \\leq \\gamma\\}$. Then we have $\\mathcal B(\\gamma) \\subseteq \\mathcal F$ for some $\\gamma > 0$.\n\\end{assumption}\n\n\\begin{assumption}[Boundedness]\n\\label{asm:bounded}\n$\\sup_{\\mu \\in \\mathcal F} \\|\\mu\\|_{\\infty} < \\infty$.\n\\end{assumption}\n\n\\begin{definition}[Rademacher Complexity]\n$\\mathcal R_n(\\mathcal F) = \\mathbb{E}[\\sup_{f \\in \\mathcal F} \\frac{1}{n} \\sum_{i=1}^n \\epsilon_i f(Z_i)]$, where $\\epsilon_i$ are iid Rademacher random variables.\n\\end{definition}\n\n\\begin{assumption}[Complexity]\n\\label{asm:complexity}\nFor each $t \\in \\{1\\ldots,m\\}$ we have $\\mathcal R_n(\\mathcal F_t) = o(1)$.\n\\end{assumption}\n\nThese assumptions are satisfied for many commonly-used families of functions, such as RKHS spaces and families of neural networks. We shall prove this claim for RKHS spaces in \\cref{sec:algo}.\n\nIn order to justify that we can control $J$, first we will show that these assumptions allow us to reverse the order of minimization and maximization in our optimization problem. This means we can reduce problem to finding weights to control any particular $\\mu$ rather than controlling all of $\\mathcal{F}$.\n\n\n\\begin{lemma}\n\\label{lem:minimax}\nLet $B(W, \\mu) = \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m f_{it} \\nu_t(X_i,T_i)$. Then under \\cref{asm:symmetry,asm:compact,asm:ltwo} for every $M>0$ we have the bound \n\\[ \\min_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) \\leq \\sup_{\\mu \\in \\mathcal F} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2}M^2. \\]\n\\end{lemma}\n\nWe prove this in \\cref{apx:minimax}.\n\nNext, we note that \\cref{lem:minimax} means that we can choose of weights given $\\mu$ to set $B(W,\\mu) = 0$, and therefore we have our desired control as long as we can justify that these weights have controlled euclidean norm. Using this strategy and optimizing for the weights of this kind with minimum euclidean norm, we are able to prove the following:\n\n\\begin{lemma}\n\\label{lem:jconvergence}\nUnder \\cref{asm:norm,asm:symmetry,asm:compact,asm:ltwo,asm:nondegen,asm:bounded,asm:complexity} we have $\\inf_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) = O_p(1\/n)$.\n\\end{lemma}\n\nWe prove this in \\cref{apx:jconvergence}. This is the key lemma in proving our main consistency theorem:\n\n\\begin{theorem}\n\\label{thm:consistency}\nUnder \\cref{asm:norm,asm:symmetry,asm:compact,asm:ltwo,asm:nondegen,asm:bounded,asm:complexity} and assuming that $\\mu \\in \\mathcal F$ we have $\\hat{\\tau}^{\\pi}_{W^*} = \\tau^{\\pi} + O_p(1\/\\sqrt{n})$.\n\\end{theorem}\n\nThis theorem follows immediately from our previous results, since $\\mu \\in \\mathcal F$ and \\cref{lem:jconvergence} imply that $J(W^*, \\mu) = O_p(1\/n)$. This combined with \\cref{thm:jbound} imply that $\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_{W^*} - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ] = O_p(1\/n)$, which in turn combined with \\cref{lem:control} gives us our result.\n\n\n\n\n\n \n\n\n\\section{Algorithms for Optimal Kernel Balancing}\n\\label{sec:algo}\n\n\\subsection{Kernel Function Class}\n\nWe now provide an algorithm for optimal balancing when our function class consists of vectors of RKHS functions. Formally, given a kernel $K$ and corresponding RKHS norm $\\|\\cdot\\|_K$, we define the space $\\fk$ as follows:\n\n\\begin{definition}[Kernel Class]\n\\label{def:kernel-class}\n$\\fk = \\{\\mu : ||\\mu|| \\leq 1\\}$, where $||(\\mu_1,\\ldots,\\mu_m)|| = \\sqrt{\\sum_{t=1}^m ||\\mu_t||_K^2}$.\n\\end{definition}\n\n\\begin{theorem}\n\\label{thm:fk}\nAssuming $K$ is a Mercer kernel \\citep{zhou2002covering} and is bounded, $\\fk$ satisfies \\cref{asm:norm,asm:symmetry,asm:compact,asm:ltwo,asm:nondegen,asm:bounded,asm:complexity}.\n\\end{theorem}\n\nWe prove this in \\cref{apx:fk}.\n\nWe can remark that the commonly used Gaussian kernel is both Mercer and bounded, so it satisfies the conditions of \\cref{thm:fk}. Given this, and assuming that $\\fk$ covers the real mean outcome function $\\mu$, we can apply \\cref{thm:consistency} to see that solving \\cref{eq:optim} using $\\fk$ gives consistent evaluation.\n\nWe can note that the $\\fk$ having maximum norm 1 is without loss of generality, because if we wanted the maximum norm to instead be $\\gamma$ we could replace the $\\Sigma$ matrix by $\\Gamma = \\frac{1}{\\gamma} \\Sigma$ in our objective function, resulting in an equivalent re-scaled optimization problem. To make this explicit, we will replace the $\\Sigma$ matrix in the objective with $\\Gamma$ in this section, where it is assumed that $\\Gamma$ is freely chosen as a hyperparameter.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Kernel Balancing Algorithm}\n\\label{sec:qpalgo}\n\nIn order to optimize \\cref{eq:optim} over a class of kernel functions as defined by \\cref{def:kernel-class}, we can observe that the definition of $J(W, \\mu)$ looks very similar to the adversarial objective of \\citet{kallus2018balanced}, except that we have $\\nu_t(X_i,T_i)$ terms instead of $\\mu_t(X_i)$ terms. This motivates the idea that, given our identified posterior model $\\varphi(z;x,t)$, we may be able to employ a similar quadratic programming (QP)-based approach. The following theorem makes this explicit, by defining a QP objective for $W$ that we can approximate by sampling from $\\varphi$:\n\n\\begin{theorem}\n\\label{thm:continuous-algorithm}\nDefine $Q_{ij} = \\mathbb{E}[K(Z_i, Z_j')]$, $G_{ij} = \\frac{1}{n^2} (Q_{ij} \\delta_{T_i T_j} + \\Gamma_{ij})$, and $a_i = \\frac{2}{n^2} \\sum_{j=1}^n Q_{ij} \\pi_{T_j}(X_i)$, where for each i $Z_i$ and $Z_i'$ are iid shadow variables. Then for some $c$ that is constant in $W$ we have the identity\n\\[ \\sup_{\\mu \\in \\fk} J(W, \\mu) = W^T G W - a^T W + c.\\]\n\\end{theorem}\n\nWe prove this in \\cref{apx:continuous-algorithm}.\n\nGiven this our balancing algorithm is natural and straightforward, and is summarized by \\cref{algo:continuous}. Note that we provide an optional weight space constraint $\\mathcal{W}$ in this algorithm, since standard weighted estimator approaches for policy evaluation regularize by forcing constraints such as $W \\in n \\Delta^n$. Under this kind of constraint our unconstrained QP becomes a LCQP. However that our theory does not support this constraint, and that we find it hurts performance in practice, especially when $\\Gamma$ is large, so we do not use this constraint in our experiments.\n\n\n\n\\begin{algorithm}\n  \\caption{Optimal Kernel Balancing}\\label{algo:continuous}\n  \\begin{algorithmic}[1]\n  \\Require \\parbox[t]{0.9\\textwidth}{\n     Data $(X_{1:n}, T_{1:n})$, policy $\\pi$, kernel function $K$, posterior density $\\varphi$, regularization matrix $\\Gamma$, number samples $B$, optional weight space $\\mathcal{W}$ (defaults to $\\mathbb{R}^n$ if not provided)\n  }\n  \\Ensure Optimal balancing weights $W_{1:n}$\n      \\For {$i \\in \\{1, \\ldots, n\\}$}\n        \\State \\parbox[t]{0.9\\textwidth}{\\textbf{Sample Data.} Draw $B$ data points $Z_i^b$ from the posterior $\\varphi(\\cdot \\ ;X_i,T_i)$}\n      \\EndFor\n      \\State \\textbf{Estimate Q.} Calculate $Q_{ij} = \\frac{1}{B^2} \\sum_{b=1}^B \\sum_{c=1}^B K(Z_i^b, Z_i^c)$\n      \\State \\textbf{Calculate QP Inputs.} Calculate $G_{ij} = Q_{ij} \\delta_{T_i T_j} + \\Gamma_{ij}$, and $a_i = 2 \\sum_{j=1}^n Q_{ij} \\pi_{T_j}(X_i)$,\n      \\State \\textbf{Solve Quadratic Program.} Calculate $W = \\argmin_{W \\in \\mathcal{W}} W^T G W - a^T W$\n  \\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Problem}\n\n\\subsection{Setting and Assumptions}\n\\label{sec:setting}\n\nWe consider a contextual decision making setting with $m$ possible treatments (aka actions or interventions).\nEach unit is associated with \na set of potential outcomes $Y(1),\\dots,Y(m)\\in\\Rl$ corresponding to the reward\/loss for each treatment, an observed treatment $T\\in\\{1,\\dots,m\\}$, an observed outcome $Y=Y(T)$, true but latent confounders $Z\\in\\mathcal Z \\subseteq \\mathbb{R}^p$, and observed covariates $X\\in\\mathcal X \\subseteq \\mathbb{R}^q$.\nOur data consists of iid observations $X_{i},T_{i},Y_{i}$ of $X,T,Y$.\nBoth the latent confounders and potential outcomes of unassigned treatments are unobserved.\nNote that $Y_i=Y_i(T_i)$ encapsulates the assumptions of {consistency} between observed and potential outcomes and non-interference between units.\n\nA \\emph{policy} is a rule for assigning the probability of each treatment option given the observed covariates $X$. Given a policy $\\pi$, we use the notation $\\pi_t(x)$ to indicate the probability of assigning treatment $t$ when observed covariates are $x$. We define the \\emph{value} of a policy, $\\tau^{\\pi}$, as the expected outcome that would be obtained from following the policy in the population. Formally:%\n\\begin{definition}[Policy Value]\n$\\tau^{\\pi} = \\mathbb{E}[\\sum_{t=1}^m \\pi_t(X) Y(t)]$.\n\n\\end{definition}\n\\begin{wrapfigure}{r}{0.225\\textwidth}\n\\vspace{-2.75\\baselineskip}\n\\begin{minipage}{0.225\\textwidth}\\small\\centering%\n\\begin{tikzpicture}[%\n    >=latex',\n    circ\/.style={draw, shape=circle, node distance=1cm, line width=1pt}]\n    \\node[circ] (X) at (-0.5,0.8) {$X$};\n    \\node[circ][draw,dashed] (U) at (1,0.8) {$Z$};\n    \\node[circ] (Y) at (2,-.3) {$Y$};\n    \\node[circ] (T) at (0,-.3) {$T$};\n    \\draw[line width=1pt,->] (T) -- (Y);\n    \\draw[line width=1pt,->] (U) -- (T);\n    \\draw[line width=1pt,->] (U) -- (Y);\n    \\draw[line width=1pt,->] (U) -- (X);\n    \\draw[line width=1pt,->] (X) -- (T);\n\\end{tikzpicture}\n\\end{minipage}%\n\\captionof{figure}{DAG representation of problem.}\\label{fig:proxies}\n\\vspace{-\\baselineskip}\n\\end{wrapfigure}\n\nWe encapsulate the assumption that $Z$ are sufficient for unconfoundedness and that $X$ is a proxy for $Z$ in the following assumption. \\Cref{fig:proxies} provides a representation of this setting using a causal DAG \\citep{pearl2000causality}.\nNote importantly that we do \\emph{not} assume ignorability given $X$. \n\\begin{assumption}[$Z$ are true confounders]\n\\label{asm:ignorability}\nFor every $t\\in\\{1,\\dots,m\\}$, $Y(t)$ is independant of $(X,T)$, given $Z$.\n\\end{assumption}\n\nWe next define the average mean outcome given $Z$ and its conditional expectations given observables:\n\\begin{align*}\n\\mu_t(z) &= \\mathbb{E}[Y(t) \\mid Z=z],\\\\\n\\nu_t(x,t') &= \\mathbb{E}[\\mu_t(Z) \\mid X=x, T=t']= \\mathbb{E}[Y(t) \\mid X=x, T=t'], \\\\\n\\rho_t(x) &= \\mathbb{E}[\\mu_t(Z) \\mid X=x]= \\mathbb{E}[Y(t) \\mid X=x].\n\\end{align*}\nWe further define the propensity function and its conditional expectation given observables:\n\\begin{align*}\ne_t(z) &= \\Prb{T=t\\mid Z=z},\\\\\n\\eta_t(x) &= P(T=t \\mid X=x)=\\Eb{e_t(Z)\\mid X=x}.\n\\end{align*}\n\nFinally, we denote by $\\varphi(z; x,t)$ the conditional density of $Z$ given $X=x,T=t$. This density represents the latent variable model underlying the observables. For example, this can be a Gaussian mixture model, a PCA-type model as in \\citep{kallus2018causal}, or a deep variational autoencoder as in \\citep{louizos2017causal}. Because we focus on how one might use such a latent model rather than the estimation of this model, we just assume we have some oracle for calculating its values. (Note that for fair comparison, in experiments in \\cref{sec:experiments}, we similarly let the outcome regression methods use this oracle.)\n\n\n\n\n\n\n\n\n\nWe further make the following regularity assumptions:\n\\begin{assumption}[Weak Overlap]\n\\label{asm:overlap}\n$\\Eb{e^{-2}_t(Z)}<\\infty$\n\\end{assumption}\n\n\\begin{remark}\n\\label{rmk:overlap}\nGiven \\cref{asm:overlap} it trivially follows that for every $t \\in \\{1,\\ldots,m\\}$, $x \\in \\mathcal X$, $z \\in \\mathcal Z$ that $e_t(z) > 0$ and $\\eta_t(x) > 0$.\n\\end{remark}\n\n\\begin{assumption}[Bounded Variance]\n\\label{asm:variance}\nThe conditional variance of our potential outcomes given $X,T$ is bounded: $\\mathbb{V}[Y(t) \\mid X, T]\\leq\\sigma^2$.\n\\end{assumption}\n\n\\subsection{The Policy Evaluation Task}\n\nThe problem we consider is to estimate the policy value $\\tau^\\pi$ given a policy $\\pi$ and data $X_{1:n},T_{1:n},Y_{1:n}$. One standard approach to this is the \\emph{direct method} \\citep{qian2011performance}, which given an estimate $\\hat \\rho_t$ of $\\rho_t$ predicts the policy value as\n\\begin{equation}\n\\label{eq:direct}\n\\hat{\\tau}^\\pi_{\\hat\\rho} = \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m \\pi_t(X_i) \\hat{\\rho}_t(X_i).\n\\end{equation}\n\nHowever this method is known to be biased and doesn't generalize well \\citep{beygelzimer2009offset}. Furthermore given \\cref{asm:ignorability} $\\hat \\rho$ is not straightforward to estimate, since the mean value of $Y$ observed in our logged data given $X=x$ and $T=t$ is  $\\nu_t(x,t)$ not $\\rho_t(x)$, so fitting $\\hat \\rho$ would require controlling for the effects of the unobserved $Z$.\n\nAn alternative to this is to come up with weights $W_{1:n}=f_W(X_{1:n},T_{1:n})$ according to some function $f_W$ of the observed covariates and treatments, in order to re-weight the outcomes to look more like those that would be observed under $\\pi$. Using these weights we can define the \\emph{weighted} estimator\n\\begin{equation}\\label{eq:weighted}\n\\hat{\\tau}^{\\pi}_W = \\frac{1}{n} \\sum_{i=1}^n W_i Y_i.\n\\end{equation}\n\nThis weighted estimator has the advantage that it does not require modeling the outcome distributions. Furthermore we could combine the weights $W_{1:n}$ with an outcome model $\\hat \\rho_t$ to calculate the \\emph{doubly robust} estimator \\citep{dudik2011doubly}, which is defined as\n\\begin{equation}\n\\label{eq:dr}\n\\hat{\\tau}^{\\pi}_{W,\\hat{\\rho}} = \\frac{1}{n} \\sum_{i=1}^n \\sum_{t=1}^m \\pi_t(X_i) \\hat{\\rho}_t(X_i) + \\frac{1}{n} \\sum_{i=1}^n W_i (Y_i - \\hat{\\rho}_{T_i}(X_i)).\n\\end{equation}\nThe doubly robust estimator is known to be consistent when either the weighted or direct estimator is consistent and can attain local efficiency \\citep{chernozhukov2016double,robins2000robust,robins1994estimation,scharfstein1999adjusting}. \n\nVarious approaches exist for coming up with weights for either the weighted or doubly robust estimators, which we discuss below. However none of these methods are applicable given \\cref{asm:ignorability}, and so we develop a theory for weighting using proxy variables in \\cref{sec:method}.\n\n\n\\subsection{Related Work}\n\n\nOne of the most standard approaches for policy evaluation is using the weighted or doubly robust estimator defined in \\cref{eq:weighted,eq:dr}, using inverse propensity score (IPS) weights. These are given by $W_i = \\pi_{T_i}(X_i) \/ e_{T_i}(Z_i)$ \\citep{bottou2013counterfactual}, where $e_t$ are known or estimated logging probabilities. \nSince these weights can be extreme, both normalization \\citep{austin2015moving,lunceford2004stratification,swaminathan2015self} and clipping \\citep{elliott2008model,ionides2008truncated,swaminathan2015counterfactual} are often employed. In addition some other approaches include recursive partitioning \\citep{kallus2016recursive}.\nNone of these methods are applicable to our setting however, since we do not know the true confounders $Z_{1:n}$.