diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzneq" "b/data_all_eng_slimpj/shuffled/split2/finalzneq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzneq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction.}\n\nFully self-consistent N-body simulations, where each galaxy is \nrepresented by a large number of particles, are a useful, albeit \nexpensive,\ntool for studying the evolution of galaxy groups and clusters. However, \nfor simulations of large clusters of galaxies, like the Coma cluster, the \nnecessary computing time is prohibitive. As a substitute people have \nconsi\\-dered explicit simulations, in which each galaxy is represented by a \nsingle point and the physics of the interactions is modelled by explicit \nprescriptions for merging conditions. In particular,\na variety of recipes are explored for the conditions the two galaxies \nmust fulfill in order to merge. In general, these merging conditions are \nbased on self-consistent si\\-mulations of two-galaxy collisions, and do not \ninclude the tidal\nforces between the galaxies or collisions involving more than two \ngalaxies. It is thus not a priori certain that they will perform well in \nsimulations of group or cluster evolution. In some cases (Merritt, 1983; \nRichstone and Malumuth, 1983; Mamon 1987), the authors also introduce \nother effects like dynamical friction and tidal forces from the\nbackground. The main advantage of this type of approach is that it is \ninexpensive in computing time and therefore allows one to explore a wide\nparameter space. In any case, a considerable fraction of the results on \nthe dynamics of galaxy groups are\nbased on the explicit approach. We may cite works by Jones and Efstathiou \n(1979), Roos and Norman (1979), Aarseth and Fall (1980), Cooper and \nMiller (1981), Roos (1981), Roos and Aarseth (1982), Merritt (1983), \nRichstone and Malumuth (1983), Malumuth and Richstone (1984), Saarinen \nand Valtonen (1985), Mamon (1987), Navarro et al. (1987) and Schindler \nand B\\\"ohringer (1993).\n\nNot many self-consistent simulations of groups with more than 10 galaxies \ncan be found in the literature. We can cite the articles by Carnevalli et \nal. (1981), Ishizawa et al. (1983), Ishizawa (1986), Rhee and Roos \n(1990), Barnes (1992), Funato et al. (1993) and Bode et al. (1994). The \nfirst works of this kind used Aarseth's (1971) N-body code and a limited \nnumber of points, typically $10-20$, to\nrepresent each galaxy, and only recently it has become possible to use \nthe order of 1000 particles per galaxy.\n\nOur aim is to compare the two approaches to see whether, and under what \nconditions, one can use explicit simulations and have confidence in the \nresults. For this purpose, we have evolved a set of initial conditions in \ntwo different ways. One way is to use an N-body code where physics is \nincluded explicitly, the other, to use self-consistent simulations and a \ntreecode (Barnes and Hut 1986, Hernquist 1987 for a vectorised \nversion), representing each galaxy either by $100$ or by $900$ points. \nIn section 2 we describe our initial conditions and the different merging \ncriteria used so far in the literature. In section 3 we compare the \nresults of fully self-consistent numerical simulations to those of \nexplicit simulations made with the various merging criteria, both without \n(section 3.1) and with dynamical friction (section 3.2). This comparison \nled us to propose a new merging criterion (section 3.3), whose \nperformance we also compare with the fully self-consistent simulations. \nIn this section we consider only groups with no common all-encompassing\ndark matter halo. Simulations including such a halo are presented in \nsection 4, where again we compare the results of self-consistent and \nexplicit simulations. We summarise and discuss our results in section 5. \n\n\n\\section{Initial conditions and merging criteria} \n\nWe have considered five different initial conditions, labeled A, B, C, D \nand H, each for systems consisting of 50 galaxies. In simulations A, B, D \nand H the radial distances from the\ngalaxy centers to the center of the group were picked at random between 0 \nand $R_{out}$.\nFor simulation C the central part of the sphere contained no galaxy, i.e. \nthe radial distances were picked between $0.5R_{out}$ and $R_{out}$. For \nsimulations\nA to D all the mass is in the individual galaxies, while in simulation H \nwe included a common live halo, centered on the center of the group, and \ncontaining half of the total mass. The halo density distribution is a \nPlummer one with a core radius equal to half $R_{out}$. Run A starts in \nfree-fall, and we will often refer to it as the collapsing group. The \nvelocity dispersions in the remaining three runs were chosen to be \nindependent of radius, gaussian, isotropic, and such that the system of \ngalaxies starts off in virial equilibrium. Simulation D is similar \nto B but more compact, as the radius of the sphere containing all the \ngalaxies is half that of run B. The particles in a given galaxy were \ninitially taken to follow a Plummer distribution of core radius equal to \n0.2 and of unit mass. When evolved in isolation, an individual\ngalaxy first shows a low amplitude relaxation in the very first few\ntime steps due to the fact that the simulations have a softening, while \nthe analytical Plummer sphere does not. After that, and for a time equal \nto that during which the group simulations were run, the galaxies do not\nevolve any further. Thus, during that time, for the representations with \n900 particles per\ngalaxy, the radii containing 25\\%, 50\\% and 75\\% of the mass of the \ngalaxy vary only by a couple of\npercent. For the representation with 100 particles the radii containing \n25\\% and 50\\% of the mass vary by 4-5\\%, and only the radius containing \n75\\% of the mass varies significantly, particularly in the later phases \nof the evolution.\n\nMore\ninformation on the initial conditions for the simulations is summarised \nin Table~1.\n{\\sl Column}~1 contains the name of the simulation, {\\sl Column}~2 gives \n$R_{out}$, the\nradius of the sphere containing the group at the start of the simulation, \n{\\sl Column}~3 shows the initial mean separation between the galaxies and \n{\\sl Column}~4 gives the ratio between the initial velocity dispersion of \nthe galaxies considered as point masses, $\\sigma_{cl}$, divided by the \nvelocity dispersion of the particles within a single galaxy, \n$\\sigma_{gal}$. {\\sl Column}~5 contains the crossing time defined as\n\\begin{equation}\nt_{cr}=\\left( {\\frac{2R_h^3}{GM}}\\right) ^{1\/2}, \\end{equation}\n\n\\noindent where $R_h$ represents the half mass radius. Finally, {\\sl \nColumn}~6 contains the ratio between $t_{tot},\\\/$ the total duration of \nthe simulation and $t_{cr}$. All through this paper our units are such \nthat the gravitational constant $G~=~1$. \n\n\\begin{table}\n\\begin{center}\n\\caption{Initial conditions of the simulations} \\vskip 0.25cm\n\\begin{tabular}{llllll}\n\\hline\nRun & $R_{out}$ & $$ & $\\sigma _{cl}\/\\sigma _{gal}$ & $t_{cr}$ & \n$t_{tot}\/t_{cr}$ \\\\ \\hline\nA & 30 & 8.0 & 0.0 & 15.3 & 2.0 \\\\\nB & 20 & 6.8 & 1.4 & 4.5 & 6.7 \\\\\nC & 20 & 10.3 & 1.0 & 11.6 & 2.6 \\\\\nD & 10 & 3.4 & 1.9 & 1.6 & 18.7 \\\\\nH & 20 & 7.4 & 2.7 & 8.9 & 3.4\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe self-consistent simulations were run using the vectorised version \n(Hernquist 1988) of the Barnes-Hut tree algorithm (Barnes and Hut 1986), \nwith a softening of 0.05 and an opening angle $\\theta=0.7$. In explicit \nsimulations each galaxy is represented by a single point to which is \nassociated a mass, an internal energy and a core radius. These parameters \nmay change during the evolution of the system due to the different \ninteractions suffered by the point-galaxies and we used the recipes of \nAarseth and Fall (1980) to follow their time evolution. The explicit \nsimulations are of course much faster than the self-consistent\nones. A complete self-consistent simulation with $900$ points per galaxy \ntook $521663$ seconds in a Cray YMP 2L computer. The self-consistent \nsimulation with $100$ particles per galaxy lasted only the $5\\%$ of this \ntime and the explicit simulation only $0.2\\%$. \n\nIn order to compare the results of the different kind of simulations we \nconsider the time evolution of the following global parameters of the \ngroups:\n\n\\begin{enumerate}\n\\item Number of galaxies: $N_{gal}$\n\\item Half mass radius: $R_h$, where $M(R_h)=1\/2\\,M_{tot}$ \\item Three \ndimensional velocity dispersion: \\end{enumerate}\n\n$$\\sigma_v^2=\\sum_{i=1}^{N_{gal}}\\frac{m_i\\mid {\\bf v_i}-<{\\bf v}> \n\\mid^2}{M_{tot}-m_i(t=0)},\\,\\,\n{\\rm where}\\,\\,<{\\bf v}>=\\sum_{i=1}^{N_{gal}}\\frac{m_i {\\bf v_i}} \n{M_{tot}}.$$\n\n\\noindent where all quantities are evaluated at each timestep, except for \n$m_i(t=0) = 1$ which is the mass of all individual galaxies at the \nstart of the simulations and is taken to be $m_i(t=0) = 1$.\n\nIn our explicit simulations we consider, in a first stage, only merging \nbetween galaxies. In a second set of simulations we include also the \neffect of dynamical friction. In this way we can check the importance of \nboth effects. Merging between galaxies is usually described in the \nliterature using an explicit condition involving the separation and \nrelative velocities of the pair of galaxies. If this condition is \nfulfilled, the two galaxies are merged in a single one in this timestep, \ntaking into account the conservation of mass, energy and momentum \n(Aarseth \\& Fall 1980). If this condition is not fulfilled both galaxies \nsurvive and continue their motion. \n\nWe found in the literature various criteria which have been used to \ndecide whether two galaxies are going to merge and we used all of them in \nturn in our explicit simulations. The condition of Roos and Norman (1979, \nhereafter condition RN) is: \\begin{equation}\nv(r_p)\\leq 3.1\\sigma (1-0.3\\frac{r_p}{R_g}) \n\\left(\\frac{1+m_2\/m_1}{2}\\right)^{1\/4}\n\\end{equation}\n\n\\noindent\nwhere $m_2 \\leq m_1$ and $r_p\/R_g < 1$. $r_p$ is the minimum separation \nbetween the galaxies, $v(r_p)$\nis their relative velocity at $r_p$, and $R_g$ is the larger of their \nradii. This criterion was obtained empirically from collisions between \ngalaxies described by fewer than $100$ particles. \n\nAarseth and Fall (1980, hereafter condition AF) used the criterion: \n\\begin{equation}\n{\\left[ \\frac{r_p}{{2.6(\\epsilon _1+\\epsilon _2)}}\\right] }^2+ \n{\\left[\\frac{v(r_p)}{{1.16v_e(r_p)}}\\right] }^2\\leq 1,\n \\end{equation}\n\n\\noindent\nwhich is a simple fit to the results of the simulations of van Albada and \nvan Gorkom (1977), White (1978) and Roos and Norman (1979). The core \nradius of galaxy {\\it i} is $\\epsilon _i$, while $v_e(r_p)$ is the escape \nvelocity of the system composed of the two galaxies before merging at \npericenter:\n\\begin{equation}\nv_e^2(r_p) = 2 G (m_1 + m_2)(r_p^2 + \\epsilon_1^2 + \\epsilon_2^2)^{-1\/2}.\n\\end{equation}\n\nFarouki and Shapiro (1982, hereafter condition FS) obtained a similar \ncondition for the merging of two rotating galaxies with massive halos and \nspins aligned with the orbital angular momentum: \\begin{equation}\n{\\left[ \\frac{r_p}{{5.5(\\epsilon _1+\\epsilon _2)}}\\right] }^2+ {\\left[ \n\\frac{v(r_p)}{{1.1v_e(r_p)}}\\right] }^2\\leq 1. \\end{equation}\n\nThis condition predicts more mergings than the criterion from Aarseth and \nFall (1980) for two reasons. It favours collisions between galaxies \nfurther apart and it forces the spins to be aligned. This criterion is \nnot directly applicable to our case, where we use initially nonrotating \nPlummer spheres, but we include it for the sake of completeness.\n\nFinally Richstone and Malumuth (1983, hereafter condition RM) use the \ndifferent criterion:\n\\begin{equation}\nr_pv(r_p)\\leq \\left[ 8\/3\\,G^2\\,(m_1\\,+\\,m_2)(m_1+m_2) \n\\right] ^{1\/4},\n\\end{equation}\n\n\\noindent\nwhich is a generalisation of a criterion propo\\-sed by Tre\\-maine (1980)\nfor the case of different masses. The value $$ is the mean \nquadratic radius of a galaxy. For the case of a Plummer sphere \n$=\\epsilon ^2\/2$, and this is the value we have used in our \nsimulations.\n\nTo save computer time we do not need to apply the adopted merging \ncriterion to all galaxy pairs at all times. Following Navarro {\\it et \nal.} (1987), we check whether the condition is fulfilled only if the \nseparation between two galaxies is smaller than $3(r_{h_1}+r_{h_2})$, \nwhere $r_{h_i}$ is the half mass radius of the galaxy {\\it i}. This \nseparation is sufficiently large so that merging events are not missed, \nwhile speeding up considerably the computations. \n\nAs the simulations evolve a central giant ``galaxy\" is formed as a result \nof the mer\\-gings and\/or tidal stripping of the galaxies in the group. \nDynamical friction between this and the remaining individual galaxies \ninfluences the\nevolution and we have therefore included this effect in the explicit \nsimulations, using the well known\nChandrashekar (1943) formula for the deceleration: \\begin{equation}\n{\\bf a_v} = - \\frac{4\\pi G^2\\, m_{gal} \\ln \\Lambda \\rho({\\bf r})} {v^3} \nF(v){\\bf v}\n\\end{equation}\nwhere\n\\begin{equation}\nF(v) = erf(X) - \\frac{2X}{\\sqrt{\\pi}}e^{-X^2} \\end{equation}\nand $erf(X)$ is the error function, $X=v\/\\sqrt{2}\\sigma$, $\\sigma$ is the \nvelocity dispersion of the\nobjects in the background, and $m_{gal}$ is the mass of the galaxy \ntravelling at speed ${\\bf v}$;\n$\\rho({\\bf r})$ is the density of the central galaxy, considered as a \nPlummer sphere, at the position of the\nsecondary galaxy, ${\\bf r}$ being the relative separation of their \ncenters, and $\\Lambda = b_{max} \/ b_{min}$, where $b_{max}$ and $b_{min}$ \nare the maximum and minimum\nimpact parameters of encounters contributing to the drag. When we include \na common halo we apply Eq. (6) twice, once for the central giant \n``galaxy\" and the other for the halo, adding these two accelerations. \n\nThe self-consistent simulations where analyzed as follows. First, we \nneed to define the central giant ``galaxy\", which we will refer to in \nthis paragraph simply as the central object. In order to do so, we \nanalyze at each timestep\nseparately each subsystem composed of the particles that were bound at \n$t=0$ in a single galaxy. Using the positions and