\n\n\n\nAn alternative to approaches based on fixed formulae for computing the importance weights is to compute weights that optimize an imbalance objective function \\citep{athey2018approximate,kallus2017framework,kallus2018optimal}. For policy evaluation, \\citet{kallus2018balanced} propose to choose weights that adversarially minimize the conditional mean squared error of policy evaluation in the worst case of possible mean outcome functions in some reproducing kernel Hilbert space (RKHS) class, by solving a linearly constraint quadratic program (LCQP). Our work follows a very similar style to this, however instead of using the true confounders we only assume access to proxies, and we prove our theory for more general families of functions. \n\nFinally there has been a long history of work in causal inference using proxies for true confounders \\citep{wickens1972note,frost1979proxy}. As in our problem setup, much of this work is based on the model of using an identified latent variable model for the proxies \\citep{wooldridge2009estimating,pearl2012measurement,kuroki2014measurement,miao2016identifying,edwards2015all}. Some recent work on this problem involves using techniques such as matrix completion \\citep{kallus2018balanced} or variational autoencoders \\citep{louizos2017causal} to infer confounders from the proxies. In addition, there is a variety of work that studies sufficient conditions for the identifiability of latent confounder models \\citep{cai2008identifying,pearl2012measurement,miao2016identifying}. Our work is complementary to this line of research in that we assume access to an identified latent confounder model, but do not study how to identify such models. Furthermore our work is novel in combining proxy variable models with optimal balancing and applying it to finding importance weights for policy evaluation.\n\n\n\n\n\n\n\n\\section{Experiments}\n\\label{sec:experiments}\n\n\\subsection{Experimental Setup}\n\\label{sec:exp-setup}\n\nWe now present a brief set of experiments to explore our methodology. The aim of these experiments is to be a proof of concept of our theory. We seek to show that given an identified posterior model $\\varphi$ policy as discussed in \\cref{sec:setting}, evaluation using the weights defined by \\cref{eq:optim} can give unbiased policy evaluation even in the face of sufficiently strong confounding where standard benchmark approaches that rely on ignorability given $X$ fail. We experiment with the following generalized linear model-style scenario:\n\n\\begin{center}\n\\begin{tabular}{lll}\n$Z \\sim \\mathcal{N}(0,1)$ \\qquad & \\qquad $X \\sim \\mathcal{N}(\\alpha^T Z + \\alpha_0, \\sigma^2_X)$ \\qquad & \\qquad $P_T = \\beta^T Z + \\beta_0$ \\\\\n$T \\sim \\op{softmax}(P_T)$ \\qquad & \\qquad $W(t) \\sim \\mathcal{N}(\\zeta(t)^T Z + \\zeta_0(t), \\sigma^2_Y)$ \\qquad & \\qquad $Y(t) = g(W(t))$ \n\\end{tabular}\n\\end{center}\n\nIn our experiments $Z$ is $1$-dimensional, $X$ is $10$-dimensional, and we have two possible treatment levels ($m=2$). We experiment with a parametric policy $\\pi$ and multiple link functions $g$ as follows:\n$$\\pi_t(X) = \\frac{\\exp(\\psi_t^T X)}{\\exp(\\psi_1^T X) + \\exp(\\psi_2^T X)}$$\n$$\n\\text{\\textbf{step}: } g(w) = 3 \\mathbbm{1}_{\\{w \\geq 0\\}} - 6 \\qquad\n\\text{\\textbf{exp}: } g(w) = \\exp(w) \\qquad\n\\text{\\textbf{cubic}: } g(w) = w^3 \\qquad\n\\text{\\textbf{linear}: } g(w) = w \n$$\n\nWe experiment with the following methods in this evaluation:\n\\begin{enumerate}\n\\item \\textbf{OptZ} Our method, using $\\Gamma = \\gamma \\op{Identity}(n)$ for $\\gamma \\in \\{0.001, 0.2, 1.0, 5.0\\}$.\n\\item \\textbf{IPS} IPS weights based on $X$ using estimated $\\hat\\eta_t$.\n\\item \\textbf{OptX} The optimal weighting method of \\citet{kallus2018balanced} with same values of $\\Gamma$ as our method.\n\\item \\textbf{DirX} Direct method by fitting $\\hat \\rho_t(x)$ incorrectly assuming ignorability given $X$.\n\\item \\textbf{DirZ} Direct method by first fitting $\\hat\\mu_t$ using posterior samples from $\\varphi$, then using the estimate $\\hat \\rho_t(x) = (1\/D) \\sum_{i=1}^D \\hat\\mu_t(z_i')$, where $z_i'$ are sampled from $\\varphi(\\cdot;x,t)$.\n\\item \\textbf{D:W} Doubly robust estimation using direct estimator $\\textbf{D}$ and weighted estimator $\\textbf{W}$.\n\\end{enumerate}\n\nFinally we detail all choices for scenario parameters in \\cref{apx:exp-scenario}, and provide implementation details of our methods in \\cref{apx:exp-benchmark}.\\footnote{Code available online at \\url{https:\/\/github.com\/CausalML\/LatentConfounderBalancing}.}\n\n\n\n\\subsection{Results}\nWe display results for our experiments using the \\textbf{step} link function in \\cref{tab:results-ours,tab:results-benchmarks}. For each of $n \\in \\{200, 500, 1000, 2000\\}$ we estimate the RMSE of policy evaluation using each method, as well as doubly robust evaluation using our best performing weights, by averaging over 64 runs. In addition, in \\cref{tab:results-ours-bias,tab:results-benchmarks-bias} we display the estimated bias from the evaluations. It is clear that the naive methods that assume ignorability given $X$ all hit a performance ceiling, where bias converges to some non-zero value. In particular for $\\text{IPS}$ we separately ran it on up to one million data points and found that the bias converged to $0.418\\pm0.001$. One the other hand, for our method it appears like we have consistency. This is particularly evident when we look at \\cref{tab:results-ours-bias}, as bias seems to be approximately converging to zero with vanishing variance. We can also observe that doubly robust estimation using either direct method does not appear to improve performance.\n\nIt is noteworthy that the \\textbf{DirZ} benchmark method fails horribly, despite being a correctly specified regression estimate. From our experience we observed that it is difficult to train the $\\mu_t$ functions accurately if there is a high amount of overlap in the $\\varphi(\\cdot;x,t)$ posteriors for fixed $t$. Therefore we postulate that in highly confounded settings this benchmark inherently difficult to train using a finite number of samples from $\\varphi(\\cdot;x,t)$, and the result seems to collapse to degenerate solutions.\n\nFinally we note that we observed similar trends to this using our other link functions, and other doubly robust estimators. We present more extensive tables of results in \\cref{apx:exp-results}. In addition we present some results there on the negative impact on our method's performance using the constraint $W \\in n\\Delta^n$, as mentioned in \\cref{sec:qpalgo}.\n\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$ & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.001}$\\\\\n\\hline\n200 & $.39\\pm.07$ & $.24\\pm.02$ & $.36\\pm.02$ & $.81\\pm.02$ & $.57\\pm.06$ & $.41\\pm.07$\\\\\n500 & $.19\\pm.02$ & $.18\\pm.02$ & $.23\\pm.02$ & $.49\\pm.02$ & $.55\\pm.03$ & $.20\\pm.02$\\\\\n1000 & $.11\\pm.01$ & $.11\\pm.01$ & $.13\\pm.01$ & $.27\\pm.01$ & $.49\\pm.02$ & $.11\\pm.01$\\\\\n2000 & $.08\\pm.01$ & $.08\\pm.01$ & $.09\\pm.01$ & $.17\\pm.01$ & $.48\\pm.01$ & $.08\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for for policy evaluation using our weights.}\n\\label{tab:results-ours}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.47\\pm.03$ & $2.0\\pm.03$ & $2.1\\pm.03$ & $2.3\\pm.02$ & $2.5\\pm.02$ & $.52\\pm.02$ & $2.6\\pm.02$\\\\\n500 & $.48\\pm.03$ & $2.0\\pm.02$ & $2.1\\pm.02$ & $2.3\\pm.02$ & $2.6\\pm.02$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n1000 & $.39\\pm.02$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n2000 & $.40\\pm.01$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.45\\pm.02$ & $2.6\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods.}\n\\label{tab:results-benchmarks}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$ & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.001}$\\\\\n\\hline\n200 & $.03\\pm.39$ & $.11\\pm.21$ & $.29\\pm.21$ & $.78\\pm.18$ & $.43\\pm.38$ & $.05\\pm.40$\\\\\n500 & $.09\\pm.17$ & $.10\\pm.15$ & $.17\\pm.16$ & $.47\\pm.15$ & $.51\\pm.19$ & $.10\\pm.18$\\\\\n1000 & $.02\\pm.11$ & $.05\\pm.09$ & $.08\\pm.09$ & $.25\\pm.09$ & $.47\\pm.13$ & $.04\\pm.11$\\\\\n2000 & $.03\\pm.07$ & $.05\\pm.06$ & $.07\\pm.07$ & $.16\\pm.07$ & $.47\\pm.09$ & $.03\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for policy evaluation using our weights.}\n\\label{tab:results-ours-bias}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccccccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.40\\pm.25$ & $1.9\\pm.21$ & $2.1\\pm.20$ & $2.3\\pm.19$ & $2.5\\pm.18$ & $.49\\pm.18$ & $2.6\\pm.14$\\\\\n500 & $.43\\pm.21$ & $2.0\\pm.16$ & $2.1\\pm.15$ & $2.3\\pm.14$ & $2.6\\pm.13$ & $.45\\pm.16$ & $2.6\\pm.12$\\\\\n1000 & $.37\\pm.12$ & $2.0\\pm.10$ & $2.1\\pm.09$ & $2.3\\pm.09$ & $2.5\\pm.08$ & $.46\\pm.15$ & $2.6\\pm.11$\\\\\n2000 & $.39\\pm.10$ & $2.0\\pm.08$ & $2.1\\pm.07$ & $2.3\\pm.07$ & $2.5\\pm.07$ & $.42\\pm.17$ & $2.6\\pm.11$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods.}\n\\label{tab:results-benchmarks-bias}\n\\end{table}\n\n\n\n\n\\section{Omitted Proofs}\n\n\n\\subsection{Proof of \\cref{thm:ips}}\n\\label{apx:ips}\n\n\nFirst we note that $W(X,T)Y - \\sum_t \\pi_t(X) Y(t)] = \\sum_t (W(X,t) \\delta_{Tt} - \\pi_t(X)) Y(t)$. Then analyzing each summand separately, we can obtain:\n\\begin{align}\n\\mathbb{E}&[W(X,t) \\delta_{Tt} Y(t) - \\pi_t(X) Y(t)] \\nonumber \\\\\n&= \\mathbb{E}[ \\mathbb{E}[ W(X,t) \\delta_{Tt} Y(t) - \\pi_t(X) Y(t) \\mid X] ] \\nonumber \\\\\n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\delta_{Tt} Y(t) \\mid X] - \\pi_t(X) \\mathbb{E}[Y(t) \\mid X] ] \\nonumber \\\\\n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\mathbb{E}[\\delta_{Tt} Y(t) \\mid X,T] \\mid X] - \\pi_t(X) \\mathbb{E}[\\mathbb{E}[Y(t) \\mid X,Z] \\mid X] ] \\nonumber \\\\\n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\delta_{Tt} \\mathbb{E}[Y(t) \\mid X,T=t] \\mid X] - \\pi_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X] ] \\nonumber \\\\ \n&= \\mathbb{E}[ W(X,t) \\mathbb{E}[\\delta_{Tt} \\mid X] \\mathbb{E}[\\mu_t(Z) \\mid X,T=t] - \\pi_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X] ] \\nonumber \\\\ \n&= \\mathbb{E}[ W(X,t) \\eta_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X,T=t] - \\pi_t(X) \\mathbb{E}[\\mu_t(Z) \\mid X] ] \\nonumber \\\\ \n&= \\mathbb{E}\\left[ W(X,t) \\eta_t(X) \\nu_t(X,t) - \\pi_t(X) \\sum_{t'} \\eta_{t'}(X) \\nu_t(X,t') \\right] \\nonumber \n\\end{align}\n\nTherefore the solution to the equation $\\mathbb{E}[W(X,t) \\delta_{Tt} Y(t) - \\pi_t(X) Y(t)] = 0$ is given by:\n\\begin{equation*}\nW(X,t) \\eta_t(X) \\nu_t(X,t) - \\pi_t(X) \\sum_{t'} \\eta_{t'}(X) \\nu_t(X,t') = \\Omega_t(X)\n\\end{equation*}\nwhere $\\Omega_t(X)$ is any arbitrary function of $X$ with mean zero. Solving this for $W$ gives\n\\begin{equation*}\nW(X,t) = \\frac{\\pi_t(X) \\sum_{t'} \\eta_{t'}(X) \\nu_t(X,t') + \\Omega_t(X)}{\\eta_t(X) \\nu_t(X,t)},\n\\end{equation*}\nand finally replacing $X$ with $x$ gives the required solution.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:jbound}}\n\\label{apx:jbound}\nDefine \n$$\\tau^{\\pi}_{SAPE} = \\frac{1}{n} \\sum_i \\sum_t \\pi_t(X_i) \\mu_t(Z_i).\n$$\nUsing $(x+y)^2\\leq2x^2+2y^2$, we have\n\\begin{align*}\n\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]\n\\leq 2\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE})^2 \\mid X_{1:n},T_{1:n} ]+2\\mathbb{E}[ (\\tau^{\\pi}_{SAPE} - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ].\n\\end{align*}\nNoting that $\\tau^{\\pi} = \\mathbb{E}[\\tau^{\\pi}_{SAPE}]$, \\Cref{asm:variance} implies that \n$\\mathbb{E}[ (\\tau^{\\pi}_{SAPE} - \\tau^{\\pi})^2]=O(1\/n)$.\nMarkov's inequality yields $\\mathbb{E}[ (\\tau^{\\pi}_{SAPE} - \\tau^{\\pi})^2 \\mid X_{1:n},T_{1:n} ]=O_p(1\/n)$.\n\nLet $CMSE(W,\\mu)=\\mathbb{E}[ (\\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE})^2 \\mid X_{1:n},T_{1:n} ]$. We proceed to bound $CMSE$.\nBy iterating expectations we can obtain:\n\\begin{align}\nCMSE(W, \\mu) &= \\mathbb{E}[ ( \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} )^2 \\mid X, T ] \\nonumber \\\\\n&= \\mathbb{E}[ \\mathbb{E}[ ( \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} )^2 \\mid X, T, Z \\mid X, T ] \\nonumber \\\\\n&= \\mathbb{E}[ \\mathbb{E}[ \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} \\mid X, T, Z]^2  \\mid X, T ] + \\mathbb{E}[ \\mathbb{V}[ \\hat{\\tau}^{\\pi}_W - \\tau^{\\pi}_{SAPE} \\mid X, T, Z] \\mid X, T] \\nonumber \\\\\n&= \\mathbb{E}[ ( \\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) )^2 \\mid X, T ] + \\mathbb{E}[ \\mathbb{V}[ \\frac{1}{n} \\sum_i W_i Y(T_i)  \\mid X, T, Z] \\mid X, T] \\nonumber \\\\\n&\\leq \\mathbb{E}[ ( \\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) )^2 \\mid X, T ] + \\frac{\\sigma^2}{n^2} \\|W\\|_2^2 \\nonumber\n\\end{align}\nwhere $\\sigma$ is the bound defined in \\cref{asm:variance}.\n\nNext observe that for any (possibly correlated) random variables $A_1,\\ldots,A_n$ and numbers $p_1,\\ldots,p_n \\in \\R n$ such that $\\sum_ip_i=1$, we have $\\mathbb{V}[\\sum_i p_i A_i] \\leq \\max_i \\mathbb{V}[A_i]$. Given this, we can simplify the first term above further, as follows:\n\\begin{align}\n\\mathbb{E}[ ( \\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) )^2 \\mid X, T ] &= (\\frac{1}{n} \\sum_{i,t} f_{it} \\mathbb{E}[\\mu_t(Z_i) \\mid X_i, T_i])^2 + \\mathbb{V}[\\frac{1}{n} \\sum_{i,t} f_{it} \\mu_t(Z_i) \\mid X, T] \\nonumber \\\\\n&= (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{1}{n^2} \\sum_i \\mathbb{V}[ \\sum_t f_{it} \\mu_t(Z_i) \\mid X, T] \\nonumber \\\\\n&\\leq (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{1}{n^2} \\sum_i \\max_t f_{it}^2 \\mathbb{V}[\\mu_t(Z_i) \\mid X, T] \\nonumber \\\\\n&\\leq (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{2\\sigma^2}{n^2} \\sum_i \\max_t W_i^2 \\delta_{T_i t} + \\pi_t(X_i)^2 \\nonumber \\\\\n&\\leq (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{2\\sigma^2}{n^2} \\sum_i (W_i^2 + 1) \\nonumber \\\\\n&= (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_t(X_i,T_i))^2 + \\frac{2\\sigma^2}{n^2} \\|W\\|_2^2 + \\frac{2 \\sigma^2}{n} \\nonumber\n\n\\end{align}\nwhere for the second inequality we used the fact that $(x+y)^2 \\leq 2x^2 + 2y^2$. This gives us $CMSE(W, \\mu) \\leq J(W, \\mu) + O_p(1\/n)$, and combining this with the above gives $\\mathbb{E}[(\\hat{\\tau}^{\\pi}_W~-~\\tau^{\\pi})^2~\\mid~X_{1:N},~T_{1:N}] \\leq 2J(W, \\mu) + O_p(1\/n)$ as required.