velocities of these \nparticles we discard from the subsystem all particles with positive \nenergy\nrelative to it and consider them as part of the central object. The \nparticles that still form a bound subsystem will define the state of the \ngalaxy at this timestep. If after this process a galaxy contains less \nthan $10\\%$ of the particles it had at $t=0$, we discard this subsystem \nas a galaxy and we add all its particles to the central object, thus \nconsidering that the initial galaxy has been definitely disrupted. For \neach of the remaining galaxies we use the $35\\%$ of its most bound \nparticles to define its position and velocity. Finally, we also consider \npossible mergings between the remaining galaxies, as well as between \nthese galaxies\nand the central object. Two galaxies were merged in a single one if the \nfollowing conditions are satisfied:\n\\begin{eqnarray*}\n\\Delta r & < & a(r_{c1}+r_{c2}) \\\\\n\\Delta v & < & b(\\sigma_1 + \\sigma_2)\n\\end{eqnarray*}\n\\noindent where $r_{ci}$ is the radius of the sphere containing the \n$35\\%$ most bound particles and $\\sigma_i$ its velocity dispersion. The \nconstants $a=1.4$\nand $b=0.6$ were selected in order to have smooth central objects. The \nparameters of this object were calculated with just the $10\\%$ most bound \nparticles and not with the $35\\%$ as with the rest of the galaxies. This \nensured that we do not consider a merger between the central galaxy and \nanother galaxy while they still form two separate objects. We finally \nused the positions and velocities of the remaining galaxies\nto define the global parameters of the system.\n\n\\section{Simulations without common dark matter halo} \n\n\\subsection{Evolution without dynamical friction.} \n\nIn Fig.~1 we show the evolution of the number of galaxies $N_{gal}$ as a \nfunction of time for all the simulations without distributed dark matter. \nIn the first column we compare the self-consistent simulations with $900$ \nand $100$ particles per galaxy with the explicit simulations obtained \nwith the AF and RM conditions.\nIn the second column, the evolution of the number of galaxies in the \nself-consistent simulations is compared with the explicit simulations \nusing the FS and RN conditions. \n\nWe note that the explicit simulations perform rather unequally. The \nresults depend on the type of initial conditions and on the mergng \ncondition used to describe the interactions. Globally we can say that the \nAF and RM conditions\nseem to follow the time evolution of $N_{gal}$ much better than the FS \nand RN conditions. In the first stages of the evolution of the collapsing \ngroup (Run A), the less tightly bound and virialised group (Run B) and \nfor Run C, which is a virialised group with no central mass \nconcentration, both AF and RM conditions describe the time evolution of \nthe number of galaxies rather well. This is not true, however, for Run D \n(tightly bound and virialised group), for which the AF condition \noverestimates the number of mergings from the start, while the RM \ncondition does the opposite. As the evolution proceeds the discrepancies \nbetween the self-consistent simulations and the explicit simulations \nbecome more evident. For all initial conditions the FS and RN conditions \noverestimate the number of mergings from the start. The sole exception is \nthe explicit simulation with the FS condition in the case of Run C, where \nthe agreement with the self-consistent simulation is quite good. \n\nFor the time evolution of the half-mass radius, $R_{h}$, we find similar \nresults. This can be seen in Fig.~2, where the panels refer to the same \ninitial conditions as in Fig.~1. In general, the explicit simulations \ncontrolled by the AF and RM conditions show a better global behaviour \nthan the simulations governed by the FS and RN conditions. This is due to \nthe high number of mergings predicted by the latter conditions. In the \ncase of Run A, all explicit simulations follow well the collapse phase. \nWhen most of the mass is accumulated in the central area, the number of \nencounters is relatively large and there are strong interactions with the \ngiant central galaxy. At this moment, self-consistent and explicit \nsimulations separate. The AF and RM conditions allow some galaxies to \navoid merging with the giant galaxy in the first passage and the system \nexperiences an expansion which is not shown in self-consistent \nsimulations. On the other hand, the FS and RN conditions predict a much \nhigher rate of mergers than the self-consistent simulations and we are \nleft too early with only a single giant galaxy. In the case of Run~B, the \nAF and RM conditions describe very well the state of the system during \nthe first part of the simulations. As the simulation evolves, however, \nsome galaxies reach the central parts where they suffer an hyperbolic \nencounter with the central mass concentration of the giant galaxy instead \nof merging with it, as is the case in the self-consistent simulations, \nbecause the merging criteria strongly disfavour merging in high speed \ncollisions. This makes the system expand, an effect which is not seen in \nthe self-consistent simulations. This does not happen for Run C where \nthere is no such central mass concentration and explicit and \nself-consistent simulations follow the same evolution, except for minor \ndifferences and a strong deviation for the case of condition RN. Run D is \nthe most difficult case for the explicit simulations. In this situation \ngalaxies move at higher speeds than in Run B or Run C. Surprisingly, in \nthe case of self-consistent simulations, this does not make merging with \nthe central object more difficult, as one might expect naively in the \nfirst instance. However, the RM conditions predicts more hyperbolic \nencounters than the self-consistent simulations, giving strong \noscillations of the half mass radius. On the other hand, the AF condition \nseem to describe the situation quite well. The number of galaxies \npredicted by the RN and FS conditions\nare well below the numbers predicted by the self-consistent simulations, \nagain due to the high number of mergers predicted by these conditions.\n\nFinally, in Fig.~3 we show similar comparisons, now for the three \ndimensional velocity dispersion.