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{lem:gbound}}\n\\label{apx:gbound}\n\nFirst, we will use the notation $f(z;x,t)$ for the conditional measure of $Z$ given $X=x$ and $T=t$, and observe that according to Bayes rule we have:\n\\begin{equation}\n\\frac{f(z;x,t'')}{f(z;x,t')} = \\frac{e_{t''}(z)}{e_{t'}(z)} \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\nonumber\n\\end{equation}\n\nDefine $\\mathbb{E}_{xt}$ and $\\mathbb{P}_{xt}$ as shorthand for expectation and probability given $X=x,T=t$ respectively. Then given the above, for any $M > 0$ we can bound\n\\begin{align*}\n\\nu_t(x,t'') &= \\frac{\\eta_{t''}(x)}{\\eta_{t'}(x)} \\mathbb{E}[\\mu_t(Z) \\mid X=x, T=t''] \\\\\n&= \\int_{\\mathcal{Z}} f(z;x,t'') \\mu_t(z) dz \\\\\n&= \\frac{\\eta_{t''}(x)}{\\eta_{t'}(x)} \\int_{\\mathcal{Z}} \\frac{e_{t''}(z)}{e_{t'}(z)} \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} f(z;x,t') \\mu_t(z) dz \\\\\n&= \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\int_{\\mathcal{Z}} \\frac{e_{t''}(z)}{e_{t'}(z)} f(z;x,t') \\mu_t(z)  dz \\\\\n&\\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\left( M \\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} \\leq M} \\mu_t(Z)] + \\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\frac{e_{t''}(z)}{e_{t'}(z)} \\mu_t(Z)] \\right) \\\\\n\\end{align*}\n\nNow we can use the fact that $\\mu_t$ is $b$-bounded to bound the first term by\n\\begin{align*}\nM \\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} \\leq M} \\mu_t(Z)] &= M \\nu_t(x,t') - M\\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\mu_t(Z)] \\\\\n&\\leq M \\nu_t(x,t') + Mb\\mathbb{P}_{xt'}[\\frac{e_{t''}(z)}{e_{t'}(z)} > M],\n\\end{align*}\nand in addition applying Cauchy Schwartz we can bound the second term by\n\\begin{align*}\n\\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\frac{e_{t''}(z)}{e_{t'}(z)} \\mu_t(Z)]  &\\leq \\sqrt{\\mathbb{E}_{xt'}[\\indicator{\\frac{e_{t''}(z)}{e_{t'}(z)} > M} \\mu_t(Z)^2] \\mathbb{E}_{xt'}[\\left(\\frac{e_{t''}(z)}{e_{t'}(z)}\\right)^2]} \\\\\n&\\leq \\sqrt{b^2 \\mathbb{P}_{xt'}[\\frac{e_{t''}(z)}{e_{t'}(z)} > M] \\mathbb{E}_{xt'}[\\left(\\frac{e_{t''}(z)}{e_{t'}(z)}\\right)^2]} \\\\\n&\\leq b \\sqrt{\\mathbb{P}_{xt'}[\\frac{1}{e_{t'}(z)} > M] \\mathbb{E}_{xt'}[\\left(\\frac{1}{e_{t'}(z)}\\right)^2]}.\n\\end{align*}\n\nNow define $g(x,t) = \\mathbb{E}_{xt}[\\left(\\frac{1}{e_t(z)}\\right)^2]$. By \\cref{asm:overlap} we know $g(x,t)$ is finite for every $x$ and $t$. Also by Markov's inequality we know that $\\mathbb{P}_{xt}[\\frac{1}{e_t(z)} > M] \\leq \\frac{g(x,t)}{M^2}$. Therefore putting all of the above together we can obtain\n\\begin{align}\n\\nu_t(x,t'') &\\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\left(M \\nu_t(x,t') + \\frac{2bg(x,t')}{M} \\right) \\nonumber \\\\\n&\\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\left(M |\\nu_t(x,t')| + \\frac{2bg(x,t')}{M} \\right) \\nonumber \n\\end{align}\nThis inequality is valid every $M$, so we can pick $M$ to make it as tight as possible. Choosing $M = \\sqrt{\\frac{2bg(x,t')}{\\nu_t(x,t')}}$ gives us:\n\\begin{equation}\n\\nu_t(x,t'') \\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\sqrt{8bg(x,t')|\\nu_t(x,t')|} \\nonumber \n\\end{equation}\n\nFinally note that, since by symmetry $\\mathbb{E}[\\mu(Z) \\mid X,T] = -\\mathbb{E}[-\\mu(Z) \\mid X,T]$, we can strengthen this inequality to the following\n\\begin{equation*}\n|\\nu_t(x,t'')| \\leq \\frac{\\eta_{t'}(x)}{\\eta_{t''}(x)} \\sqrt{8bg(x,t')|\\nu_t(x,t')|},\n\\end{equation*}\nand noting that $g(x,t') = \\mathbb{E}[e^{-2}_t(Z) \\mid X=x, T=t']$ gives us our final result.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{lem:minimax}}\n\\label{apx:minimax}\n\n\nFirst note that by \\cref{asm:compact} $\\mathcal{F}$ is compact. Also $J(W,\\cdot)$ is continuous for every $W$, since by \\cref{asm:ltwo} we know that the norm on each $\\mathcal F_t$ dominates the norm on $\\ltwo(\\mathcal Z)$ and this continuity result would be trivial if $\\mathcal F_t = \\ltwo(\\mathcal Z)$. This means that by the Extreme Value theorem we can replace the supremum over $\\mu$ with a maximum over $\\mu$ in the quantity we are bounding. Given this, we will proceed by bounding $\\min_W \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)$ using von Neumann's minimax theorem to swap the minimum and the maximum, and then use this to establish the overall bound for $J(W,\\mu)$. \n\nNext we can observe that $B(W, \\mu)$ is linear, and therefore both convex and concave, for each of $W$ and $\\mu$. Next, by \\cref{asm:compact} $\\mathcal{F}$ is convex and compact, and following the same argument as above $B(W,\\cdot)$ is continuous for every $W$. In addition, $B(W, \\mu)$ is also clearly continuous in $W$ for fixed $\\mu$, and the set $\\{W : \\|W\\|_2 \\leq M\\}$ is obviously compact and convex for any constant $M$. Thus by von Neumann's minimax theorem we have the following for every finite $M$:\n\\begin{equation}\n\\label{eq:minimax}\n\\min_{\\|W\\|_2 \\leq M} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu) = \\max_{\\mu \\in \\mathcal{F}} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu)\n\\end{equation}\n\nGiven this, we can bound $\\min_W \\max_{\\mu \\in \\mathcal{F}} \\hat{J}(W, \\mu)$ as follows, which is valid for any $M$:\n\\begin{align}\n\\min_{W} \\max_{\\mu \\in \\mathcal{F}} \\hat{J}(W, \\mu) &\\leq \\min_{W} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)^2 + \\frac{1}{n^2} W^T \\Sigma W \\nonumber \\\\\n&\\leq \\min_{W} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2} \\|W\\|^2_2 \\nonumber \\\\\n&\\leq \\min_{\\|W\\|_2 \\leq M} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2}\\|W\\|^2_2 \\nonumber \\\\\n&\\leq (\\min_{\\|W\\|_2 \\leq M} \\max_{\\mu \\in \\mathcal{F}} B(W, \\mu))^2 + \\frac{\\sigma^2}{n^2}M^2 \\nonumber \\\\\n&= (\\max_{\\mu \\in \\mathcal{F}} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu))^2 + \\frac{\\sigma^2}{n^2}M^2 \\nonumber \\\\\n&= \\max_{\\mu \\in \\mathcal{F}} \\min_{\\|W\\|_2 \\leq M} B(W, \\mu)^2 + \\frac{\\sigma^2}{n^2}M^2 \\nonumber\n\\end{align}\n\nIn these inequalities we use the fact that $\\min_W \\max_{\\mu} B(W, \\mu)^2 = (\\min_W \\max_{\\mu} B(W, \\mu))^2$ and $\\max_{\\mu} \\min_W B(W, \\mu)^2 = (\\max_{\\mu} \\min_W B(W, \\mu))^2$ due to the symmetry of $\\mu$ in $\\mathcal{F}$ implied by \\cref{asm:symmetry}.\n\n\n\n\n\n\n\\subsection{Proof of \\cref{lem:jconvergence}}\n\\label{apx:jconvergence}\n\nLet $\\prod$ denote Cartesian product. First we note that without loss of generality we can prove this lemma in the case that $\\mathcal F = \\prod_t \\mathcal F_t$, since in general $F \\subseteq \\prod_t \\mathcal F_t$ so $\\sup_{\\mu \\in \\mathcal F} \\inf_W J(W,\\mu) \\leq \\sup_{\\mu \\in \\prod_t \\mathcal F_t} \\inf_W J(W,\\mu)$, and it is easy to verify that all of our assumptions would still hold on the larger set $\\prod_t \\mathcal F_t$. \n\nNow define the set $\\mathcal{H}^0 = \\{\\mu \\in \\prod_t \\ltwo(\\mathcal Z) : \\mathbb{E}[\\nu_T(X,T)^2] = 0\\}$. Each coordinate $\\mathcal H_t^0$ of $\\mathcal H^0$ is a subspace of $\\ltwo(\\mathcal Z)$, so we can also define its orthogonal complement $\\mathcal{H}_t^+$. Also, we have separability since from \\cref{sec:setting} we know that $\\mathcal Z \\subseteq \\mathbb{R}^q$, so any function $f \\in \\ltwo(\\mathcal Z)$ can be uniquely represented as $f = f^0 + f^+$ where $f^0 \\in \\mathcal{H}_t^0$ and $f^+ \\in \\mathcal{H}_t^+$. This means that for each $\\mu_t \\in \\mathcal F_t$, we can similarly uniquely represent $\\mu_t = \\mu_t^0 + \\mu_t^+$, and we can easily extend this to a unique representation of the vector $\\mu = \\mu^0 + \\mu^+$. Now in the case that $\\mathbb{E}[\\nu_T(X,T)^2] = 0$ we have $\\nu_{T_i}(X_i,T_i) = 0$ almost surely for all $i$, and it follows from \\cref{lem:gbound} that $\\nu_{T_i}(X_i,t) = 0$ almost surely also for all $i$ and $t$. Therefore any component of $\\mu$ in $\\mathcal{H}^0$ has no effect on the function $J(W,\\mu)$ which we are bounding, so without loss of generality we can restrict our attention to the following space:\n\\begin{equation*}\n\\mathcal{F}^+ = \\prod_t (\\mathcal{H}_t^+ \\cap \\mathcal F_t).\n\\end{equation*}\n\nBy construction the only function in $\\mathcal{F}^+$ such that $\\mathbb{E}[\\nu_T(X,T)^2] = 0\\}$ is the zero function, which we can also ignore in our bounds below, since when $\\mu = 0$ we can easily obtain $J(W,\\mu) = 0$ by choosing $W = 0$. Furthermore, by \\cref{asm:ltwo} we know that for each $t$ the $\\mathcal F_t$ norm dominates the $\\ltwo(\\mathcal Z)$ norm, so it must be the case that that each space $\\mathcal{F}_t^+$ is closed, since $\\mathcal{H}_t^+$ is a closed subspace of $\\ltwo(\\mathcal Z)$ due to it being an orthogonal complement. Thus it follows easily from \\cref{asm:norm} that $\\mathcal{F}^+$ is closed, given that its norm is an $\\mathbb{R}^m$ norm on top of the corresponding $\\mathcal{F}^+_t$ norms and $m$ is finite.\n\nNow, based on \\cref{lem:minimax}, it is sufficient to pick weights in response to $\\mu$ that control for a single mean outcome function. Instead of actually constructing a particular set of weights, we take the approach of viewing this as a convex optimization problem. Specifically, given $\\mu$, we calculate the minimum euclidean norm of all weights that set the bias term $B(W,\\mu)$ to zero exactly. This can be formulated as the following convex optimization program\n\\begin{align*}\n&\\min_W \\sum_i W_i^2 \\\\\n&\\text{s.t.} \\sum_i W_i \\nu_{T_i}(X_i, T_i) = \\sum_{i,t} \\pi_t(X_i) \\nu_t(X_i, T_i).\n\\end{align*}\n\nGiven the program only has linear constraints with equality, it satisfies Slater's condition, and therefore satisfies strong duality, which we will use to find the optimal value of this program. First we calculate the Lagrangian as\n\\begin{equation*}\n\\mathcal L_n(W, \\lambda) = \\sum_i W_i^2 + \\lambda \\left( \\sum_i W_i \\nu_{T_i}(X_i, T_i) - \\sum_{i,t} \\pi_t(X_i) \\nu_t(X_i, T_i) \\right).\n\\end{equation*}\n\nIt can easily be verified by taking derivatives that for any $\\lambda \\in \\mathbb{R}$ this function is minimized by setting $W_i = -\\frac{\\lambda}{2} \\nu_{T_i}(X_i,T_i)$, and plugging this value in gives the dual formulation of the program as\n\\begin{align*}\n&\\max_{\\lambda} -\\frac{\\lambda^2}{4} \\sum_i \\nu_{T_i}(X_i,T_i)^2 - \\lambda \\sum_{i,t} \\pi_t(X_i) \\nu_t(X_i,T_i),\n\\end{align*}\nwhich is unconstrained. Again by taking derivatives we can maximize this function, and we find the maximum value is given by\n\\begin{equation*}\n\\lambda = -2\\frac{\\sum_i \\pi_t(X_i) \\nu_t(X_i, T_i)}{\\sum_i \\nu_{T_i}(X_i, T_i)^2},\n\\end{equation*}\nand finally plugging this value into the dual objective function we see that the euclidean norm of the weights $W^*$ solving the convex program above is given by\n\\begin{equation*}\n\\|W^*\\|_2^2 = \\frac{\\left(\\sum_i \\pi_t(X_i) \\nu_t(X_i, T_i)\\right)^2}{\\sum_i \\nu_{T_i}(X_i, T_i)^2}.\n\\end{equation*}\n\nNow define $\\mathbb{E}_n$ as the mean with respect to the empirical distribution of the logged data. Then this objective value can be reformulated as\n\\begin{equation*}\n\\|W^*\\|_2^2 = \\frac{n\\mathbb{E}_n[\\sum_t \\nu_t(X,T)]^2}{\\mathbb{E}_n[\\nu_{T}(X,T)^2]}.\n\\end{equation*}\n\nTherefore choosing $M = \\|W^*\\|$, combining this result with \\cref{lem:minimax} gives us\n\\begin{align*}\n\\min_W \\sup_{\\mu \\in \\mathcal{F}^+} J(W, \\mu) &\\leq \\sup_{\\mu \\in \\mathcal{F}^+} \\frac{1}{n} \\left( \\frac{\\sigma^2\\mathbb{E}_n[\\sum_t \\nu_t(X,T)]^2}{\\mathbb{E}_n[\\nu_{T}(X,T)^2]} \\right) \\\\\n&= \\sup_{\\mu \\in \\mathcal{F}^+} \\frac{1}{n} \\left( \\frac{\\sigma^2\\mathbb{E}_n[\\sum_t \\nu_t(X,T)]^2}{\\mathbb{E}[\\nu_{T}(X,T)^2] + (\\mathbb{E}_n[\\nu_{T}(X,T)^2] - \\mathbb{E}[\\nu_{T}(X,T)^2])} \\right).\n\\end{align*}\n\nGiven this we will proceed by arguing that we can bound the denominator away from zero. We can note that $\\mu$ appears in both the numerator and denominator on the same scale, so without loss of generality we can further restrict our attention to $\\mu$ with fixed norm. By \\cref{asm:nondegen} we know that we can rescale every $\\mu \\in \\mathcal F$ to have norm $\\gamma$ for some $\\gamma > 0$. Given this we will restrict ourselves to the set $\\mathcal{F}^+_{\\gamma} = \\{\\mu \\in \\mathcal{F}^+ : \\|\\mu\\| = \\gamma\\}$. Since $\\mathcal{F}^+_{\\gamma}$ is the intersection of two closed sets it must be closed. Furthermore by \\cref{asm:compact} it is also compact, so it satisfies the conditions for the extreme value theorem. By construction $\\mathbb{E}[\\nu_T(X,T)^2] > 0$ for every $\\mu \\in \\mathcal{F}^+_{\\gamma}$, so putting the above together we have $\\inf_{\\mu \\in \\mathcal{F}^+_{\\gamma}} \\mathbb{E}[\\nu_T(X,T)^2] > 0$. We will define this value to be $\\alpha$. \n\nNow the numerator in the above bound is clearly bounded above by some $\\beta > 0$ uniformly over $\\mu \\in \\mathcal F$, since by \\cref{asm:bounded} we know that every $\\mu_t \\in \\mathcal F_t$ is uniformly bounded by some global constant, and therefore all $\\nu$ terms are bounded by some constant $b$. Given this all that remains to be shown is that $\\sup_{\\mu \\in \\mathcal F} |\\mathbb{E}_n[\\nu_{T}(X,T)^2] - \\mathbb{E}[\\nu_{T}(X,T)^2]|$ converges in probability to zero. In order to show this we will define the following terms:\n\\begin{align*}\nD_n &= \\mathbb{E}_n[\\nu_T(X,T)^2] \\\\\nE_n &= \\sup_{\\mu \\in \\mathcal{F}} |D_n - \\mathbb{E}[D_n]|\n\\end{align*}\n\nWe need to show that $E_n$ converges uniformly to zero. Define $D_n'$ as an arbitrary recalculation of $D_n$ replacing $(X_{1:n}, T_{1:n})$ with $(X'_{1:n}, T'_{1:n})$, which differ from the originals at most in a single coordinate $i$, and define $E_n' = \\sup_{\\mu \\in \\mathcal{F}} |D_n' - \\mathbb{E}[D_n']|$. Furthermore as argued above all $\\nu$ terms are bounded above by some constant $b$, so each $\\nu_{T_i}(X_i,T_i)^2$ is bounded by $b^2$. Given this we can obtain\n\\begin{align*}\n|E_n - E_n'| &= |\\sup_{\\mu \\in \\mathcal{F}} |D_n - \\mathbb{E}[D_n]| - \\sup_{\\mu \\in \\mathcal{F}} |D_n' - \\mathbb{E}[D_n']| | \\\\\n&\\leq \\sup_{\\mu \\in \\mathcal{F}} |(D_n - \\mathbb{E}[D_n]) - (D_n' - \\mathbb{E}[D_n'])| \\\\\n&= \\sup_{\\mu \\in \\mathcal{F}} |D_n - D_n'| \\\\\n&= \\frac{1}{n} \\sup_{\\mu \\in \\mathcal{F}} |\\nu_{T_i}(X_i,T_i)^2 - \\nu_{T_i'}(X_i',T_i')^2| \\\\\n&\\leq \\frac{2b^2}{n}\n\\end{align*}\nGiven this we can apply McDiarmid's inequality to obtain the following bound:\n\\begin{equation*}\n\\label{eq:mcdiarmid}\nP(|E_n - \\mathbb{E}[E_n]| \\leq \\epsilon) \\leq 2\\exp \\left( -\\frac{n\\epsilon^2}{2b^4} \\right) \n\\end{equation*}\n\nThis implies that $E_n - \\mathbb{E}[E_n] = o_p(1)$. Next we show that $\\mathbb{E}[E_n] = o_p(1)$ also. We do this using a symmetrization argument as follows, where $D'_n$ is defined identically to $D_n$ using iid shadow variables $(X'_i,T'_i)$ in place of $(X_i,T_i)$ for each $i$, and $\\epsilon_i$ are iid Rademacher random variables:\n\\begin{align*}\n\\mathbb{E}[E_n] &= \\mathbb{E}\\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| D_n - \\mathbb{E}[D_n] \\right| \\right] \\\\\n&= \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\nu_{T_i}(X_i,T_i) - \\mathbb{E}[\\nu_{T_i'}(X_i',T_i')] \\right| \\right] \\\\\n&\\leq 2 \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\nu_{T_i}(X_i,T_i)  \\right| \\right]  \\\\\n&\\leq 2 \\sum_t \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\delta_{T_i t} \\nu_{t}(X_i,T_i)  \\right| \\right] \\\\\n&\\leq 4 \\sum_t \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\nu_{t}(X_i,T_i)  \\right| \\right] \\\\\n&\\leq 4 \\sum_t \\mathbb{E} \\left[ \\sup_{\\mu \\in \\mathcal{F}} \\left| \\frac{1}{n} \\sum_i \\epsilon_i \\mu_t(Z_i)  \\right| \\right] \\\\\n&\\leq 4 \\sum_t \\mathcal R_n(\\mathcal F_t) \\\\\n\\end{align*}\nwhere in the third inequality we appeal to the Rademacher comparison lemma \\citep[Thm.