\nThe larger number of mergers predicted by the FS and RN conditions nearly \nalways gives lower velocity dispersions than the self-consistent \nsimulations as well as strong oscillations due to small number\nstatistics. On the other hand, the AF and RM conditions give a better \ngeneral description of the evolution of the three dimensional velocity \ndispersion. This is specially true for Run A, where all the\nmotion is nearly radial and only small discrepancies appear at the end of \nthe simulations. For the case of Run B and the RM condition, the \nhyperbolic encounters which lead to a higher half mass radius of the \nsystem, give also higher velocity dispersions, because some galaxies \nwhich merge in the self-consistent simulations can escape in the explicit \nones. The AF condition describes this time evolution much better. The \nvelocity dispersion of Run C is well described for both conditions until \nshortly before the end of the simulation, when both conditions predict \nhigher velocity dispersions than the self-consistent simulations. In the \ncase of Run D the RM condition has again\nsome difficulty in describing the behaviour of the self-consistent \nsimulations. This is also due to the high number of large deflections of \nthe secondary galaxies. The AF condition follows well the evolution of \nthe three dimensional velocity dispersion in this situation. \n\nWe would like to note at this point that the self-consistent simulations \nwith $100$ particles per galaxy and with $900$ particles per galaxy do \nnot show major differences. The number of galaxies as a function of time \ndoes not change appreciably between these two simulations and this for \nall the initial conditions, i.e. both for virialised and collapsing \ngroups. In this sense our results differ from those of van Kampen (1995), \nwho found that the small virialised clumps formed during the simulations associated with the \ngalaxies do not resist the passage through the central part of the \ncluster. This could be due to the somewhat lower number of particles per \ngalaxy, since the typical galaxies in van Kampen's simulations are \ncomposed of 10-50 points (van Kampen 1995).\n\nSimilarly good agreement between the 900 and 100 points per galaxy \nsimulations is found for the\nvelocity dispersion. Somewhat bigger differences, in particular for run \nB, can be seen for the half-mass radius, but even these are not \nexcessive.\n\n\\subsection{Simulations with dynamical friction.} \n\nFigure~4 compares the time evolution of the number of galaxies in the \nself-consistent simulations and in the explicit simulations when the \neffect of dynamical friction is included. Since this slows down the \ngalaxies and thus\nfavours merging, the number\nof galaxies, $N_g$ will diminish faster. This is clearly seen in all the \npanels of Fig.~4. As can be seen from the left hand panels, this worsens \nthe predictions of the RN and FS conditions. The right hand panels show \nthat the agreement is now better for the RM condition, and worst for the \nAF one. For the case of Run A there is a systematic deviation between the \nAF condition and the self-consistent simulations. On the other hand, the \nRM condition which had, in the absence of dynamical friction, predicted a \nlow number of mergings is, in this case, in much better agreement with \nthe self-consistent case. The same can be said about Run B, while in Run \nC the effect of dynamical frictions is not noticeable. This is not \nsurprising as we take into account only the effect of dynamical friction \nwith the most massive galaxy which, in this case, is practically \nnonexistent. For the most difficult case, Run D, the AF condition falls \nbelow the results of the self-consistent simulations while the RM \ncondition gives good agreement. \n\nThe evolution of the half mass radius is also affected by the inclusion \nof dynamical friction, as is shown in Fig.~5, where we plot the evolution \nof $R_h$ as a function of time. For the explicit simulations with the RN \nand FS conditions dynamical friction does not alter the strong \ndisagreement with the\nself-consistent simulations. This happens because the explicit \nsimulations with these conditions allow too many mergings and we are left \nwith a single supergiant galaxy at the center of the system which \ncontains a large fraction of the mass and some small satellites. On the \nother hand, there is now a much better agreement between the explicit \nsimulations made with the AF and RM conditions and the self-consistent \ncases. For Run A neither condition shows a secondary bouncing of the \nsystem. The dynamical\nfriction acts as a braking mechanism that favours merging between the \nsecondary galaxies and the central one and a lower number of satellites \nsurvive in this situation. In Run B, the hyperbolic encounters of the \nsatellite galaxies with the central giant are not present and there is no \nlater expansion of the system as in the explicit simulations without \ndynamical friction. The explicit simulations with both\nthe AF and RM conditions predict too small a half mass radius. For Run C, \nas there\nis no giant galaxy, dynamical friction is unimportant and all the \nsimulations again show the same general behaviour. In Run D the galaxies \nmove faster because the system is more tightly bound. The explicit \nsimulations with the RM condition and no dynamical friction were not \ncapable of describing the evolution of the self-consistent simulations. \nThe inclusion of dynamical friction gives a much better agreement between \nthese two simulations. On the other hand, the explicit simulations with \nthe AF conditions seem to be systematically below the predictions of the \nself-consistent simulations. \n\nAs can be seen in Fig.~6, the three dimensional velocity dispersion shows \nmarked diffe\\-rences between the explicit simulations and the \nself-consistent ones. As was the case in the absence of dynamical \nfriction the explicit simulations with the RN and FS conditions do not \ntrack well the self-consistent results. The dynamical friction effect is \nbarely noticeable in this case, except for some tendency towards lower \nvelocity dispersions. As the RN and FS conditions predict many mergings, \nwe are left with a giant galaxy in the center and a low number\nof satellites orbiting around it. The dispersions are then low but they \nare more subject to fluctuations and have stronger oscillations. \nIncluding dynamical friction in the explicit simulations with the AF and \nRM conditions does not substantially improve their\nresults as can be seen if we compare Fig.~6 with Fig.~3. For runs A and C \nthe situation is further improved and the explicit simulations follow the \nself-consistent ones very well. Bigger differences between the explicit \nsimulations with and without dynamical friction are found for the \nvirialised groups (Run B and D). The values predicted by the AF condition \nare now always near the values obtained with the self-consistent \nsimulations. However, this is not the case for the RM parametrization. \nFor Run B, there are marked differences between these explicit \nsimulations and the last phase of the self-consistent simulations. For \nthe case of Run D the RM condition gives a systematically higher velocity \ndispersion than the self-consistent simulations.\n\n\\subsection{A new merging criterion}\n\nAs we have seen, none of the merging criteria proposed so far in the \nliterature is capable of describing the time evolution of the global \nproperties of groups of galaxies in the variety of situations considered \nin this paper. We can\nsay that, in general, the AF and RM conditions perform better that the FS \nand RN ones, but even they\nfail to describe the evolution of some of the groups. This has motivated \nour search for a more adequate merging criterion. \n\nWe searched for a formula of a form similar to the one proposed by \nAarseth and Fall (1980), namely:\n\\begin{equation}\n{\\left[ \\frac{(m_1+m_2)r_p}{{a(m_1\\epsilon _1+m_2\\epsilon _2)}} \n\\right]}^2+{\\left[\\frac{v(r_p)}{{bv_e(r_p)}}\\right] }^2\\leq 1. \n\\end{equation}\nFor the part concerning the velocities, we keep the same expression as in \nthe Aarseth and Fall formula, which performs quite well in the case of \nthe time evolution of the three dimensional velocity dispersions. For the \npart concerning the cores of the galaxies and the separation at \npericenter we use a mass weighted expression with the aim of taking into \naccount possible differences in collisions between galaxies of different \nmasses as in the expression due to Richstone and Malumuth (1983). The \nconstants $a$ and $b$ are free parameters and will be determined using \nthe self-consistent simulations as a reference. This expression can be \nviewed as the equation of the points within an ellipse centered at the \norigin in the plane defined by $(m_1+m_2)r_p\/ (m_1\\epsilon_1+ m_2\\epsilon \n_2)$ and $v(r_p)\/v_e(r_p)$. Then $a$ and $b$ are the semimajor axes of \nthis ellipse. Increasing the value of $a$ means increasing the axis of \nthe ellipse corresponding to the relative separation at pericenter and \nthus allowing mergings in more distant collisions. On the other hand, \nif we increase the value of $b$ we allow merging in faster \ncollisions. With this in mind, we fitted the values of $a$ and $b$ to \nthe self-consistent simulations using as the basis for our exploration \nthe values used by Aarseth and Fall (1980). After some trials and \ncomparisons with the self-consistent simulations we obtained the \nfollowing merging criterion: \\begin{equation}\n{\\left[ \\frac{(m_1+m_2)r_p}{{2.5(m_1\\epsilon _1+m_2\\epsilon _2)}} \\right] \n}^2+\n{\\left[\\frac{v(r_p)}{{1.18v_e(r_p)}}\\right] }^2\\leq 1. \\end{equation}\nThe effect of this new criterion is shown in Figs.~7, Fig.~8 and Fig.~9, \nwhere we compare the time evolution of the global parameters of the \nself-consistent simulations with that of the explicit simulations using \nthe AF and RM criteria and our new one. The dynamical friction with the \nmost massive galaxy is also included in these cases. \n\nIn Fig.~7 we show the time evolution of the number of galaxies $N_g$. In \nthe first column, we repeat the comparison between the self-consistent \nsimulations\nand the explicit simulations with the AF and RM criteria and dynamical \nfriction. In the second column, we have the comparison between the \nself-consistent simulations and the explicit simulations with dynamical \nfriction and our new merging criterion. As can be seen, while the \nexplicit simulations with the RM criterion mimic quite well the \nself-consistent simulations, this is not true for the AF condition. On \nthe other hand, our new criteria follows quite well the evolution of the \nnumber of galaxies given by the self-consistent simulations for all \ninitial conditions.\n\nIn Fig.~8 we show the time evolution of the half mass radius. For the \ncase of Run A both AF and RM conditions follow quite well the \nself-consistent simulations until the point of maximum collapse. After \nthis point, the half mass radius given by these explicit simulations \nfalls below the\nself-consistent case. Our new condition, however, follows the \nself-consistent simulations with $900$ particles very well. For the case \nof Run B, the AF and RM conditions end below the self-consistent case. \nOur new criterion performs better, following the self-consistent \nsimulations, but with some oscillations. For runs C and D we can say that \nall three criteria give similar results. \n\nFigure~9 which gives the time evolution of the three dimensional velocity\n\ndispersion, is the most interesting one. We have seen that the AF and RM \nconditions give good results for the case of the collapsing group (Run A) \nand this is true also for our new criterion. However, the AF and RM \nexplicit simulations do not work well for the case of a virialised group \n(Run B). The AF condition ends with a higher velocity dispersion and the \nRM with a smaller velocity disperson compared to the self-consistent \ncase; on the other hand, our new criterion performs much better than \neither. This is specially true for the most difficult case, Run D, the \nvirialised and tightly bound group. In this case our new criterion \nperforms much better than the AF and RM criteria.\n\n\\section{Simulations with a dark matter halo encompassing the whole \ngroup}\n\nSeveral observations suggest that clusters and groups of galaxies \nmay contain much matter not bound to the galaxies. This led us to run a\nself-consistent simulation (Run H), where part of the mass of the system \nis distributed in a background. In the corresponding explicit\nsimulations the background is included as a rigid Plummer potential with \nthe same parameters as the live background in the initial conditions of \nthe self-consistent\nsimulation. The explicit simulations include dynamical friction with the \nmost massive galaxy and with the Plummer halo. \n\nThe evolution of the group leads to a system where the central part of \nthe galaxy distribution has contracted, while the outer one has expanded. \nThis results in an increase of the half-mass radius and a lowering of the \nvelocity dispersion, as shown in Fig.~10. The upper panels give the time \nevolution of the number of galaxies in the system $N_g$, the middle ones \nthat of the half mass radius $R_h$ and the lower ones that of the three \ndimensional velocity dispersion. In the left panels the self-consistent \nsimulations are compared to\nthe explicit simulations with the AF and RM conditions and in the right \npanels with simulations using our new criterion. As we can see, the \nnumber of\ngalaxies diminishes slower in simulations including a common halo than in \nthe case of virialised\nsimulations with no distributed dark matter. The AF and RM conditions \nunderestimate the real number of mergers, and so, though to a lesser \nextent, does our new criterion.\nFor the time evolution of the half mass radius there are strong \ndiscrepancies between the self-consistent simulations and the explicit \nsimulations using any of the merging criteria including the new \ncriterion proposed in the previous section. \n\nThe three dimensional velocity dispersion of the galaxies is well \ndescribed by the explicit simulations using any of the merging criteria. \nThis global parameter systematically decreases during the simulation as \nthe galaxies that move faster near the center disappear and form the \ngiant central object. The slope of this evolution flattens off toward the \nend of the simulations. This behaviour is not well followed by the \nexplicit simulations using the AF or RM criterion. On the other hand, \nFig.~10 shows that our new merging criterion is able to reproduce these \nminor details better.\n\n\\section{Summary.}\n\nIn this paper we compared self-consistent simulations of galaxy groups \nwith simulations where the physics of the interactions is modelled by \nmerger rules. We used two sets of self-consistent simulations, one in \nwhich the galaxies were modelled with 900 points and the other with 100 \npoints. Insofar as the global dynamical\nparameters are concerned, the evolution of galaxy groups is similar in \nthose two cases. This shows that simulations with a relatively low number \nof particles can be used to follow the evolution of global dynamical \nproperties of groups or clusters. However, from the work of van Kampen \n(1995)\nit can be inferred that using lower that 100 points per galaxy can be \ndangerous.