~4.12]{ledoux2013probability}. Thus since from \\cref{asm:complexity} we know that the Rademacher complexity of each set $\\mathcal R_n(\\mathcal F_n)$ vanishes, it follows that $\\mathbb{E}[E_n] = o_p(1)$. Putting everything from above together we get\n\\begin{equation*}\n\\min_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) \\leq \\frac{1}{n} \\frac{\\beta}{\\alpha + o_p(1)},\n\\end{equation*}\nso we have $\\min_W \\sup_{\\mu \\in \\mathcal{F}} J(W, \\mu) \\leq O_p(1\/n)$ as required.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:fk}}\n\\label{apx:fk}\n\nFirst, \\cref{asm:norm} follows trivially from the definition of $\\fk$. Next, \\cref{asm:symmetry} and $\\cref{asm:nondegen}$ follow from the fact that $\\fk$ consists of all functions in $\\op{span}(\\mathcal{F}_t)$ with norm at most 1, as does the fact that it is a closed space. Given that $K$ is a Mercer kernel, balls in the corresponding RKHS have finite covering number \\citep{zhou2002covering}, and it follows easily from this that $\\fk$ has finite covering numbers, as its covering number must be bounded above by the sum of the covering numbers of the spaces $\\fkt$. So $\\fk$ is closed and totally bounded with respect to its norm, and therefore compact, which gives us \\cref{asm:compact}. Clearly each $\\mathcal F_t$ is contained in $\\ltwo(\\mathcal Z)$ since RKHS spaces are square integrable, and the fact that the $K$ norm dominates the $\\ltwo$ follows from Mercer's Theorem, which implies that $\\|f\\|_K^2 = \\sum_{i=1}^{\\infty} f_i^2 \/ \\sigma_i$, where $f_i$ is the $i$th eigenvalue of $f$ for some orthonormal basis of $\\ltwo(\\mathcal Z)$, and $\\sigma_i \\geq 0$ converges to zero. This gives us \\cref{asm:ltwo}. Next, by construction each $\\fkt$ consists of all functions in the RKHS up to norm 1. Therefore assuming the kernel $K$ is bounded, it is trivial to verify that function application must be globally bounded, since for any function $f \\in \\fkt$ we have $f(x) \\leq <f, K_x> \\leq \\|f\\| \\sqrt{K(x,x)} \\leq \\sqrt{K(x,x)}$, which gives us \\cref{asm:bounded}. Finally, given this characterization of $\\fkt$ as the 1-ball of the RKHS, it has vanishing Rademacher complexity \\citep[Thm.~2.1]{mendelson2003performance}, so we have \\cref{asm:complexity}.\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:continuous-algorithm}}\n\\label{apx:continuous-algorithm}\n\nFirst we will find a closed form expression for $\\sup_{\\mu \\in \\fk} (\\frac{1}{n} \\sum_{i,t} f_{it} \\nu_{t}(X_i,T_i))^2$. In this derivation we will use the shorthand $\\varphi_{ti}$ for the conditional density of $\\mu_t$ given $X_i$ and $T_i$, and $T_K$ for the kernel intergral operator defined according to $T_K f = \\int_{\\mathcal Z} K(\\cdot,z) f(z) dz$. In this derivation we will make use of the fact that $\\ltwoprod{f}{g} = \\rkhsprod{f}{T_K g}$ for any square integrable $f$ and $g$. Given all this we can obtain:\n\\begin{align*}\n\\sup_{\\mu \\in \\fk} \\left( \\frac{1}{n} \\sum_{i,t} f_{it} \\nu_{t}(X_i,T_i) \\right)^2 &= \\sum_t \\sup_{\\mu_t \\in \\fkt} \\left( \\frac{1}{n} \\sum_i f_{it} \\ltwoprod{\\mu_t}{\\varphi_i} \\right)^2 \\\\\n&= \\sum_t \\sup_{\\mu_t \\in \\fkt} \\ltwoprod{\\mu_t}{\\frac{1}{n} \\sum_i f_{it} \\varphi_i}^2 \\\\\n&= \\sum_t \\sup_{\\mu_t \\in \\fkt} \\rkhsprod{\\mu_t}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}^2 \\\\\n&= \\sum_t \\frac{\\rkhsprod{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}^2}{\\|T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i\\|_K} \\\\\n&= \\sum_t \\rkhsprod{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i} \\\\\n&= \\sum_t \\ltwoprod{\\frac{1}{n} \\sum_i f_{it} \\varphi_i}{T_K \\frac{1}{n} \\sum_i f_{it} \\varphi_i} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\ltwoprod{\\varphi_i}{T_K \\varphi_j} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\int_{\\mathcal Z} \\varphi_i(z) (\\int_{\\mathcal Z'} K(z,z') \\varphi_j(z') dz') dz \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\int_{\\mathcal Z} \\int_{\\mathcal Z'} \\varphi_i(z) \\varphi_j(z') K(z,z') dz' dz \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t} f_{it} f_{jt} \\mathbb{E}[K(Z_i,Z_j')] \n\\end{align*}\n\nGiven this, and recalling that $f_{it} = W_i \\delta_{T_i t} - \\pi_t(X_i)$ we can derive a closed form for our minimization objective, as follows:\n\\begin{align*}\n\\sup_{\\mu \\in \\fk} J(W, \\mu) &= \\frac{1}{n^2} \\sum_{i,j,t}  f_{it} f_{jt} Q_{ij} + \\frac{1}{n^2} \\sum_{i,j} W_i W_j \\Gamma_{ij} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j,t}  Q_{ij} (W_i W_j \\delta_{T_i t} \\delta_{T_j t} - 2 W_j \\delta_{T_j t} \\pi_t(X_i) + \\pi_t(X_i) \\pi_t(X_j)) \\\\\n&\\qquad\\qquad + \\frac{1}{n^2} \\sum_{i,j} W_i W_j \\Gamma_{ij} \\\\\n&= \\frac{1}{n^2} \\sum_{i,j} W_i W_j (Q_{ij} \\delta_{T_i T_j} + \\Gamma_{ij}) - \\frac{2}{n^2} \\sum_j W_j (\\sum_i Q_{ij} \\pi_{T_j}(X_i)) \\\\\n&\\qquad\\qquad + \\frac{1}{n^2} \\sum_{i,j,t} Q_{ij} \\pi_t(X_i) \\pi_t(X_j)\n\\end{align*}\n\nFinally we can conclude by noting that this corresponds to the quadratic program formulation given in the question with $c = \\frac{1}{n^2} \\sum_{i,j,t} Q_{ij} \\pi_t(X_i) \\pi_t(X_j)$.\n\n\n\n\n\n\n\n\\section{Additional Experimentation Details}\n\n\\subsection{Experiment Scenario}\n\\label{apx:exp-scenario}\n\nAll our experiments were conducted using the setup described in \\cref{sec:exp-setup}. We used the following parameter values for our data-generating distribution:\n\\begin{align*}\n\\alpha &= [1.0, -2.0, -1.0, 2.0, 4.0, 0.0, -2.0, -1.0, -3.0, 1.0] \\\\\n\\alpha_0 &= 0 \\\\\n\\sigma^2_X &= 4.0 \\\\\n\\beta &= [0.5, -0.5] \\\\\n\\beta_0 &= 0 \\\\\n\\zeta(0) &= 1.0 \\\\\n\\zeta(0)_0 &= 0 \\\\\n\\zeta(1) &= -0.5 \\\\\n\\zeta(1)_0 &= 0 \\\\\n\\sigma^2_Y &= 0.01 \n\\end{align*}\n\nIn addiiton, the policy $\\pi$ we are evaluating takes the form as described in \\cref{sec:exp-setup}, and we used the following parameter values for this policy:\n\\begin{align*}\n\\psi_0 &= [-0.1, 0.2, 0.2, -0.1, -0.1, -0.1, 0.1, 0.1, 0.1, -0.1] \\\\\n\\psi_1 &= - \\psi_0 \n\\end{align*}\n\n\n\\subsection{Method Implementation Details}\n\\label{apx:exp-benchmark}\n\nIn all methods where we sampled from the posterior $\\varphi(\\cdot;x,t)$, this sampling was done using STAN \\citep{carpenter2017stan}, solving QPs and LCQPs was done using the Python package \\texttt{quadprog},\\footnote{https:\/\/pypi.org\/project\/quadprog\/} all stochastic gradient descent (SGD) learning was performed using the Adam \\citep{kingma2014adam} optimizer with a learning rate of 0.001.\n\n\\paragraph{\\textbf{OptZ}} We ran \\cref{algo:continuous} with $B = 50$.\n\n\\paragraph{\\textbf{IPS}} Since the propensity scores $\\eta_t(x)$ are not not tractable to compute analytically, we trained a neural network $\\hat \\eta$ to estimate this function. This was done by sampling batches of $(Z,X)$ pairs from the data model, and training the network using SGD to predict the vector of probabilities $P_T = \\beta^T Z + \\beta_0$ from $X$, using cross-entropy loss. We used a neural network with two hidden layers of size $200$ for $\\hat \\eta$, and trained for $2000$ iterations with a batch size of 32. We found in practice this training was very stable and gave accurate results. \n\n\\paragraph{\\textbf{DirX}} For each $t$ we trained a neural network $\\hat \\rho_t$ to predict $\\nu_t(x,t)$ by taking the set of $(X,T,Y)$ triplets in our training data where $T=t$, and training the network using SGD to predict $Y$ from $X$ using MSE loss. Based on pilot experiments we used a network architecture with a single hidden layer of size $100$, and trained using a batch size of 128. We used $80\\%$ of our data for training, and used the remaining $20\\%$ for the purpose of early stopping. We trained for a maximum of $500$ epochs, or until we made no progress on development data for $20$ epochs. \n\n\\paragraph{\\textbf{DirZ}} For each $t$ we trained a neural network $\\hat \\mu_t$ to predict $\\mu_t$. This was done by taking the set of $(X,T,Y)$ triplets in our training data, and for each sampling $200$ $Z$ values from the posterior using our identified model given $X$ and $T$. This gives us a set of $(Z,T,Y)$ triplets $200$ times as large as our original training set. We then trained each $\\hat \\mu_t$ network by taking the set of these triplets where $T=t$, and optimized the network using SGD on this data predicting $Y$ from $Z$. We used the same settings for this optimization as with the \\emph{direct-naive} method, except we allowed up to $1000$ epochs. Note that for both training and inference we limited ourselves to sampling $200$ $Z$ values per data point due to computational limitations.\n\n\n\\section{Additional Experiment Results}\n\\label{apx:exp-results}\n\nWe present here our additional experiment results. In these results \\textbf{SimplexOptZ} refers to our method using the simplex constraints discussed in \\cref{sec:qpalgo}.\n\n\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.39\\pm.07$ & $.24\\pm.02$ & $.36\\pm.02$ & $.81\\pm.02$\\\\\n500 & $.19\\pm.02$ & $.18\\pm.02$ & $.23\\pm.02$ & $.49\\pm.02$\\\\\n1000 & $.11\\pm.01$ & $.11\\pm.01$ & $.13\\pm.01$ & $.27\\pm.01$\\\\\n2000 & $.08\\pm.01$ & $.08\\pm.01$ & $.09\\pm.01$ & $.17\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{step} link}\n\\label{tab:step-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.57\\pm.06$ & $.42\\pm.03$ & $.39\\pm.03$ & $.43\\pm.03$\\\\\n500 & $.55\\pm.02$ & $.46\\pm.02$ & $.39\\pm.02$ & $.37\\pm.02$\\\\\n1000 & $.49\\pm.02$ & $.45\\pm.01$ & $.39\\pm.01$ & $.32\\pm.01$\\\\\n2000 & $.48\\pm.01$ & $.47\\pm.01$ & $.42\\pm.01$ & $.34\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{step} link}\n\\label{tab:step-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.41\\pm.07$ & $.29\\pm.02$ & $.50\\pm.02$ & $1.1\\pm.03$\\\\\n500 & $.20\\pm.02$ & $.21\\pm.02$ & $.31\\pm.02$ & $.70\\pm.02$\\\\\n1000 & $.11\\pm.01$ & $.13\\pm.01$ & $.18\\pm.01$ & $.42\\pm.01$\\\\\n2000 & $.08\\pm.01$ & $.09\\pm.01$ & $.13\\pm.01$ & $.26\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{step} link}\n\\label{tab:step-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.30\\pm.02$ & $.25\\pm.02$ & $.38\\pm.02$ & $.91\\pm.02$\\\\\n500 & $.18\\pm.02$ & $.19\\pm.02$ & $.24\\pm.02$ & $.54\\pm.02$\\\\\n1000 & $.12\\pm.01$ & $.11\\pm.01$ & $.13\\pm.01$ & $.29\\pm.01$\\\\\n2000 & $.07\\pm.01$ & $.08\\pm.01$ & $.10\\pm.01$ & $.18\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{step} link}\n\\label{tab:step-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.47\\pm.03$ & $2.0\\pm.03$ & $2.1\\pm.03$ & $2.3\\pm.02$ & $2.5\\pm.02$ & $.52\\pm.02$ & $2.6\\pm.02$\\\\\n500 & $.48\\pm.03$ & $2.0\\pm.02$ & $2.1\\pm.02$ & $2.3\\pm.02$ & $2.6\\pm.02$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n1000 & $.39\\pm.02$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.48\\pm.02$ & $2.6\\pm.01$\\\\\n2000 & $.40\\pm.01$ & $2.0\\pm.01$ & $2.1\\pm.01$ & $2.3\\pm.01$ & $2.5\\pm.01$ & $.45\\pm.02$ & $2.6\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{step} link}\n\\label{tab:step-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.03\\pm.39$ & $.11\\pm.21$ & $.29\\pm.21$ & $.78\\pm.18$\\\\\n500 & $.09\\pm.17$ & $.10\\pm.15$ & $.17\\pm.16$ & $.47\\pm.15$\\\\\n1000 & $.02\\pm.11$ & $.05\\pm.09$ & $.08\\pm.09$ & $.25\\pm.09$\\\\\n2000 & $.03\\pm.07$ & $.05\\pm.06$ & $.07\\pm.07$ & $.16\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{step} link}\n\\label{tab:step-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.43\\pm.38$ & $.35\\pm.24$ & $.31\\pm.24$ & $.37\\pm.22$\\\\\n500 & $.51\\pm.19$ & $.42\\pm.18$ & $.35\\pm.18$ & $.33\\pm.17$\\\\\n1000 & $.47\\pm.13$ & $.44\\pm.11$ & $.37\\pm.10$ & $.30\\pm.11$\\\\\n2000 & $.47\\pm.09$ & $.46\\pm.08$ & $.41\\pm.08$ & $.33\\pm.08$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{step} link}\n\\label{tab:step-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.05\\pm.40$ & $.19\\pm.22$ & $.45\\pm.22$ & $1.1\\pm.21$\\\\\n500 & $.10\\pm.18$ & $.14\\pm.16$ & $.26\\pm.16$ & $.68\\pm.17$\\\\\n1000 & $.04\\pm.11$ & $.09\\pm.10$ & $.15\\pm.10$ & $.41\\pm.10$\\\\\n2000 & $.03\\pm.07$ & $.06\\pm.07$ & $.10\\pm.07$ & $.25\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{step} link}\n\\label{tab:step-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.04\\pm.30$ & $.12\\pm.21$ & $.31\\pm.21$ & $.89\\pm.20$\\\\\n500 & $.08\\pm.15$ & $.10\\pm.15$ & $.18\\pm.16$ & $.51\\pm.16$\\\\\n1000 & $.01\\pm.12$ & $.06\\pm.09$ & $.09\\pm.09$ & $.27\\pm.10$\\\\\n2000 & $.03\\pm.07$ & $.05\\pm.06$ & $.07\\pm.07$ & $.17\\pm.07$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{step} link}\n\\label{tab:step-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.40\\pm.25$ & $1.9\\pm.21$ & $2.1\\pm.20$ & $2.3\\pm.19$ & $2.5\\pm.18$ & $.49\\pm.18$ & $2.6\\pm.14$\\\\\n500 & $.43\\pm.21$ & $2.0\\pm.16$ & $2.1\\pm.15$ & $2.3\\pm.14$ & $2.6\\pm.13$ & $.45\\pm.16$ & $2.6\\pm.12$\\\\\n1000 & $.37\\pm.12$ & $2.0\\pm.10$ & $2.1\\pm.09$ & $2.3\\pm.09$ & $2.5\\pm.08$ & $.46\\pm.15$ & $2.6\\pm.11$\\\\\n2000 & $.39\\pm.10$ & $2.0\\pm.08$ & $2.1\\pm.07$ & $2.3\\pm.07$ & $2.5\\pm.07$ & $.42\\pm.17$ & $2.6\\pm.11$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{step} link}\n\\label{tab:step-bias-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.07\\pm.01$ & $.04\\pm.00$ & $.04\\pm.00$ & $.07\\pm.00$\\\\\n500 & $.04\\pm.00$ & $.03\\pm.00$ & $.03\\pm.00$ & $.04\\pm.00$\\\\\n1000 & $.02\\pm.00$ & $.02\\pm.00$ & $.02\\pm.00$ & $.02\\pm.00$\\\\\n2000 & $.01\\pm.00$ & $.01\\pm.00$ & $.01\\pm.00$ & $.01\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{exp} link}\n\\label{tab:exp-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.13\\pm.01$ & $.10\\pm.01$ & $.12\\pm.01$ & $.11\\pm.01$\\\\\n500 & $.10\\pm.01$ & $.09\\pm.01$ & $.10\\pm.01$ & $.12\\pm.01$\\\\\n1000 & $.08\\pm.00$ & $.08\\pm.00$ & $.08\\pm.00$ & $.10\\pm.00$\\\\\n2000 & $.07\\pm.00$ & $.07\\pm.00$ & $.08\\pm.00$ & $.09\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{exp} link}\n\\label{tab:exp-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.15\\pm.02$ & $.15\\pm.01$ & $.25\\pm.01$ & $.44\\pm.01$\\\\\n500 & $.10\\pm.01$ & $.11\\pm.01$ & $.18\\pm.01$ & $.32\\pm.01$\\\\\n1000 & $.07\\pm.01$ & $.08\\pm.01$ & $.12\\pm.01$ & $.23\\pm.01$\\\\\n2000 & $.04\\pm.00$ & $.05\\pm.00$ & $.08\\pm.00$ & $.16\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{exp} link}\n\\label{tab:exp-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.05\\pm.00$ & $.11\\pm.01$ & $.22\\pm.01$ & $.39\\pm.02$\\\\\n500 & $.04\\pm.00$ & $.08\\pm.01$ & $.15\\pm.01$ & $.28\\pm.01$\\\\\n1000 & $.02\\pm.00$ & $.05\\pm.00$ & $.09\\pm.00$ & $.19\\pm.00$\\\\\n2000 & $.02\\pm.00$ & $.03\\pm.00$ & $.07\\pm.00$ & $.14\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{exp} link}\n\\label{tab:exp-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.12\\pm.01$ & $.76\\pm.02$ & $.81\\pm.02$ & $.92\\pm.02$ & $1.0\\pm.02$ & $.10\\pm.01$ & $1.0\\pm.01$\\\\\n500 & $.11\\pm.00$ & $.76\\pm.01$ & $.82\\pm.01$ & $.92\\pm.01$ & $1.0\\pm.01$ & $.10\\pm.01$ & $1.0\\pm.01$\\\\\n1000 & $.10\\pm.00$ & $.74\\pm.01$ & $.79\\pm.01$ & $.90\\pm.01$ & $1.0\\pm.01$ & $.09\\pm.01$ & $1.1\\pm.01$\\\\\n2000 & $.10\\pm.00$ & $.73\\pm.00$ & $.78\\pm.00$ & $.88\\pm.00$ & $.99\\pm.01$ & $.10\\pm.01$ & $1.0\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{exp} link}\n\\label{tab:exp-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.01\\pm.06$ & $.00\\pm.04$ & $-0.00\\pm.04$ & $-0.05\\pm.04$\\\\\n500 & $.01\\pm.04$ & $.01\\pm.03$ & $.00\\pm.03$ & $-0.03\\pm.03$\\\\\n1000 & $.00\\pm.02$ & $.01\\pm.02$ & $-0.00\\pm.02$ & $-0.01\\pm.02$\\\\\n2000 & $.01\\pm.01$ & $.01\\pm.01$ & $.00\\pm.01$ & $.00\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{exp} link}\n\\label{tab:exp-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.07\\pm.10$ & $.08\\pm.06$ & $.10\\pm.05$ & $.11\\pm.04$\\\\\n500 & $.07\\pm.07$ & $.07\\pm.05$ & $.09\\pm.05$ & $.11\\pm.04$\\\\\n1000 & $.06\\pm.05$ & $.07\\pm.03$ & $.07\\pm.03$ & $.09\\pm.02$\\\\\n2000 & $.06\\pm.04$ & $.06\\pm.03$ & $.07\\pm.03$ & $.09\\pm.03$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{exp} link}\n\\label{tab:exp-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.05\\pm.14$ & $.12\\pm.10$ & $.23\\pm.09$ & $.43\\pm.09$\\\\\n500 & $.03\\pm.10$ & $.09\\pm.07$ & $.16\\pm.07$ & $.32\\pm.07$\\\\\n1000 & $.02\\pm.06$ & $.06\\pm.05$ & $.10\\pm.05$ & $.23\\pm.06$\\\\\n2000 & $.01\\pm.04$ & $.03\\pm.03$ & $.07\\pm.03$ & $.16\\pm.04$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{exp} link}\n\\label{tab:exp-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.02\\pm.05$ & $.10\\pm.06$ & $.20\\pm.08$ & $.38\\pm.11$\\\\\n500 & $.02\\pm.04$ & $.07\\pm.04$ & $.14\\pm.05$ & $.27\\pm.06$\\\\\n1000 & $.01\\pm.02$ & $.04\\pm.02$ & $.08\\pm.03$ & $.19\\pm.04$\\\\\n2000 & $.01\\pm.01$ & $.03\\pm.02$ & $.06\\pm.02$ & $.14\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{exp} link}\n\\label{tab:exp-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.10\\pm.06$ & $.74\\pm.13$ & $.80\\pm.13$ & $.91\\pm.15$ & $1.0\\pm.16$ & $.07\\pm.07$ & $1.0\\pm.08$\\\\\n500 & $.11\\pm.03$ & $.76\\pm.07$ & $.81\\pm.08$ & $.92\\pm.09$ & $1.0\\pm.10$ & $.07\\pm.08$ & $1.0\\pm.09$\\\\\n1000 & $.10\\pm.03$ & $.74\\pm.05$ & $.79\\pm.05$ & $.90\\pm.06$ & $1.0\\pm.06$ & $.06\\pm.07$ & $1.0\\pm.10$\\\\\n2000 & $.10\\pm.02$ & $.73\\pm.03$ & $.78\\pm.03$ & $.88\\pm.04$ & $.99\\pm.04$ & $.07\\pm.07$ & $1.0\\pm.09$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{exp} link}\n\\label{tab:exp-bias-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.47\\pm.04$ & $.35\\pm.02$ & $.39\\pm.02$ & $.48\\pm.02$\\\\\n500 & $.36\\pm.05$ & $.27\\pm.02$ & $.30\\pm.02$ & $.40\\pm.02$\\\\\n1000 & $.25\\pm.02$ & $.22\\pm.01$ & $.25\\pm.01$ & $.37\\pm.01$\\\\\n2000 & $.14\\pm.01$ & $.14\\pm.01$ & $.17\\pm.01$ & $.27\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.58\\pm.07$ & $.41\\pm.03$ & $.42\\pm.02$ & $.38\\pm.02$\\\\\n500 & $.37\\pm.04$ & $.33\\pm.02$ & $.35\\pm.02$ & $.37\\pm.02$\\\\\n1000 & $.31\\pm.02$ & $.31\\pm.02$ & $.33\\pm.02$ & $.39\\pm.01$\\\\\n2000 & $.21\\pm.02$ & $.23\\pm.02$ & $.26\\pm.01$ & $.32\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.49\\pm.04$ & $.42\\pm.03$ & $.54\\pm.03$ & $.76\\pm.02$\\\\\n500 & $.38\\pm.05$ & $.30\\pm.02$ & $.38\\pm.02$ & $.59\\pm.02$\\\\\n1000 & $.27\\pm.02$ & $.25\\pm.02$ & $.32\\pm.02$ & $.52\\pm.02$\\\\\n2000 & $.16\\pm.01$ & $.16\\pm.01$ & $.22\\pm.01$ & $.39\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.45\\pm.04$ & $.41\\pm.03$ & $.52\\pm.03$ & $.70\\pm.03$\\\\\n500 & $.37\\pm.05$ & $.30\\pm.02$ & $.37\\pm.02$ & $.55\\pm.02$\\\\\n1000 & $.26\\pm.02$ & $.24\\pm.02$ & $.31\\pm.02$ & $.50\\pm.02$\\\\\n2000 & $.14\\pm.01$ & $.15\\pm.01$ & $.20\\pm.01$ & $.35\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.46\\pm.04$ & $1.1\\pm.02$ & $1.1\\pm.03$ & $1.3\\pm.03$ & $1.4\\pm.03$ & $.36\\pm.02$ & $1.4\\pm.01$\\\\\n500 & $.38\\pm.02$ & $1.1\\pm.01$ & $1.2\\pm.01$ & $1.3\\pm.01$ & $1.4\\pm.01$ & $.34\\pm.02$ & $1.4\\pm.01$\\\\\n1000 & $.39\\pm.01$ & $1.1\\pm.01$ & $1.2\\pm.01$ & $1.3\\pm.01$ & $1.4\\pm.01$ & $.35\\pm.02$ & $1.4\\pm.01$\\\\\n2000 & $.35\\pm.01$ & $1.1\\pm.01$ & $1.2\\pm.01$ & $1.3\\pm.01$ & $1.4\\pm.01$ & $.39\\pm.02$ & $1.4\\pm.01$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{cubic} link}\n\\label{tab:cubic-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.01\\pm.47$ & $.16\\pm.31$ & $.29\\pm.27$ & $.45\\pm.16$\\\\\n500 & $-0.01\\pm.36$ & $.12\\pm.24$ & $.22\\pm.21$ & $.37\\pm.15$\\\\\n1000 & $.03\\pm.25$ & $.12\\pm.18$ & $.20\\pm.15$ & $.35\\pm.12$\\\\\n2000 & $.02\\pm.14$ & $.07\\pm.12$ & $.13\\pm.11$ & $.26\\pm.08$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{cubic} link}\n\\label{tab:cubic-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.13\\pm.57$ & $.26\\pm.32$ & $.33\\pm.27$ & $.34\\pm.17$\\\\\n500 & $.12\\pm.35$ & $.22\\pm.25$ & $.28\\pm.21$ & $.34\\pm.15$\\\\\n1000 & $.18\\pm.26$ & $.24\\pm.19$ & $.29\\pm.15$ & $.37\\pm.11$\\\\\n2000 & $.14\\pm.16$ & $.18\\pm.14$ & $.22\\pm.12$ & $.31\\pm.09$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.04\\pm.49$ & $.25\\pm.34$ & $.45\\pm.29$ & $.74\\pm.20$\\\\\n500 & $.03\\pm.38$ & $.17\\pm.25$ & $.32\\pm.21$ & $.57\\pm.16$\\\\\n1000 & $.04\\pm.26$ & $.16\\pm.19$ & $.28\\pm.16$ & $.50\\pm.13$\\\\\n2000 & $.03\\pm.16$ & $.10\\pm.13$ & $.19\\pm.12$ & $.37\\pm.10$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.02\\pm.45$ & $.22\\pm.35$ & $.40\\pm.34$ & $.65\\pm.28$\\\\\n500 & $.01\\pm.37$ & $.15\\pm.26$ & $.29\\pm.23$ & $.52\\pm.18$\\\\\n1000 & $.04\\pm.25$ & $.15\\pm.19$ & $.27\\pm.17$ & $.47\\pm.14$\\\\\n2000 & $.02\\pm.14$ & $.08\\pm.13$ & $.17\\pm.11$ & $.34\\pm.09$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.26\\pm.38$ & $1.1\\pm.20$ & $1.1\\pm.20$ & $1.2\\pm.21$ & $1.4\\pm.22$ & $.32\\pm.16$ & $1.4\\pm.12$\\\\\n500 & $.31\\pm.22$ & $1.1\\pm.10$ & $1.2\\pm.10$ & $1.3\\pm.10$ & $1.4\\pm.11$ & $.29\\pm.18$ & $1.4\\pm.10$\\\\\n1000 & $.37\\pm.14$ & $1.1\\pm.07$ & $1.2\\pm.07$ & $1.3\\pm.08$ & $1.4\\pm.08$ & $.32\\pm.16$ & $1.4\\pm.10$\\\\\n2000 & $.34\\pm.09$ & $1.1\\pm.05$ & $1.2\\pm.06$ & $1.3\\pm.06$ & $1.4\\pm.06$ & $.34\\pm.18$ & $1.4\\pm.11$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{cubic} link}\n\\label{tab:cubic-bias-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.09\\pm.01$ & $.08\\pm.01$ & $.13\\pm.01$ & $.23\\pm.01$\\\\\n500 & $.06\\pm.01$ & $.06\\pm.00$ & $.09\\pm.00$ & $.16\\pm.00$\\\\\n1000 & $.04\\pm.01$ & $.04\\pm.00$ & $.06\\pm.00$ & $.12\\pm.00$\\\\\n2000 & $.02\\pm.00$ & $.03\\pm.00$ & $.04\\pm.00$ & $.08\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights, with \\textbf{linear} link}\n\\label{tab:linear-rmse}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.15\\pm.01$ & $.13\\pm.01$ & $.13\\pm.01$ & $.13\\pm.01$\\\\\n500 & $.14\\pm.01$ & $.13\\pm.01$ & $.13\\pm.01$ & $.13\\pm.00$\\\\\n1000 & $.13\\pm.01$ & $.13\\pm.00$ & $.13\\pm.00$ & $.13\\pm.00$\\\\\n2000 & $.12\\pm.00$ & $.12\\pm.00$ & $.12\\pm.00$ & $.12\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{linear} link}\n\\label{tab:linear-rmse-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.11\\pm.01$ & $.11\\pm.01$ & $.18\\pm.01$ & $.35\\pm.01$\\\\\n500 & $.06\\pm.01$ & $.08\\pm.01$ & $.13\\pm.01$ & $.24\\pm.01$\\\\\n1000 & $.05\\pm.01$ & $.06\\pm.00$ & $.09\\pm.00$ & $.18\\pm.00$\\\\\n2000 & $.03\\pm.00$ & $.04\\pm.00$ & $.06\\pm.00$ & $.12\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{linear} link}\n\\label{tab:linear-rmse-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.09\\pm.01$ & $.09\\pm.01$ & $.15\\pm.01$ & $.29\\pm.01$\\\\\n500 & $.06\\pm.01$ & $.07\\pm.01$ & $.10\\pm.01$ & $.19\\pm.01$\\\\\n1000 & $.04\\pm.01$ & $.04\\pm.00$ & $.07\\pm.00$ & $.14\\pm.00$\\\\\n2000 & $.02\\pm.00$ & $.03\\pm.00$ & $.05\\pm.00$ & $.09\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{linear} link}\n\\label{tab:linear-rmse-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.15\\pm.01$ & $.57\\pm.01$ & $.60\\pm.01$ & $.66\\pm.01$ & $.72\\pm.01$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n500 & $.15\\pm.01$ & $.57\\pm.00$ & $.60\\pm.00$ & $.66\\pm.00$ & $.72\\pm.00$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n1000 & $.14\\pm.00$ & $.57\\pm.00$ & $.60\\pm.00$ & $.66\\pm.00$ & $.72\\pm.00$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n2000 & $.14\\pm.00$ & $.57\\pm.00$ & $.60\\pm.00$ & $.66\\pm.00$ & $.72\\pm.00$ & $.13\\pm.00$ & $.76\\pm.00$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of RMSE for benchmark methods, with \\textbf{linear} link}\n\\label{tab:linear-rmse-bench}\n\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{OptZ}_{0.001}$ & $\\textbf{OptZ}_{0.2}$ & $\\textbf{OptZ}_{1.0}$ & $\\textbf{OptZ}_{5.0}$\\\\\n\\hline\n200 & $.03\\pm.09$ & $.06\\pm.06$ & $.11\\pm.05$ & $.23\\pm.04$\\\\\n500 & $.02\\pm.05$ & $.04\\pm.05$ & $.08\\pm.04$ & $.15\\pm.04$\\\\\n1000 & $.01\\pm.04$ & $.03\\pm.03$ & $.05\\pm.03$ & $.11\\pm.03$\\\\\n2000 & $.01\\pm.02$ & $.02\\pm.02$ & $.04\\pm.02$ & $.08\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights, with \\textbf{linear} link}\n\\label{tab:linear-bias}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirX:OptZ}_{0.001}$ & $\\textbf{DirX:OptZ}_{0.2}$ & $\\textbf{DirX:OptZ}_{1.0}$ & $\\textbf{DirX:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.12\\pm.09$ & $.12\\pm.06$ & $.12\\pm.05$ & $.13\\pm.04$\\\\\n500 & $.13\\pm.06$ & $.12\\pm.05$ & $.12\\pm.04$ & $.12\\pm.04$\\\\\n1000 & $.12\\pm.04$ & $.12\\pm.03$ & $.12\\pm.03$ & $.12\\pm.03$\\\\\n2000 & $.12\\pm.03$ & $.12\\pm.03$ & $.12\\pm.02$ & $.12\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirX}$, with \\textbf{linear} link}\n\\label{tab:linear-bias-drx}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{DirZ:OptZ}_{0.001}$ & $\\textbf{DirZ:OptZ}_{0.2}$ & $\\textbf{DirZ:OptZ}_{1.0}$ & $\\textbf{DirZ:OptZ}_{5.0}$\\\\\n\\hline\n200 & $.04\\pm.10$ & $.09\\pm.07$ & $.17\\pm.06$ & $.34\\pm.05$\\\\\n500 & $.02\\pm.06$ & $.06\\pm.05$ & $.12\\pm.05$ & $.24\\pm.05$\\\\\n1000 & $.02\\pm.04$ & $.05\\pm.04$ & $.08\\pm.04$ & $.18\\pm.04$\\\\\n2000 & $.01\\pm.03$ & $.03\\pm.03$ & $.06\\pm.03$ & $.12\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for doubly robust estimator using our weights and $\\textbf{DirZ}$, with \\textbf{linear} link}\n\\label{tab:linear-bias-drz}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{ccccc}\n\\hline\nn & $\\textbf{SimplexOptZ}_{0.001}$ & $\\textbf{SimplexOptZ}_{0.2}$ & $\\textbf{SimplexOptZ}_{1.0}$ & $\\textbf{SimplexOptZ}_{5.0}$\\\\\n\\hline\n200 & $.02\\pm.08$ & $.07\\pm.06$ & $.14\\pm.06$ & $.29\\pm.06$\\\\\n500 & $.02\\pm.05$ & $.05\\pm.05$ & $.09\\pm.05$ & $.19\\pm.04$\\\\\n1000 & $.01\\pm.04$ & $.03\\pm.03$ & $.06\\pm.03$ & $.14\\pm.03$\\\\\n2000 & $.01\\pm.02$ & $.02\\pm.02$ & $.04\\pm.02$ & $.09\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for weighted estimator using our weights and constraining $W \\in n \\Delta^n$, with \\textbf{linear} link}\n\\label{tab:linear-bias-simplex}\n\n\\begin{center}\n\\footnotesize\n\\begin{tabular}{cccccccc}\n\\hline\nn & $\\textbf{IPS}$ & $\\textbf{OptX}_{0.001}$ & $\\textbf{OptX}_{0.2}$ & $\\textbf{OptX}_{1.0}$ & $\\textbf{OptX}_{5.0}$ & $\\textbf{DirX}$ & $\\textbf{DirZ}$\\\\\n\\hline\n200 & $.13\\pm.08$ & $.57\\pm.05$ & $.60\\pm.05$ & $.66\\pm.05$ & $.72\\pm.05$ & $.13\\pm.04$ & $.76\\pm.04$\\\\\n500 & $.14\\pm.05$ & $.57\\pm.04$ & $.60\\pm.03$ & $.66\\pm.03$ & $.72\\pm.03$ & $.12\\pm.04$ & $.76\\pm.02$\\\\\n1000 & $.14\\pm.04$ & $.57\\pm.02$ & $.60\\pm.02$ & $.66\\pm.02$ & $.72\\pm.02$ & $.13\\pm.03$ & $.76\\pm.03$\\\\\n2000 & $.13\\pm.03$ & $.57\\pm.02$ & $.60\\pm.02$ & $.66\\pm.02$ & $.72\\pm.02$ & $.13\\pm.03$ & $.76\\pm.02$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Convergence of bias for benchmark methods, with \\textbf{linear} link}\n\\label{tab:linear-bias-bench}\n\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMachine learning algorithms have been increasingly used as diagnostic tools in biomedical research\\cite{gulshan2016,esteva2017,golden2017}. The widespread availability of smartphones and other health tracking devices generates high volumes of sensor data, and makes machine learning uniquely well posed to impact clinical research using digital health tools. In clinical applications, gender, age, and other demographic characteristics of the study participants often play the role of confounders. Confounding is particularly prevalent in mobile health studies run under uncontrolled conditions outside clinical and laboratory settings, where we have little control over the demographic and clinical characteristics of the cohort of participants that self-select to participate in a study.\n\nIn the context of predictive modeling, we define a confounder as a variable that causes spurious associations between the features and response variable. In machine learning applications, the presence of confounding can lead to ambiguous inference and poor generalizability of models. Confounding is usually present when the joint probability distribution of the confounder and response variables is different in the data available to develop the learner (which we from now on denote as the \"development dataset\") relative to the population where the learner will be applied (denoted as the \"target population\")\\cite{rao2017}. For example, consider a diagnostic application where most cases are old aged while most controls are young, but where age is not associated with disease status in the target population (e.g., the target population is composed of older patients only). If the classifier can more efficiently detect age-related signals than disease-related signals, then it will likely perform poorly when deployed in the target population.\n\nConfounding adjustment is an active area of research in machine learning. The goal is to prevent an algorithm from learning the confounding signal. Since any variable that confounds the feature-response relationship has to be associated with both the features and the response, most of the methods proposed in the literature can be divided into two approaches: (i) methods  that remove the association between the confounder and the response; or (ii) methods that remove the association between the confounder and the features. A standard example of the first approach is to match subjects from the development data in order to obtain a subsample that more closely resembles the target population. This strategy, however, results in a smaller number of participants to train and evaluate the machine learning algorithm, and, in highly unbalanced situations, might lead to the exclusion of most of the participants from the analyses. Alternative methods that make more efficient use of the data include inverse probability weighting approaches\\cite{linn2016,rao2017}, which weight the training samples in order to make model training better tailored to the target population. A canonical example of the second approach is to separately regress each feature on the confounders, and use the residuals as the predictors in the machine learning algorithm. Other approaches that do not fall into categories (i) or (ii) include penalized learners\\cite{li2011} and backdoor adjustment\\cite{landeiro2016}.\n\nIn this paper, we present statistical methods to detect and quantify the influence of observed confounders,  and to estimate the actual (i.e., unconfounded) predictive performance of a learner. We use a large Parkinsons digital health study cohort to illustrate how our methods can be used to evaluate the effectiveness of standard confounding adjustment methods.