\n\nAs far as the explicit simulations are concerned, we show that the \nconditions used in the literature to\nsimulate the merging between galaxies are of unequal quality. Of these \nconditions, in the case\nwhere there is neither dynamical friction nor tidal forces, the best are \nthose of Tremaine (1980), modified for the case of different masses by \nRichstone and Malumuth (1983), and the one by Aarseth and Fall (1980). \nWhen we include dynamical friction effects the AF condition predicts too \nmany mergers but still maintains good predictions for the rest of the \nglobal parameters. The condition proposed by Richstone and Malumuth \n(1980) does better as far as the number of galaxies and $R_h$ are \nconcerned, but considerably worse for the velocity dispersion. \n\nAs none of these criteria seems to be a good guide for the time evolution \nof the groups as compared with the self-consistent simulations, we have \nfitted a new\ncriterion to the results of self-consistent simulations. This new \ncriterion is:\n\\begin{equation}\n{\\left[ \\frac{(m_1+m_2)r_p}{{2.5(m_1\\epsilon _1+m_2\\epsilon _2)}} \\right] \n}^2+\n{\\left[\\frac{v(r_p)}{{1.18v_e(r_p)}}\\right] }^2\\leq 1,\n \\end{equation}\nand is inspired in the expressions given by Aarseth and Fall (1980) and \nRichstone and Malumuth (1980). This new criterion mimics relatively well \nthe time evolution of the global parameters of the groups in as wide a\nvariety of situations as those presented by our simulations A to D. \nHowever it performed not so well in case H which has a common halo, but \nthis can be explained by the different nature of the simulations \nimplying that even this new criterion has only a limited range of \napplicability.\n\nOur comparisons show that some of the older results on the dynamics \nof groups and clusters of galaxies should be viewed with caution. For \ninstance, Roos (1981) studied the evolution of expanding systems of \ngalaxies to simulate the evolving universe. As he used the RN criterion \nin his simulation the predicted merger rate can be too high. In the same \nway, when Roos and Aarseth (1982) used this criterion to study the \nevolution of the luminosity function of a cluster of galaxies, their \nfinal luminosity functions can be artificialy peaked towards\nhigh luminositues. Similarly, Valtonen et al. (1984), Saarinen and \nValtonen (1985) and Perea et al. (1990) use explicit simulations to \ncriticize the virial mass obtained for galaxy clusters. We have, however, \nseen that this kind of simulation is biased toward higher velocity \ndispersions. Finally, the explicit simulations on compact groups by Mamon \n(1987) using a diffuse intergalactic background may also be biased.\n\nThus we can conclude that there is no ideal substitute for fully \nself-consistent N-body simulations. However, in cases when one needs to \nlook only at global quantities describing the system and is not \ninterested in fine structure and details, a first exploration of \nparameter space can be done using explicit simulations and the criterion \nproposed in this paper. This performs particularly well in cases where \nthe group has no common halo.\n\n{\\bf Acknowledgements.}\nWe thank Albert Bosma and Kevin Prendergast for reading and improving the \nmanuscript, and our referee, Joshua Barnes for his useful \nsuggestions and criticism which improved the quality of this paper. We \nalso thank L. Hernquist for making available to us his vectorised version \nof the treecode.\nSome of the simulations discussed in this paper were made at the C98 of \nthe IDRIS (Institut du d\\'eveloppement et des ressources en informatique \nscientifique, Orsay, France). \n\n\\noindent\n{\\Large \\bf References.}\n\n\\noindent\nAarseth, S.J. 1971 ApSS 14,20\\\\\nAarseth, S.J., Fall, S.M. 1980 ApJ 236,43\\\\ Barnes, J. 1992 in {\\sl \nMorphological and physical classification of galaxies} G. Longo et \nal.\\linebreak \\hspace*{0.5cm} (eds.) 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S.M. Fall et D. \n\\linebreak \\hspace*{0.5cm} Lynden-Bell. Cambridge University Press\\\\ van \nAlbada, T.S., van Gorkom, J.H. 1977 A\\&A 54,121\\\\ van Kampen, E. 1995 \nMNRAS 273,295\\\\\nValtonen, M.J., Innanen, K.A., Huang, T.-Y., Saarinen, S. 1985 A\\&A \n143,182\\\\ White, S.D.M. 1978 MNRAS 184,185\\\\\n\n\\newpage\n\n\\noindent\n{\\Large \\bf Figure Captions.}\n\n\\noindent {\\bf Fig.~1} Comparison of the time evolution of the number of \ngalaxies in the self-consistent simulations with $100$ particles per \ngalaxy (thin line) and $900$ particles per galaxy (thick line) with the \nexplicit simulations for the same initial conditions and without \ndynamical friction. In the left panels we use the AF and RM merging \nconditions and in the right panels we use the FS and RN ones. The initial \nconditions of each simulation are described in Table~1.\n\n\\noindent {\\bf Fig.~2} Comparison of the time evolution of the half mass \nradius of the system in self-consistent simulations and in explicit \nsimulations without dynamical friction for the same initial conditions. \nThe symbols are as in Fig.~1. \n\n\\noindent {\\bf Fig.~3} Time evolution of the three dimensional velocity \ndispersion of the galaxies consi\\-dered as point masses for the \nself-consistent and the explicit simulations. The symbols are as in \nFig.~1.\n\n\\noindent {\\bf Fig.~4} Time evolution of the number of galaxies $(N_g)$ \nin the self-consistent simulations compared with the evolution of this \nnumber in the explicit simulations with dynamical friction included. The \nthick lines correspond to the self-consistent simulations with $900$ \nparticles per galaxy and the thin lines to the simulations with $100$ \npoints per galaxy. In the first column, we show the comparison with the \nexplicit simulations using the AF criterion and using the RM criterion. \nIn the second column, we show the same comparisons with the explicit \nsimulations using the FS condition and using the RN condition. \n\n\\noindent {\\bf Fig.~5} Same as for Fig.~4 but for the time evolution of \nthe half mass radius of the system.\n\n\\noindent {\\bf Fig.~6} Same as for Fig.~5 but for the time evolution of \nthe three dimensional velocity dispersion. \n\n\\noindent {\\bf Fig.~7} Comparison of the explicit simulations using the \nAF and the RM criteria with the explicit simulations using the new \ncriterion. The performance of each criterion is compared with the \nself-consistent simulations. In the first column, we show the time \nevolution of the number of galaxies in the self-consistent simulations \nwith $900$ particles per galaxy (thick lines) and with $100$ particles \nper galaxy (thin lines) compared with the explicit\nsimulations using the AF criterion and using the RM criterion. In the \nsecond column, we compare the time evolution of $N_g$ for the \nself-consistent simulations with the results of the explicit simulations \nusing the new criterion. In all cases we include dynamical friction.\n\n\\noindent {\\bf Fig.~8} Same as Fig.~7 but for the time evolution of the \nhalf mass radius of the system.\n\n\\noindent {\\bf Fig.~9} Same as Fig.