\n\n\n\\section{Methods}\n\nWe adopt restricted permutations\\cite{good2000,rao2017} to isolate the contribution of the confounder from the predictive performance of a learner. The key idea is to shuffle the response data within the levels of a categorical\/ordinal confounder (as illustrated in the Supplementary Figure \\ref{fig:stratifiedperm}) in order to destroy the direct association between the response and the features while still preserving the indirect association due to the confounder. Algorithm \\ref{alg:stratifiedShuffling} describes the procedure for an arbitrary performance metric, $m$ (such as the area under the receiver operating characteristic curve, AUC, or root mean square error).\n\n\\begin{algorithm}\n\\caption{Restricted Monte Carlo permutation null distribution for performance metric $m$}\\label{alg:stratifiedShuffling}\n\\begin{algorithmic}[1]\n\\State \\textbf{Input}: Number of permutations, $b$. Development data set feature matrix, response vector, and confounder vector, $\\boldsymbol{X}$, $\\boldsymbol{y}$, $\\boldsymbol{c}$. Training and test set indexes, $i_{train}$, $i_{test}$\n\\State Split $\\boldsymbol{X}$, $\\boldsymbol{y}$ and $\\boldsymbol{c}$ into training and test sets\n\\For{$i = 1, 2, \\ldots, b$}\n  \\State $\\boldsymbol{y}^{\\ast}_{train} \\leftarrow \\mbox{RestrictedShuffle}(\\boldsymbol{y}_{train}, \\boldsymbol{c}_{train})$, and $\\boldsymbol{y}^{\\ast}_{test} \\leftarrow \\mbox{RestrictedShuffle}(\\boldsymbol{y}_{test}, \\boldsymbol{c}_{test})$\n  \\State Train a machine learning algorithm on the $\\boldsymbol{X}_{train}$ and $\\boldsymbol{y}_{train}^{\\ast}$ data\n  \\State Evaluate the algorithm on the $\\boldsymbol{X}_{test}$ and $\\boldsymbol{y}_{test}^{\\ast}$ data\n  \\State Record the value of the performance metric, $m^\\ast_i$, on the shuffled data\n\\EndFor\n\\State \\textbf{Output}: $m^\\ast_1$, $m^\\ast_2$, \\ldots, $m^\\ast_b$\n\\end{algorithmic}\n\\end{algorithm}\n\nBuilding upon the restricted permutation null distribution, we developed two statistical tools to deal with confounding. First, we estimate the ``unconfounded\" predictive performance of a learner by building a mapping from the restricted permutation null to the standard permutation null (where the standard permutation null distribution is generated by shuffling the labels in the usual unconstrained manner). As fully described in the Supplement, for any performance metric that can be expressed as a (generalized) U-statistic\\cite{hoeffding1948,lehmann1951,serfling1980,delong1988} (e.g., AUC), or expressed as a simple average (e.g., mean square error, mean absolute error, and classification accuracy), we have that an asymptotic estimate of the unconfounded performance metric is given by,\n\\begin{equation}\n\\hat{m}_u \\, = \\, (m_o - a_{\\hat{\\pi}^{\\ast}}) (s_{\\hat{\\pi}^{\\ast\\ast}}\/s_{\\hat{\\pi}^{\\ast}}) + a_{\\hat{\\pi}^{\\ast\\ast}}~,\n\\label{eq:correctedmetric}\n\\end{equation}\nwhere $m_o$ represents the uncorrected metric value; $a_{\\hat{\\pi}^{\\ast}}$ and $s^2_{\\hat{\\pi}^{\\ast}}$ represent the sample average and variance of the restricted permutation null; and $a_{\\hat{\\pi}^{\\ast\\ast}}$ and $s^2_{\\hat{\\pi}^{\\ast\\ast}}$ represent the analogous quantities for the standard permutation null. It is important to point out that we don't view the unconfounded metric estimation as an adjustment method (in the sense that it does not prevent an algorithm from learning the confounding signal in the first place). It simply quantifies the amount of response signal learned by the algorithm, after the algorithm has had a chance to learn both confounding and response signals.\n\n\nSecond, by noticing that the location of the restricted permutation null provides a natural measure of the amount of confounding signal learned by the algorithm, we adopt the average of the restricted permutation null as a test statistic, and develop a statistical test to compare the hypotheses,\n\\begin{align}\nH_0^c &: \\mbox{the algorithm has not learned the confounding signal~,} \\\\\nH_1^c &: \\mbox{the algorithm has learned the confounding signal~,} \\nonumber\n\\end{align}\nand detect confounding learning per se. In the Supplement we show that, under $H_0^c$, the average of the restricted permutation null distribution is asymptotically distributed as a $N(a_{\\hat{\\pi}^{\\ast\\ast}}, s^2_{\\hat{\\pi}^{\\ast\\ast}}\/n)$ distribution, where $n$ represents the test set sample size.\n\nFor the AUC metric, additional analytical results are available. It is well known\\cite{bamber1975,mason2002} that the standard permutation null can be approximated by,\n\\begin{equation}\nN(0.5 \\, , \\, (n_n + n_p + 1)\/(12 \\, n_n \\, n_p))~,\n\\label{eq:standnullapprox}\n\\end{equation}\nwhere $n_n$ and $n_p$ represent the number of negative and positive labels in the test set. Hence, $a_{\\hat{\\pi}^{\\ast\\ast}} \\approx 0.5$ and $s^2_{\\hat{\\pi}^{\\ast\\ast}} \\approx (n_n + n_p + 1)\/(12 \\, n_n \\, n_p)$ and the estimator in (\\ref{eq:correctedmetric}) becomes,\n\\begin{equation}\nauc_u = (auc_o -  a_{\\hat{\\pi}^{\\ast}})(n_n + n_p + 1)\/(12 \\, n_n \\, n_p \\, s_{\\hat{\\pi}^\\ast}) + 0.5~,\n\\label{eq:auccorrectionformula}\n\\end{equation}\nthe null distribution under $H_0^c$ reduces to $N(0.5, (n_n + n_p + 1)\/(12 \\, n_n \\, n_p \\, n))$, and,\n\\begin{equation}\np = 1 - \\Phi\\big((a_{\\hat{\\pi}^{\\ast}} - 0.5)\/[(n_n + n_p + 1)\/(12 \\, n_n \\, n_p \\, n)]^{1\/2}\\big)~,\n\\label{eq:confoundingpval}\n\\end{equation}\ncorresponds to the confounding p-value, where $\\Phi(.)$ represents the c.d.f. of standard normal variable.\n\n\\section{Real data illustrations}\n\nA key practical application of our tools is to evaluate if an adjustment method is working as expected. This is important in practice since most of these methods rely on assumptions, and it is generally unclear how robust they are to violations of these assumptions. Here we illustrate the application of our tools to two confounding adjustment methods: sample matching, and approximate inverse probability weighting (IPW) based on the propensity score\\cite{propensityscore1983}.\n\nOur development data was collected in a digital health study on Parkinsons disease\\cite{bot2016,trister2016} and consist of features generated from 30 second inertial sensor readings captured during walking. We focused on walking, as walking patterns are influenced by age and gender\\cite{ko2011} in addition to Parkinson's disease. The development data was split into training and test sets with similar joint distributions for the age, gender, and disease status (Supplementary Figure \\ref{sfig:agegenderassociations}). We apply the adjustment methods to both training and test sets, and the analyses are based on a combined gender\/discretized age\\footnote{While, in theory, we can only perform restricted permutations using categorical\/ordinal confounders, in practice we can discretize and evaluate continuous confounders as well. Clearly, if the discretization is too coarse the discretized confounder might not be able to fully capture the association between the confounder and the response, and we might end up underestimating the amount of confounding learned by the algorithm. In practice, one should experiment with distinct discretizations, as illustrated in Supplementary Figure \\ref{sfig:agediscretization}.} confounder with levels: young male, young female, middle age male, middle age female, senior male, and senior female.\n\nFigure \\ref{fig:adjustmentcomparison} shows the results based on logistic regression (top panels) and random forest (bottom panels) classifiers. In all panels, the blue histograms represent the restricted permutation null distributions generated by Algorithm 1, the red curves represent the normal approximation for the standard permutation null distribution presented in equation 3, the orange line shows the unconfounded estimate of AUC computed using equation \\ref{eq:auccorrectionformula}, and  the cyan line represents the observed AUC.\n\nFor the sake of comparison, panels a and d report the results when no confounding adjustment is performed. Both logistic regression and random forest classifiers are clearly learning confounding signal since the restricted permutation nulls are centered around 0.7, and the confounding test p-values (equation \\ref{eq:confoundingpval}) are highly significant ($p<10^{-16}$). Hence, the high AUC scores (cyan lines above 0.81) reflect the classifiers' ability to detect both disease and confounding signals, while the unconfounded estimates (orange scores around 0.66) are considerably more modest.\n\nPanels b and e show the results based on a matched subset of participants. The fact that the restricted permutation nulls are centered around 0.5, and closely match the standard permutation null density (red curve), suggests that matching effectively prevented the classifier from learning the confounding signal ($p<0.51$ and $p<0.58$, respectively) and that the classifiers are only learning the disease signal. As expected, the observed and unconfounded AUC scores match each other closely in this situation. Finally, note that the much larger spread of the null distributions (in comparison to panels a and d) is due to the smaller test set available after matching.\n\nPanels c and f report the results for the approximate IPW approach. This method makes use of the entire data set and attempts to prevent confounding learning by weighting the samples according to the inverse of their estimated propensity scores (i.e., the conditional probability that a participant has the disease given its gender and age). While several approaches have been proposed in the literature for the estimation of propensity scores\\cite{lee2010,pirracchio2014}, here, we adopt the most commonly used method based on logistic regression. The panels show that the approximate IPW approach managed to reduce the amount of confounding (the blue histograms are closer to 0.5 compared to panels a and d). However, it didn't remove it completely ($p < 10^{-16}$). This suggests that the estimated inverse probability weights did not generate a well balanced augmented data set. (Supplementary Figure \\ref{sfig:failuretobalance} confirms this is indeed the case.) Most likely, the reason for this suboptimal performance is that propensity score estimation using logistic regression makes the strong assumption that the observed associations between the confounders and disease labels can be well described by the logistic function. This example illustrates how the violation of a parametric modeling assumption can lead to an inefficient confounding adjustment.\n\n\n\\begin{figure}[h]\n  \\centering\n  \\centerline{\\includegraphics[width=\\linewidth, bb = 0 20 720 210]{adjustment_checks_glm_and_rf3.pdf}}\n  \\caption{Comparison of confounding adjustment methods.}\n  \\label{fig:adjustmentcomparison}\n\\end{figure}\n\n\n\\section{Final remarks}\n\nDigital health enabled diagnostic systems have the potential to provide low cost remote diagnostic tools to underserved communities that lack easy access to medical care. However this opportunity cannot be fully realized without (i) efficient approaches to combat confounding (without which we run the risk of making spurious inferences from the data) and (ii) rigorous methods to evaluate these adjustment methods. The tools proposed in this paper address the second need.\n\nTo the best of our knowledge, the use of restricted permutations in the context of predictive modeling has only been leveraged by\\cite{rao2017}. These authors, however, use restricted permutations to test if a machine learning algorithm has learned the response signal in the presence (or absence) of confounders, but not to detect and quantify confounding learning per se, as proposed in this paper.\n\nFor the sake of clarity, our illustrations focused on the case where confounder and response are associated in the development set but not in the target population. We can, however, still apply our methodology when response and confounder are known to be associated in the target population but have a different joint probability distribution compared to the development data. Section 9 of the Supplement provides an illustrative example based on synthetic data.\n\nWe also conducted a simulation study to evaluate the statistical properties of the confounding test (Section 10 of the Supplement). Our simulations show reasonable statistical power for the range of parameters investigated in our experiments, and well-controlled type I error rates.\n\nFinally, we point out that while this paper has focused on digital health applications, the proposed tools can be more generally applied to any other areas impacted by confounders.\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis is a continuation of papers \\cite{toric} and \\cite{toricII}.\nWe have studied the effect of each elementary birational map in the minimal model program to the derived categories \nin the case of toric pairs.\nIn this paper we consider problems of more global nature.\nWe add some cases to the list of affirmative answers to the \nderived McKay correspondence conjecture and its generalization, \nthe \\lq\\lq $K$ implies $D$ conjecture''.\n\nFirst we remark that the results in \\cite{toric} and \\cite{toricII} are \neasily extended to the relative situation (Theorem~\\ref{relative}).\nThen we remark that the derived McKay correspondence holds for any abelian quotient singularity \nin the following sense; \nthe derived category defined by a finite abelian group contained in a general linear group \nis semi-orthogonally decomposed \ninto derived categories of a relative minimal model and subvarieties of the quotient space \n(Theorem~\\ref{abelian}).  \nAs an application, we prove that such derived McKay correspondence also holds for arbitrary quotient singularities in dimension $2$ (Theorem~\\ref{GL2}).\n\nNext we prove that any proper birational morphism between toric pairs which is a $K$-equivalence\nis decomposed into a sequence of flops (Theorem~\\ref{decomposition}).\nThis is a generalization, in the toric case, of a theorem in \\cite{flop}, which states that \nany birational morphism between minimal models is decomposed into a sequence of flops.\nWe note that the latter case is easier because the log canonical divisor stays \nat the bottom and cannot be decreased when the models are minimal, \nwhile it is more difficult to preserve the level of the log canonical divisor in the former case. \n\nNow we state the theorems in details.\nWe refer the terminology to the next section.\n\nThe following is the reformulation of the results of \\cite{toric} and \\cite{toricII} to the relative case: \n\n\\begin{Thm}\\label{relative}\nLet $(X,B)$ and $(Y,C)$ be $\\mathbf{Q}$-factorial toric pairs whose coefficients \nbelong to the standard set \n$\\{1 - 1\/m \\mid m \\in \\mathbf{N}\\}$, \nlet $\\tilde X$ and $\\tilde Y$ be smooth Deligne-Mumford stacks associated to \nthe pairs $(X,B)$ and $(Y,C)$, respectively, and\nlet $f: X \\dashrightarrow Y$ be a toric rational map.\nAssume one of the following conditions:\n\n(a) $f$ is the identity morphism, and $B \\ge C$.\n\n(b) $f$ is a flip, $C=f_*B$, and $K_X+B \\ge K_Y+C$.\n\n(c) $f$ is a divisorial contraction, $C=f_*B$, and $K_X+B \\ge K_Y+C$.\n\n(d) $f$ is a divisorial contraction, $C=f_*B$, and $K_X+B \\le K_Y+C$.\n\n(e) $f$ is a Mori fiber space and $C$ is determined by $(X,B)$ and $f$ as explained in \\cite{toric}. \n\nThen the following hold:\n\n(a), (b), (c): There are toric closed subvarieties $Z_i$ ($1 \\le i \\le l$) of $Y$ such that $Z_i \\ne Y$ \nfor some $l \\ge 0$,  \nand fully faithful functors $\\Phi: D^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde X))$\nand $\\Psi_i: D^b(\\text{coh}(\\tilde Z_i)) \\to D^b(\\text{coh}(\\tilde X))$\nfor the smooth Deligne-Mumford stacks $\\tilde Z_i$ associated to the $Z_i$ \nsuch that there is a semi-orthogonal decomposition of triangulated categories \n\\[\nD^b(\\text{coh}(\\tilde X)) = \\langle \\Psi_1(D^b(\\text{coh}(\\tilde Z_1))), \\dots,\n\\Psi_l(D^b(\\text{coh}(\\tilde Z_l))), \\Phi(D^b(\\text{coh}(\\tilde Y))) \\rangle\n\\]\n\n(d): There are $Z_i \\subset Y$ ($1 \\le i \\le l$) as in (c), \nand fully faithful functors $\\Phi: D^b(\\text{coh}(\\tilde X)) \\to D^b(\\text{coh}(\\tilde Y))$\nand $\\Psi_i: D^b(\\text{coh}(\\tilde Z_i)) \\to D^b(\\text{coh}(\\tilde Y))$\nsuch that there is a semi-orthogonal decomposition\n\\[\nD^b(\\text{coh}(\\tilde Y)) = \\langle \\Psi_1(D^b(\\text{coh}(\\tilde Z_1))), \\dots,\n\\Psi_l(D^b(\\text{coh}(\\tilde Z_l))), \\Phi(D^b(\\text{coh}(\\tilde X))) \\rangle\n\\]\n\n(e):  There are fully faithful functors $\\Phi_j: D^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde X))$ \n($1 \\le j \\le m$) for some $m \\ge 2$ and a semi-orthogonal decomposition\n\\[\nD^b(\\text{coh}(\\tilde X)) = \\langle \\Phi_1(D^b(\\text{coh}(\\tilde Y))), \\dots, \n\\Phi_m(D^b(\\text{coh}(\\tilde Y))) \\rangle\n\\]\n\nMoreover if $K_X+B = K_Y+C$ in the cases (b) or (c), then $\\Phi$ is an equivalence.\n\\end{Thm} \n\nWe note that, when $f$ is a divisorial contraction, \nthe direction of the inclusion of the derived categories \nis unrelated to the direction of the morphism, \nbut is the same as the direction of the inequality of the canonical divisors.\n\nIf $X$ and $Y$ are smooth and $f$ is a blowing up of a smooth center in the case (b), or if  \n$X$ and $Y$ are smooth and $f$ is a standard flip in the case (c), then\nit is a theorem by Bondal-Orlov (\\cite{BO}).\nThe derived equivalence in the case of flops (case (c) with $K_X+B = K_Y+C$) was \nproved for general (non-toric) case under the additional assumptions as follows:\nif $\\dim X = 3$ and $X$ is smooth by Bridgeland (\\cite{Bridgeland}),\nif $\\dim X = 3$ and $X$ has only Gorenstein terminal singularities by Chen (\\cite{Chen}) \nand Van den Bergh (\\cite{VdBergh}), \nif $\\dim X = 3$ and $X$ has only terminal singularities by \\cite{DK}, \nand if $X$ is a holomorphic simplectic manifold by Kaledin (\\cite{Kaledin}).\nThere are also interesting results concerning the last case by Cautis (\\cite{Cautis}) by using the \ncategorification of linear group actions, \nand by Donovan-Segal (\\cite{DS}), Ballard-Favero-Katzarkov (\\cite{BFK}) and\nHalpern-Leistner (\\cite{HL}) by using the theory of variations of GIT quotients.\n\nAs a corollary we obtain\n\n\\begin{Cor}\\label{combine}\nLet $(X,B)$ be a $\\mathbf{Q}$-factorial toric pair whose coefficients belong to the standard set \n$\\{1 - 1\/m \\mid m \\in \\mathbf{N}\\}$.\nAssume that there is a projective toric morphism $f: X \\to Z$ to another  \n$\\mathbf{Q}$-factorial toric variety.\nThen there exist toric closed subvarieties $Z_i$ ($1 \\le i \\le m$) of $Z$ \nfor some $m \\ge 1$ such that there are fully faithful functors \n$\\Phi_i: D^b(\\text{coh}(\\tilde Z_i)) \\to D^b(\\text{coh}(\\tilde X))$ with\na semi-orthogonal decomposition \n\\[\nD^b(\\text{coh}(\\tilde X)) \\cong \\langle \\Phi_1(D^b(\\text{coh}(\\tilde Z_1))), \\dots, \n\\Phi_m(D^b(\\text{coh}(\\tilde Z_m))) \\rangle\n\\]\nwhere $\\tilde X$ and the $\\tilde Z_i$ are smooth Deligne-Mumford stacks associated \nto $(X,B)$ and the $Z_i$. \n\\end{Cor}\n\nIn other words, one can say that the derived category \n$D^b(\\text{coh}\\tilde X)$ is generated by a {\\em relative exceptional collection}.\nThe proofs of the above statements are the same as in \\cite{log crepant}, \n\\cite{toric} and \\cite{toricII} except Theorem~\\ref{relative}~(a).\n\nThe following is the derived McKay correspondence for finite abelian groups:\n\n\\begin{Thm}\\label{abelian}\nLet $G \\subset GL(n,\\mathbf{C})$ be a finite abelian subgroup acting naturally on an affine \nspace $\\mathbf{A} = \\mathbf{C}^n$, and let $X=\\mathbf{A}\/G$ be the quotient space.\nLet $f: Y \\to X$ be a $\\mathbf{Q}$-factorial terminal relative minimal model of $X$.\nThen there exist toric closed subvarieties $Z_i$ ($1 \\le i \\le m$) of $X$ with $Z_i \\ne X$ \nfor some $m \\ge 0$, with possible repetitions, \nsuch that there is a semi-orthogonal decomposition \n\\[\nD^b(\\text{coh}([\\mathbf{A}\/G])) \\cong \\langle D^b(\\text{coh}(\\tilde Z_1)), \\dots, \nD^b(\\text{coh}(\\tilde Z_m)), D^b(\\text{coh}(\\tilde Y)) \\rangle\n\\]\nwhere $\\tilde Y$ and the $\\tilde Z_i$ are smooth Deligne-Mumford \nstacks associated to $Y$ and the $Z_i$, respectively. \nMoreover if $G \\subset SL(n,\\mathbf{C})$, then $m=0$.\n\\end{Thm}\n\nIn the case of dimension $2$, we do not need the assumption that the group is abelian:\n\n\\begin{Thm}\\label{GL2}\nLet $G \\subset GL(2,\\mathbf{C})$ be a finite subgroup acting naturally on an affine \nspace $\\mathbf{A} = \\mathbf{C}^2$, let $X = \\mathbf{A}\/G$, and\nlet $f: Y \\to X$ be the minimal resolution.\nThen there exist smooth closed subvarieties $Z_i$ ($1 \\le i \\le m$) of $X$ with $Z_i \\ne X$ \nfor some $m \\ge 0$, with possible repetitions, \nsuch that there is a semi-orthogonal decomposition \n\\[\nD^b(\\text{coh}([\\mathbf{A}\/G])) \\cong \\langle D^b(\\text{coh}(Z_1)), \\dots, D^b(\\text{coh}(Z_m)),\nD^b(\\text{coh}(Y)) \\rangle.\n\\]\nMoreover if there are no quasi-reflections in $G$, the elements whose invariant \nsubspaces are of codimension $1$, then $\\dim Z_i = 0$ for all $i$.\nThus the semi-orthogonal complement of $D^b(\\text{coh}(Y))$\nin $D^b(\\text{coh}([\\mathbf{A}\/G]))$ is generated by an exceptional collection in this case.\n\\end{Thm}\n\nThe derived McKay correspondence was already proved for finite subgroups of \n$SL(2,\\mathbf{C})$, $SL(3,\\mathbf{C})$ (Bridgeland-King-Reid \\cite{BKR}), \nand $Sp(2n,\\mathbf{C})$ (Bezrukavnikov-Kaledin \\cite{BK}).\n\nFinally, we prove the following decomposition theorem for $K$-equivalent toric birational map:\n\n\\begin{Thm}\\label{decomposition}\nLet $f: X \\dashrightarrow Y$ be a toric birational map between\nprojective $\\mathbf{Q}$-factorial toric varieties which is an isomorphism in codimension $1$, and\nlet $B$ be a toric $\\mathbf{R}$-divisor on $X$\nwhose coefficients belong to the interval $(0,1)$, and let $f_*B=C$.\nAssume that $K_X+B = K_Y+C$.\nThen the birational map $f: (X,B) \\dashrightarrow (Y,C)$ is decomposed into a \nsequence of flops.\n\\end{Thm}\n\nAs a corollary of Theorems~\\ref{relative} and \\ref{decomposition}, \nwe obtain an affirmative answer to \n\\lq\\lq $K$ implies $D$ conjecture'' in the toric case:\n\n\\begin{Cor}\nAssume additionally that the coefficients of $B$ belong to a set\n$\\{1 - 1\/m \\mid m \\in \\mathbf{Z}_{>0}\\}$.\nLet $\\tilde X$ and $\\tilde Y$ be the smooth Deligne-Mumford staks \nassociated to the pairs $(X,B)$ and $(Y,C)$, respectively.\nThen there is an equivalence of triangulated categories\n$D^b(\\text{coh}(\\tilde X)) \\cong D^b(\\text{coh}(\\tilde Y))$.\n\\end{Cor}\n\n\\section{Preliminaries}\n\nWe fix the terminology in this section.\nA pair $(X,B)$ consisting of a normal algebraic variety \nand an $\\mathbf{R}$-divisor\nis said to be {\\em KLT} (resp. {\\em terminal}) \nif there is a projective birational morphism \n$p: Z \\to X$ from a smooth varity with an $\\mathbf{R}$-divisor $D$ \nwhose support is a normal crossing divisor such that \nan equality $p^*(K_X+B)=K_Z+D$ holds \nand all the coefficients of $D$ are smaller than $1$\n(resp. all the coefficients of $D - p^{-1}_*B$ are smaller than $0$),\nwhere the canonical divisors $K_X$ and $K_Z$ are defined by using the \nsame rational differential forms.\nIt is called {\\em $\\mathbf{Q}$-factorial} if any prime divisor on $X$ is a \n$\\mathbf{Q}$-Cartier divisor.\n \nA birational map $h: X \\dashrightarrow Y$ between varieties is said to be {\\em proper}\nif there exists a third variety $Z$ with proper birational morphisms $f: Z \\to X$ and \n$g: Z \\to Y$ such that $g = h \\circ f$. \n\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a proper birational map between KLT pairs.\nWe say that there is an {\\em equality of log canonical divisors}\n$K_X+B = K_Y+C$\n(resp. {\\em an inequality} $K_X+B \\ge K_Y+C$), \nif $p^*(K_X+B) = q^*(K_Y+C)$\n(resp. $p^*(K_X+B) \\ge q^*(K_Y+C)$) \nas an ineqality (resp. an equality) of $\\mathbf{R}$-divisors for some $p,q$ as above.\nWe note that such equality or inequality can be defined only \nwhen the birational map \n$f$ is fixed, and the definition is independent of the choice of $p,q$.\nSuch an equality (resp. an inequality) is called a {\\em $K$-equivalence} \n(resp. {\\em $K$-inequality}).\n$f$ is also said to {\\em crepant} if it is a $K$-equivalence.\n\nA proper birational map $f: X \\dashrightarrow Y$ is said to be \n{\\em isomorphic in codimension $1$}\n(resp. {\\em surjective in codimension $1$}) if it induces a bijection\n(resp. surjection) between the sets of prime divisors.\n\nA {\\em flop} (resp. {\\em flip}) is a proper birational map\n$f: (X,B) \\dashrightarrow (Y,C)$ between \n$\\mathbf{Q}$-factorial KLT pairs satisfying the following conditions: \n$f_*B=C$, $K_X+B=K_Y+C$ (resp. $K_X+B > K_Y+C$), and \nthere exist projective birational morphisms \n$s: X \\to W$ and $t: Y \\to W$ to a normal variety \nwhich are isomorphisms in codimension $1$ and the relative Picard numbers \n$\\rho(X\/W)$ and $\\rho(Y\/W)$ are equal to $1$.\n\nA {\\em divisorial contraction} is a projective \nbirational morphism $f: (X,B) \\to (Y,C)$ between \n$\\mathbf{Q}$-factorial KLT pairs satisfying the following conditions: \n$f_*B=C$, and the exceptional locus of $f$ consists of a single prime divisor.\nWe have always $\\rho(X\/Y)=1$, and \nwe have either $K_X+B=K_Y+C$, $K_X+B > K_Y+C$, or $K_X+B < K_Y+C$.\nThe inverse birational map $f^{-1}: (Y,C) \\dashrightarrow (X,B)$ of a \ndivisorial contraction is called a {\\em divisorial extraction}.\n\nA {\\em Mori fiber space } is a projective morphism $f: (X,B) \\to Y$ from a  \n$\\mathbf{Q}$-factorial KLT pair to a normal variety satisfying the following conditions: \n$\\rho(X\/Y)=1$, $-(K_X+B)$ is ample, and $\\dim X > \\dim Y$.\n\nFor a $\\mathbf{Q}$-factorial KLT pair $(X,B)$, there exists a sequence of \ncrepat divisorial contractions \n\\[\n(Y,C)=(X_m,B_m) \\to \\dots \\to (X_1,B_1) \\to (X_0,B_0)=(X,B)\n\\]\nsuch that $(Y,C)$ is terminal and $\\mathbf{Q}$-factorial (\\cite{BCHM}).\nThe composition $f: (Y,C) \\to (X,B)$ is called a {\\em $\\mathbf{Q}$-factorial crepant terminalization} \nof the pair $(X,B)$.\nWe note that even if $B$ has standard coefficients, $C$ may not be so.\n\nLet $f: (Y,C) \\to (X,B)$ be a projective birational morphism\nfrom a $\\mathbf{Q}$-factorial terminal pair to a KLT pair. \nIt is called a {\\em $\\mathbf{Q}$-factorial terminal relative minimal model} \nif $C=f_*^{-1}B$ and $K_Y+C$ is relatively nef over $X$.\nWe note that $f$ is not necessarily crepant.\nThe existence of such a model is also proved in \\cite{BCHM}.\n\nA rational number is said to be {\\em standard} if it is of the form  \n$1-1\/m$ for a positive integer $m$.\nA KLT pair $(X,B)$ is said to be of {\\em quotient type}, if \nthere is a quasi-finite surjective morphism $\\pi': X' \\to X$ from a \nsmooth variety $X'$, which is not necessarily irreducible, such that \n$K_{X'}=(\\pi')^*(K_X+B)$.\nIn this case $B$ has only standard coefficients, and \n$X$ is automatically $\\mathbf{Q}$-factorial.\nIf the pair $(X,B)$ is toric, $X$ is $\\mathbf{Q}$-factorial, and if \n$B$ has standard coefficients, then it is of quotient type.\n\nThere is a {\\em smooth Deligne-Mumford stack} $\\tilde X$ \nassociated to the pair $(X,B)$ of quotient type in such a way that\na sheaf $\\mathcal{F}$ on $\\tilde X$ is nothing but a sheaf $\\mathcal{F}'$ on $X'$  \nwith an isomorphism $p_1^*\\mathcal{F}' \\cong p_2^*\\mathcal{F}'$ satisfying the cocycle condition, \nwhere $p_i: X' \\times_X X' \\to X'$ ($i=1,2$) are projections.\nIn this case, there is a finite birational morphism \n$\\pi: \\tilde X \\to X$ such that $\\pi^*(K_X+B)=K_{\\tilde X}$.\n\nFor an algebraic variety or a Deligne-Mumford stack $X$, \nlet $D^b(\\text{coh}(X))$ \ndenotes the bounded derived category of coherent sheaves on $X$.\nThere are variants of this notation; if a finite group $G$ acts on $X$, then\n$D^b(\\text{coh}^G(X))$ \ndenotes the bounded derived category of $G$-equivariant coherent \nsheaves on $X$.\nThus $D^b(\\text{coh}^G(X)) \\cong D^b(\\text{coh}([X\/G]))$.\n\nA triangulated category $\\mathcal{T}$ is said to have a {\\em semi-orthogonal decomposition}\n$\\mathcal{T} = \\langle \\mathcal{T}_1, \\dots, \\mathcal{T}_m \\rangle$ by \ntriangulated full subcategories $\\mathcal{T}_i$ if the \nfollowing conditions are satisfied:\n(a) $\\text{Hom}(a,b) = 0$ for $a \\in \\mathcal{T}_i$ and $b \\in \\mathcal{T}_j$ if $i > j$, and \n(b) for any object $a \\in \\mathcal{T}$ there is a sequence of objects $a_i \\in \\mathcal{T}$ \nand $b_i \\in \\mathcal{T}_i$ for $1 \\le i \\le m$ such that \n$a=a_1$ and there are distinguished triangles $a_{i+1} \\to a_i \\to b_i$, where\n$a_{m+1}=0$.\nIn particular, if the $\\mathcal{T}_i$ are equivalent to $D^b(\\text{Spec }k)$ for all $i$, then \n$\\mathcal{T}$ is said to have a {\\em full exceptional collection}.\n\nA {\\em toric pair} $(X,B)$ consists of a toric variety and an $\\mathbf{R}$-divisor \nwhose irreducible components are invariant under the torus action.\n$X$ is $\\mathbf{Q}$-factorial if and only if the corresponding fan is simplicial.\nIn this case it has only abelian quotient singularities.\nConversely, the quotient of an affine space by any finite abelian group is a \n$\\mathbf{Q}$-factorial toric variety. \nAny $\\mathbf{Q}$-factorial crepant terminalization and \n$\\mathbf{Q}$-factorial terminal relative minimal model of a toric pair is toric.\n\nFor a pair of quotient type, we can define a derived category whose homological dimension is finite.\nTherefore we would like to ask the following:\n\n\\begin{Que}\\label{quotient-flop}\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a flop or a flip.\nIf $(X,B)$ is of quotient type, then is $(Y,C)$ too?\n\\end{Que}\n\nWe note that a divisorial contraction of a smooth $3$-fold may produce a \nsingularity of non-quotient type even if it is crepant.\nWe note also that even if $X$ is smooth and $f$ is a flop, $Y$ is not necessarily smooth, but we have\nstill the derived equivalence in some examples when we consider the associated \nDeligne-Mumford stacks (cf. \\cite{Francia}).