~7 but for the time evolution of the \nthree dimensional velocity dispersion. \n\n\\noindent {\\bf Fig.~10} Time evolution of the global parameters of the \nsimulations with distributed background. In both columns we show the \nevolution of $N_g$, $R_h$ and $\\sigma (3D)$ for the self-consistent \nsimulations with $450$ particles per galaxy (thick lines) and for the \nself-consistent simulations with $100$ particles per galaxy (thin lines). \nIn the left panel these are compared with the explicit simulations with \nthe AF condition and with the RM condition. In the right panel the \nself-consistent simulations are compared with the explicit simulations \nwith our new criterion. \n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nNowadays, advances in digital music production technology enabled the musicians to explore a greater range of sonic possibilities to work with.\nParticularly, the development of the Digital Audio Workstation (DAW) and virtual instruments greatly expanded the space of the musical creativity~\\cite{walzer2017independent}.\nAs there are a large number of virtual instruments with high timbral diversity and the quality of music is highly dependent on the timbre of the instruments, selecting appropriate musical instruments plays a crucial role in digital music production. \n\nTypical ways to retrieve proper musical instruments from a large library of instruments are listening to the audio samples of the instruments one-by-one or referring to the text description of the instruments if available.\nHowever, listening to the audio samples is time-consuming and inefficient, and the descriptions are often unavailable or insufficient to express the subtle nuance of the timbre of the musical instruments~\\cite{knees2015giantsteps}.\n\n\n\n\\begin{figure}[tb]\n\\setlength{\\belowcaptionskip}{-7pt}\n\\begin{minipage}[b]{1.0\\linewidth}\n \\centering\n \\centerline{\\includegraphics[width=8.5cm]{Figure1.png}}\n\\end{minipage}\n\\caption{Comparison between musical instrument recognition and retrieval task.}\n\\label{fig:task}\n\\end{figure}\n\n\nWe call this task of retrieving specific desired instruments from the library of musical instruments as \\textit{Musical Instrument Retrieval}.\nSince musicians often refer to existing music to describe the sound they want, we propose to use reference music as a query for musical instrument retrieval.\nIn this task, given a mixture audio query, the model has to retrieve the instruments that most closely resemble the instrument used in the mixture audio query.\nIn our experiment, for quantitative evaluation, the instrument used for mixture audio query was always included in the library.\nWe evaluated whether the model retrieved the exact instruments used in mixture audio query in terms of F1 score and mean Average Precision (mAP). \n\nMusical instrument recognition is a closely related task that has been actively studied in the field of music information retrieval~\\cite{han2016deep,avramidis2021deep,kratimenos2021augmentation,li2015automatic,lostanlen2016deep,cheuk2022jointist}.\nHowever, existing methods of musical instrument recognition rule out the unique character of each instrument and only predicts the coarse categories of the instrument so that it cannot be directly used for fine-grained musical instrument retrieval task.\nComparison between the two tasks is illustrated in Fig.~\\ref{fig:task}.\n\n\\begin{figure*}\n \\centering\n \\begin{tabular}{cc}\n \\begin{subfigure}{\\columnwidth}\n \\centering\n \\caption{Training Single-Instrument Encoder.}\n \\label{fig:single_enc}\n \\vspace*{0.25cm}\n \\includegraphics[width=\\columnwidth]{Figure2a.png}\n \\end{subfigure}\n &\n \\\\\n \\hfill\n \\vspace*{0.1cm}\n \\begin{subfigure}{\\columnwidth}\n \\centering\n \\caption{Training Multi-Instrument Encoder.}\n \\label{fig:multi_enc}\n \\vspace*{0.25cm}\n \\includegraphics[width=\\columnwidth]{Figure2b.png}\n \\end{subfigure}\n &\n \\multirow[t]{2}[0]{*}[0.7cm]{\n \\begin{subfigure}{\\columnwidth}\n \\centering\n \\caption{Retrieving similar instruments from the library using proposed method.}\n \\label{fig:single_enc}\n \\vspace*{0.25cm}\n \\includegraphics[width=\\columnwidth]{Figure2c.png}\n \\end{subfigure}\n }\n \\end{tabular}\n \\vspace*{0.1cm}\n \\caption{The overall process of the suggested method. (a) Single-Instrument Encoder is trained to classify which instrument played the input audio. We take the penultimate layer's activation of the trained network as instrument embedding. (b) Multi-Instrument Encoder extracts multiple instrument embeddings from the mixture audio. The Single-Instrument Encoder provides the set of target embeddings. (c) At inference time, we first extract the instrument embeddings of each instrument in the instrument library for a single time. Then we extract the multiple embeddings from the mixture audio query and retrieve the most similar instruments from the instrument library.}\n \\label{fig:model}\n\\end{figure*}\n\n\n\nOur proposed method employs the Single-Instrument Encoder and the Multi-Instrument Encoder.\nThe Single-Instrument Encoder extracts the instrument embedding from single-track audio.\nUsing the embeddings extracted by the Single-Instrument Encoder as the target, the Multi-Instrument Encoder is trained to extract multiple numbers of instrument embeddings from the mixture audio.\nSince we estimate the set of embeddings, which is permutation-invariant, we use permutation invariant training (PIT)~\\cite{yu2017permutation} scheme for Multi-Instrument Encoder training.\n\n\nTraining and evaluating a general instrument encoder requires a dataset consisting of a large number of different instruments.\nAt the same time, the dataset should contain ensembles of different instruments to enable the model to extract embeddings robustly to instrument combinations and performance.\nTo meet these conditions, we propose a new dataset called \\textit{Nlakh} (pronounced as en-l\u00e4k), which is a combination of the NSynth dataset~\\cite{nsynth2017} and the Lakh dataset~\\cite{raffel2016learning,lakh}.\nBased on the implementation of ~\\cite{Martel_NSynth-MIDI-Renderer_2019}, we rendered the MIDI files in the Lakh dataset by using note samples in the NSynth dataset.\n\nExperimental results show that the Single-Instrument Encoder successfully maps the different audio samples of the same instruments into close embeddings. \nResults also show that the Multi-Instrument Encoder is able to separate the mixture audio at the embedding level and retrieve desired instruments from the library successfully.\n\n\n\\section{Related Works}\n\\label{sec:relatedworks}\n\nStudies on retrieving musical instruments or extracting instrument embeddings are still in its early stages. \nRecently,~\\cite{shi2022use} has trained and evaluated a model for extracting instrument embedding from a music signal by adopting the framework of speaker verification task, but the model was limited to extracting an embedding from single-sourced audio.\nMusical instrument retrieval methods with audio query have also been studied recently, but mostly focusing on retrieving drum samples. \n~\\cite{mehrabi2018similarity} adopts deep convolutional auto-encoders to retrieve drum samples by using vocal imitations as the query.