\n\nThe following lemma is standard:\n\n\\begin{Lem}\\label{codimension}\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a proper birational map between \n$\\mathbf{Q}$-factorial terminal pairs, and assume that\n$K_X+B \\ge K_Y+C$. \nThen $f$ is surjective in codimension $1$, and $f_*B \\ge C$.\nMoreover, if $K_X+B = K_Y+C$, then $f$ is an isomorphism in \ncodimension $1$.\n\\end{Lem}\n\n\\begin{proof}\nSuppose that there is a prime divisor $C_1$ on $Y$ \nwhich is not the strict transform of a prime\ndivisor on $X$.\nLet $c_1 \\ge 0$ be the coefficient of $C_1$ in $C$.\nLet $p: Z \\to X$ and $q: Z \\to Y$ be projective birational morphisms \nfrom a smooth variety, and let $D_1$ be the strict transform of $C_1$.\nSince we have an equality $p^*(K_X+B) \\ge q^*(K_Y+C)$, \nthe coefficient of $D_1$ in $p^*(K_X+B)-K_Z$ \nshould be non-negative, a contradiction to the fact that $(X,B)$ is terminal.\nTherefore $f$ is surjective in codimension $1$.\nIf we apply the first assertion to $f$ and $f^{-1}$, then we obtain the second assertion.\n\\end{proof}\n\n\\section{Decomposition theorem for toric pairs}\n\nWe prove Theorem~\\ref{decomposition} in this section.\n\nFirst we prove a condition for the decomposition theorem that is also valid for the non-toric case.\nLet $f: (X,B) \\dashrightarrow (Y,C)$ be a birational map between projective $\\mathbf{Q}$-factorial \nKLT pairs which is an isomorphic in codimension 1 and such that $K_X+B=K_Y+C$. \nLet $p:Z \\to X$ and $q: Z \\to Y$ be projective birational morphisms\nsuch that $f \\circ p = q$.\nLet $\\{l_{\\lambda}\\}$ be a set of curves on $X$ whose classes generate the \ncone of curves $\\overline{NE}(X)$ as a closed convex cone.\nLet $L_Y$ be an ample and effective divisor on $Y$ \nsuch that $L_Y-(K_Y+C)$ is also ample, and \nlet $L_X = f_*^{-1}L_Y$ be its strict transform.\n\n\\begin{Lem}\\label{condition}\nAssume that, if $(L_X,l_{\\lambda}) < 0$ for some $\\lambda$, \nthen $((K_X+B),l_{\\lambda}) = 0$.\nThen $f$ is decomposed into a sequence of flops.  \n\\end{Lem}\n\n\\begin{proof}\nWe shall find suitable extremal rays\nin order to find a path from $(X,B)$ to $(Y,C)$.\n\nAssume first that $L_X$ is nef.\nWe claim that $L_X-(K_X+B)$ is also nef.\nOtherwise, there exists a curve $l_{\\lambda}$ such that \n$((L_X-(K_X+B)),l_{\\lambda})<0$.\nBy the assumption, it follows that $((K_X+B),l_{\\lambda})=0$.\nTherefore $(L_X,l_{\\lambda})<0$, a contradiction.\n\nBy the base point free theorem (\\cite{KMM}), $L_X$ is semiample.\nSince $f$ is an isomorphism in codimension $1$, \nthe associated morphism coincides with $f$.\nSince both $X$ and $Y$ are $\\mathbf{Q}$-factorial, we conclude that $f$ is an\nisomorphism.\n\nNext assume that $L_X$ is not nef.\nWe take a small positive number $\\epsilon$ such that \nthe pair $(X,B+\\epsilon L_X)$ is KLT.\nWe would like to run a minimal model program for this pair in order to\nmove from the pair $(X,B)$ to the pair $(Y,C)$.\nHere we have to note that there may be extremal rays of $(X,B+\\epsilon L_X)$ \nwhich do not come from the negativity of $L_X$.\n\nSince $L_X$ is not nef, \nthere exists a curve $l_{\\lambda}$ such that $(L_X,l_{\\lambda}) < 0$.\nBy the assumption, $((K_X+B),l_{\\lambda})=0$ for such curve.\nTherefore $K_X+B+\\epsilon L_X$ is not nef.\n\nThe part of the closed cone of curves $\\overline{NE}(X)$ \nwhere $L_X$ is negative is contained in the part where $K_X+B+\\epsilon L_X$ \nis negative.\nThus there exists an extremal ray $R$ of $\\overline{NE}(X)$ on which \n$L_X$ is negative.\nSince $K_X+B$ is numerically trivial on $R$.\nthe flip of $R$ for the pair $(X,B+\\epsilon L_X)$ is a flop \nfor $(X,B)$.\nSince $B+\\epsilon L_X$ is big, this process terminates by \\cite{BCHM}, and \n$f$ is decomposed into the flops.\n\\end{proof}\n\nIt is important note that $\\{l_{\\lambda}\\}_{\\lambda \\in \\Lambda}$ is not necessarily the \nset of all curves.\n\nNow we consider the toric case.\nWe take $p: Z \\to X$, $q: Z \\to Y$ and $L_Y$ to be also toric.\nLet $\\Delta_X$ and $\\Delta_Y$ be the simplicial fans corresponding to \n$X$ and $Y$, respectively, in the same vector space $N_{\\mathbf{R}}$.\nBy assumption, the sets of primitive vectors on the edges of \n$\\Delta_X$ and $\\Delta_Y$ coincide.\nSince $\\overline{NE}(X)$ is generated by toric curves, \nTheorem~\\ref{decomposition} is a consequence of the following:\n\n\\begin{Lem}\n(1) Let $w$ be a wall in $\\Delta_X$ and let $l_w$ be the corresponding curve \non $X$.\nAssume that $(L_X,l_w) \\le 0$.\nThen $w$ is not a wall in $\\Delta_Y$.\n\n(2) Let $w$ be a wall in $\\Delta_X$ which does not belong to $\\Delta_Y$.\nThen $((K_X+B),l_w)=0$.\n\\end{Lem}\n\n\\begin{proof}\n(1) Since $f$ is an isomorphism in codimension $1$, \nthe set of primitive vectors $\\{v_i\\}$ on the edges of $\\Delta_X$ and $\\Delta_Y$ coincide.\nLet $g_X$ and $g_Y$ be functions on $N_{\\mathbf{R}}$ corresponding to the divisors\n$L_X$ and $L_Y$, respectively.\n$g_X$ and $g_Y$ have the same values at the $v_i$, \nand are linear inside each simplex \nbelonging to $\\Delta_X$ and $\\Delta_Y$, respectively.\nSince $L_Y$ is ample, $g_Y$ is convex along walls of $\\Delta_Y$.\n\nAssuming that $w$ is also a wall in $\\Delta_Y$, \nwe shall derive a contradiction.\nUnder this assumption, the values of $g_X$ and $g_Y$ are equal on $w$.\n\nLet $v_1,v_2$ be two vertexes adjacent to $w$ in $\\Delta_X$.\nWe take a general point $P$ on $w$ such that the line segments $I_i$ \nconnecting $P$ to $v_i$ for $i=1,2$ pass only the interiors of simplexes and \nwalls of $\\Delta_Y$.\nThen the function $g_Y$ is convex along the $I_i$.\n\nSince $g_X$ is linear on the $I_i$ and coincides with $g_Y$ at the end points,\nwe have an inequality $g_X \\ge g_Y$ on $I_1 \\cup I_2$.\nOn the other hand, $g_X = g_Y$ on $w$.\nSince $g_Y$ is convex along $w$, we conclude that $g_X$ is also convex along \n$w$, hence $(L_X,l_w) > 0$, a contradiction.\n\n(2) Let $D_i$ be the the prime divisors on $X$ corresponding the $v_i$.\nWe write $B = \\sum d_iD_i$, and let $h$ be the function on the vector space \n$N_{\\mathbf{R}}$ which takes values $1-d_i$ at the $v_i$ and linear \ninside each simplex of $\\Delta_X$.\nSince $K_X+B=K_Y+C$, $h$ is also linear inside each simplex of $\\Delta_Y$.\nThere is a point inside $w$ which is an interior point of a simplex in \n$\\Delta_Y$.\nThen $h$ is linear across $w$.\nTherefore $((K_X+B),l_w)=0$.\n\\end{proof}\n\n\\begin{Rem}\nThe relative version is similarly proved.\nWe need to assume that $X$ and $Y$ are projective and toric over a toric variety $S$ and \n$f$ is a toric birational map over $S$.\nWe only consider curves relative over $S$, i.e., those mapped to single points on $S$.\n\\end{Rem}\n\nIf $\\dim X = 3$, then the condition of Lemma~\\ref{condition} is easily verified\neven in the non-toric case:\n\n\\begin{Prop}\\label{dim3}\nAssume that $\\dim X = 3$.\nThen any $K$-equivalent birational map between projective $\\mathbf{Q}$-factorial  KLT pairs\n$f: (X,B) \\dashrightarrow (Y,C)$ which is an isomorphism in codimension $1$\nis decomposed into a sequence of flops.\n\\end{Prop}\n\n\\begin{proof}\nAssume that $(L_X,l) \\le 0$ for a curve $l$.\nSince $\\dim X = 3$, there are only finitely many such curves $l$.\nThen $l$ is the image of a fiber of $q$ by $p$.\nTherefore we have $((K_X+B),l)=0$.\n\\end{proof}\n\n\n\\section{Derived McKay correspondence}\n\nWe prove Theorems~\\ref{abelian} and \\ref{GL2} as well as Theorem~\\ref{relative}~(a) \nin this section.\n\n\\begin{proof}[proof of Theorem~\\ref{abelian}]\nLet $B$ be a $\\mathbf{Q}$-divisor on $X$ such that $p^*(K_X+B)=K_{\\mathbf{A}}$.\nSince the action of $G$ is diagonalizable, the pair $(X,B)$ is toric.\nWe have $D^b(\\text{coh}([\\mathbf{A}\/G]) \\cong D^b(\\text{coh}(\\tilde X))$ for the \nsmooth Deligne-Mumford stack $\\tilde X$ associated to the pair $(X,B)$.\n\nThere exists a toric relative minimal model of $X$.\nAny other relative minimal model is obtained by a sequence of flops from the toric \nrelative minimal model by \\cite{flop}. \nSince the extremal rays are toric, the sequence is automatically toric, \nhence any relative minimal model is again toric.\nBy Theorem~\\ref{relative}, it is sufficient to prove the theorem for a particular relative minimal model.\n\nSince $X$ is $\\mathbf{Q}$-factorial and $(X,B)$ is KLT, we can construct a sequence \ntoric birational morphisms\n\\[\n(X_t,B_t) \\to (X_{t_1},B_{t-1}) \\to \\dots \\to (X_0,B_0)=(X,B)\n\\]\nwhich satisfy the following conditions:\n\n(1) Each step $f_i: (X_i,B_i) \\to (X_{i-1},B_{i-1})$ is a divisorial contraction such that \n$B_i = f_{i*}^{-1}B_{i-1}$ and $K_{X_i}+B_i \\le K_{X_{i-1}}+B_{i-1}$ for $1 \\le i \\le t$.\n\n(2) The pair $(X_t,B_t)$ is terminal.\n\nNext we run a toric minimal model program for $X_t$ over $X$ \nto obtain a sequence of birational maps\n\\[\nX_t=Y_0 \\dashrightarrow Y_1 \\dashrightarrow \\dots \\dashrightarrow Y_s=Y\n\\]  \nconsisting of divisorial contractions and flips such that $K_Y$ is nef over $X$. \nWe have $K_{Y_{j-1}} > K_{Y_j}$ for $1 \\le j \\le s$.\n\nWe apply Theorem~\\ref{relative} and Corollary~\\ref{combine} \nto each step of the above sequences to obtain our result. \n\\end{proof}\n\n\\begin{proof}[proof of  Theorem~\\ref{GL2}]\nLet $H$ be the normal subgroup of $G$ defined by $H = G \\cap SL(2,\\mathbf{C})$.\nThe residue group is a cyclic group $\\mathbf{Z}_r=G\/H$ for $r=[G:H]$.\nBy \\cite{BKR}, \na component of the Hilbert scheme of $H$-invariant subschemes\nof $\\mathbf{A}$ gives a crepant resolution $f: M \\to \\mathbf{A}\/H$, and \nthere is a derived equivalence $\\Phi:D^b(\\text{coh}(M)) \\to D^b(\\text{coh}^H(\\mathbf{A}))\n= D^b(\\text{coh}([\\mathbf{A}\/H]))$.\nMore precisely, $M$ is the moduli space of closed subschemes\n$P \\subset \\mathbf{A}$ which is $H$-invariant\nand such that $\\text{length }\\mathcal{O}_P = \\#H$.\nFor a generic point $[P] \\in M$, $P$ is a reduced subscheme, \nand corresponds to a generic orbit of $H$ in $\\mathbf{A}$.\nThere exists a universal subscheme \n$W \\subset M\\times \\mathbf{A}$ with projections \n$p: W \\to M$ and $q: W \\to \\mathbf{A}$ such that the \nfunctor $\\Phi=q_*p^*$ gives the equivalence.\n\nFor $[P] \\in M$ and $g \\in G$, $gP$ is also $H$-invariant, since $H$ is a normal subgroup.\nHence $[gP] \\in M$, and $G$ acts on $M$ such that its subgroup $H$ acts trivially.\nThe equivariant derived category $D^b(\\text{coh}^{\\mathbf{Z}_r}(M))$ is identified with \nthe category where the objects are $G$-equivariant complexes on which $H$ acts \ntrivially, and the morphisms are $G$-equivariant morphisms.\n\n$G$ acts on the product $M \\times \\mathbf{A}$ diagonally and preserves the subscheme $W$.\nWe claim that the functor\n\\[\n\\Phi'= q_*p^*: D^b(\\text{coh}^{\\mathbf{Z}_r}(M)) \\to D^b(\\text{coh}^G(\\mathbf{A})).\n\\]\nis an equivalence.\nIndeed if $a,b$ are $\\mathbf{Z}_r$-equivariant objects, then\nthe isomorphism \n\\[\n\\text{Hom}_M(a,b) \\cong \\text{Hom}_{\\mathbf{A}}(\\Phi(a),\\Phi(b))^H\n\\]\nimplies an isomorphism\n\\[\n\\text{Hom}_M(a,b)^{\\mathbf{Z}_r} \\cong \n(\\text{Hom}_{\\mathbf{A}}(\\Phi(a),\\Phi(b))^H)^{\\mathbf{Z}_r}\n= \\text{Hom}_{\\mathbf{A}}(\\Phi(a),\\Phi(b))^G. \n\\]\n\nLet $g: M \\to M\/\\mathbf{Z}_r$ be the map to the quotient space.\nWe define a boundary divisor $B_{M\/\\mathbf{Z}_r}$ on $M\/\\mathbf{Z}_r$ by \n$g^*(K_{M\/\\mathbf{Z}_r}+B_{M\/\\mathbf{Z}_r})=K_M$. \nSince $M$ is smooth, $M\/\\mathbf{Z}_r$ has only cyclic quotient singularities.\nLet $h: Y' \\to M\/\\mathbf{Z}_r$ be the minimal resolution, and \nlet $k: Y' \\to Y$ be the contraction morphism to the minimal resolution of $\\mathbf{A}\/G$.\nThe former is a toroidal morphism, and decomposed into toroidal divisorial contractions.\nThe latter is a composition of divisorial \ncontractions of $(-1)$-curves, which are toroidal too.\n\nThe canonical divisors satisfy the following inequalities:\n\\[\nK_Y \\le K_{Y'} \\le K_{M\/\\mathbf{Z}_r} \\le K_{M\/\\mathbf{Z}_r}+B_{M\/\\mathbf{Z}_r}\n=K_{\\mathbf{A}\/G}.\n\\]\nBy Theorem~\\ref{relative}, we obtain fully faithful embeddings\n\\[\nD^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde Y')) \\to D^b(\\text{coh}([M\/\\mathbf{Z}_r])) \n= D^b(\\text{coh}([\\mathbf{A}\/G]))\n\\]\nwhose semi-orthogonal complements are generated by the derived categories of subvarieties of \n$X=\\mathbf{A}\/G$.\n\\end{proof}\n\n\\begin{proof}[proof of Theorem~\\ref{relative}~(a)]\nIt is sufficient to consider the following situation:\n$X=Y$ is an affine toric variety corresponding to a simplicial cone \n$\\sigma \\subset N_{\\mathbf{R}}$ generated by primitive vectors $v_1,\\dots,v_n \\in N$.\nLet $B_i$ be prime divisors on $X$ corresponding to the $v_i$, and let \n$B = \\sum (1-1\/r_i)B_i$ and $C=\\sum (1-1\/s_i)B_i$, where $r_1 > s_1$ and\n$r_i=s_i$ for $i=2,\\dots,n$.\nLet $t_1=\\text{LCM}(r_1,s_1)$ and $t_i=r_i$ for $i=2,\\dots,n$.\n\nThe stacks $\\tilde X$,  $\\tilde Y$ and $\\tilde W$ are defined by the sublattices \n$N_X$, $N_Y$ and $N_W$ \nof $N$ generated by the $r_iv_i$, $s_iv_i$ and $t_iv_i$, respectively. \nLet $\\tilde B_i^X$, $\\tilde B_i^Y$ and $\\tilde B_i^W$ be prime divisors \non $\\tilde X$,  $\\tilde Y$ and $\\tilde W$ over $B_i$, respectively.\nThey correspond primitive vectors $r_iv_i$, $s_iv_i$ and $t_iv_i$, respectively.\nLet $p: \\tilde W \\to \\tilde X$ and $q: \\tilde W \\to \\tilde Y$ be the natural morphisms.\nThen the fully faithful functor $\\Phi: D^b(\\text{coh}(\\tilde Y)) \\to D^b(\\text{coh}(\\tilde X))$ is \nconstructed as $\\Phi=p_*q^*$ in \\cite{log crepant}~Theorem~4.2 (1).\nWe have \n\\[\n\\Phi(\\mathcal{O}_{\\tilde Y}(k\\tilde B_1^Y)) = \n\\mathcal{O}_{\\tilde X}(\\llcorner kr_1\/s_1 \\lrcorner \\tilde B_1^X)\n\\]\nfor $k = 1,\\dots, s_1-1$.\n\nWe consider the semi-orthogonal complement.\nIt is generated by the sheaves\n$\\mathcal{O}_{\\tilde B_1^X}(l\\tilde B_1^X)$ for $1 \\le l \\le r_1-1$ such that\n$l \\ne \\llcorner kr_1\/s_1 \\lrcorner$ for any $k$.\nWe have \n\\[\nR\\text{Hom}_{\\tilde X}(\\mathcal{O}_{\\tilde X}(\\llcorner kr_1\/s_1 \\lrcorner \\tilde B_1^X),\n\\mathcal{O}_{\\tilde B_1^X}(l\\tilde B_1^X))=0\n\\]\nfor such $k,l$, and \n\\[\nR\\text{Hom}_{\\tilde X}(\\mathcal{O}_{\\tilde B_1^X}(l\\tilde B_1^X), \n\\mathcal{O}_{\\tilde B_1^X}(l'\\tilde B_1^X))=0\n\\]\nfor such $l,l'$ with $l < l'$.\n\nWe define a toric boundary $D$ on $B_1$ \nfrom the pair $(X,B)$ as follows (cf. \\cite{toric}~\\S5).\nLet $\\bar N = N\/\\mathbf{Z}v_1$, and write $v_i + \\mathbf{Z}v_1 = d_i \\bar v_i \\in \\bar N$ for \nprimitive vectors $\\bar v_i$ ($i=2,\\dots,n$).\nThen define \n$D= \\sum_{i=2}^n (1-1\/d_ir_i)D_i$ for $D_i=B_1 \\cap B_i$. \nLet $\\tilde B_1$ be the associated smooth Deligne-Mumford stack above $(B_1,D)$.\nLet $\\tilde B_1'$ be the normalization of the fiber product $\\tilde B_1 \\times_X \\tilde X$ with \nnatural morphisms \n$p_1: \\tilde B_1' \\to \\tilde B_1$ and $p_2: \\tilde B_1' \\to \\tilde X$.\nThen as in \\cite{toric}, the semi-orthogonal complement is generated by \nsubcategories \n\\[\np_{2*}p_1^*D^b(\\text{coh}(\\tilde B_1)) \\otimes \\mathcal{O}_{\\tilde X}(l\\tilde B_1^X)\n\\]\n$1 \\le l \\le r_1-1$ such that $l \\ne \\llcorner kr_1\/s_1 \\lrcorner$ for any $k$.\nMoreover, there is no morphism between objects belonging to different $k$'s.\nTherefore we conclude the proof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}