\nFurthermore, ~\\cite{kim2020drum} conducts deep metric learning to extract the drum embeddings from a mixture audio as the query.\nIn this paper, we expand this approach for retrieving multi-pitched instruments. \n\n\n\\section{Method}\n\\label{sec:pagestyle}\n\n\nThe proposed model consists of the Single-Instrument Encoder and the Multi-Instrument Encoder.\nThe Single-Instrument Encoder extracts an instrument embedding from a single-track audio of the instrument.\nUsing the instrument embeddings computed by the Single-Instrument Encoder as a set of target embeddings, the Multi-Instrument Encoder is trained to estimate the multiple instrument embeddings.\nAs we estimate the set of embeddings, which is permutation-invariant, PIT scheme~\\cite{yu2017permutation} was used for training.\nThe overall framework of the proposed model is depicted in Fig.~\\ref{fig:model}.\n\n\\subsection{Single-Instrument Encoder}\n\\label{ssec:subhead_f}\n\nIn order to extract an instrument embedding from single-track audio with the Single-Instrument Encoder, we trained a network performing classification to match the audio samples with their instrument labels.\nWe used the network's penultimate layer's activation as the instrument embedding, which is a 1024-dimensional vector.\nFor an instrument $i_k$, the Single-Instrument Encoder $f$ extracts the embedding of the instrument $i_k$ as $f(x_{i_k})$, where $x_{i_k}$ is the single-track audio of the instrument $i_k$.\n\n\\subsection{Multi-Instrument Encoder}\n\\label{ssec:subhead}\n\n\nThe Multi-Instrument Encoder $g$ aims to estimate the embeddings of a set of instruments $I=\\{{i_1, i_2,...,i_N}\\}$ given a mixture audio $m=\\sum_{i\\in I} x_{i}$. \nThe target embeddings are the outputs of the Single-Instrument Encoder.\nWe designed the Multi-Instrument Encoder to output $M$ possible embeddings, where $M$ was set as the maximum number of instruments in a mixture audio in the training set.\n\nThe Multi-Instrument Encoder $g$ is trained to minimize the cosine embedding loss between the optimal permutation of the set of output embeddings $G=\\{ g(m)_{1,:}, g(m)_{2,:}, ..., g(m)_{M,:}\\}$ and the set of target embeddings $F=\\{ f(x_1), f(x_2), ..., f(x_N)\\}$.\nTo compensate for the difference in the number of embeddings and the indeterminacy of the instrument order, we used the idea of permutation invariant training to compute the loss function~\\cite{yu2017permutation}.\nThe minimized loss function is described as follows:\n\\begin{align*}\n \\mathcal{L} &= \\min\\limits_{\\pi} \\sum \\limits_{n=1}^N \\big( 1 - \\cos\\theta_{\\pi(n),n} \\big) \\\\\n \\cos \\theta_{\\pi(n),n} &= \\frac{g(m)_{{\\pi(n)},:} \\cdot f(x_n)}{||g(m)_{{\\pi(n)},:}|| \\cdot ||f(x_n)||}\n\\end{align*}\nwhere $\\pi:\\{1,2,\\dots,N\\} \\mapsto \\{1,2,\\dots,M\\}$ is an injective function.\n\nTo minimize the computational cost of finding the optimal permutation, we applied the optimal permutation invariant training method that utilizes the Hungarian algorithm~\\cite{dovrat2021many, kuhn1955hungarian}.\n\n\\subsection{Inference}\n\\label{section:inference}\nTo use the trained encoders for retrieval task, for each instrument $l_k$ in instrument library $L = \\{ l_1, l_2, l_3, ..., l_K\\}$, we extract the instrument embedding $f(x_{l_k})$ to construct the embedding library $E = \\{f(x_{l_1}), f(x_{l_2}), ..., f(x_{l_K}) \\} $ using the trained Single-Instrument Encoder.\nGiven the mixture audio query $m$, we extract output embeddings $\\{g(m)_{1,:}, ..., g(m)_{M,:}\\}$ using the trained Multi-Instrument Encoder.\nThen we calculate the cosine similarity $\\cos \\phi_{j,k}$ as follows.\n\\begin{align*}\n \\cos \\phi_{j,k} &= \\frac{g(m)_{j,:} \\cdot f(x_{l_k})}{||g(m)_{j,:}|| \\cdot ||f(x_{l_k})||}\n\\end{align*}\n\nFor each output embedding $g(m)_{j,:}$, we pick the instrument $l_k$ whose cosine similarity $\\cos \\phi_{j,k}$ is the largest among other instruments in $L$.\nTherefore, the set of retrieved instruments $R$ given mixture audio query $m$ can be formulated as follows.\n\\begin{align*}\n R &= \\{ l_{k'} | k' \\in \\{\\operatorname*{argmax}_k \\cos \\phi_{j,k}\\}_{j=1}^{M} \\}\n\\end{align*}\nNote that more than two output embeddings may be assigned to the same instrument.\nTherefore, the size of a set $R$ may be smaller than $M$.\n\n\n\\begin{figure}[tb]\n \\centering\n \\begin{subfigure}[b]{\\columnwidth}\n \\centering\n \\includegraphics[width=\\columnwidth]{Figure3a.png}\n \\caption{}\n \\label{fig:data1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{\\columnwidth}\n \\centering\n \\includegraphics[width=\\columnwidth]{Figure3b.png}\n \\caption{}\n \\label{fig:data2}\n \\end{subfigure}\n \\caption{The process of rendering a sample of (a) Nlakh-single and (b) Nlakh-multi }\n \\label{fig:data}\n\\end{figure}\n\n\\begin{table}[tb]\n\\centering\n\\hfill\n\\resizebox{\\columnwidth}{!}{\n\\begin{tabular}{lccc}\n\\Xhline{2\\arrayrulewidth}\n\\textbf{Dataset} & \\begin{tabular}[x]{@{}c@{}} Size \\\\ (Hours) \\end{tabular} & \\begin{tabular}[x]{@{}c@{}} Number of \\\\ Instruments \\\\ (Categories) \\end{tabular} & \\begin{tabular}[x]{@{}c@{}} Stem \\\\ Availability\\end{tabular} \\\\ \\hline \nNlakh-single (ours) & 1,397 & 1,006 & $\\checkmark$ \\\\\nNlakh-multi (ours) & 153 & 1,006 & $\\checkmark$ \\\\ \\hline\nSlakh~\\cite{manilow2019cutting} & 145 & 158 & $\\checkmark$ \\\\\nMUSDB18~\\cite{rafii2017musdb18} & 10 & (5) & $\\checkmark$ \\\\\nMedleyDB~\\cite{bittner2014medleydb} & 7 & (80) & $\\checkmark$ \\\\\nOpenMIC~\\cite{humphrey2018openmic} & 56 & (20) & - \\\\\nIRMAS~\\cite{bosch2012comparison} & 6 & (11) & - \\\\\n\\Xhline{2\\arrayrulewidth}\n\\end{tabular}\n}\n\\caption{Comparison with other datasets.}\n\\label{tab:dataset}\n\\end{table}\n\n\\section{The Nlakh Dataset}\n\\label{ssec:subhead}\nTo train and evaluate the proposed model, the dataset should have a large number of different instruments. \nAlso, the dataset should contain the ensembles of different instruments to enable the model to extract instrument embeddings robustly to instrument combinations and performance. \nHowever, no existing dataset fully met these requirements. Therefore, we propose a new dataset called \\textit{Nlakh} that combines the NSynth dataset, which provides a large number of instruments, and the Lakh dataset, which provides multi-track MIDI data.\n\nNlakh consists of \\textit{Nlakh-single} that contains single-track audio and \\textit{Nlakh-multi} that contains mixture audio with separate tracks (stem) of each instrument.\nTo make Nlakh-single, we first separated each MIDI track of the Lakh dataset and categorized the tracks by their instrument family (bass, organ, guitar, etc.) according to the MIDI program number.\nThen for each instrument of NSynth, we randomly selected a five-second-long excerpt from MIDI tracks in the corresponding instrument family.\nFor example, if the selected instrument's family is the guitar, only the MIDI files in the guitar category are used for rendering.\nWe rendered 1,000 samples for each instrument. \nIn total, there are 1,006,000 samples in Nlakh-single.\nNlakh-single is split into train\/valid set following the instrument split of NSynth (953\/53).\n\n\n\nTo make Nlakh-multi, we first find a five-second-long multi-track MIDI section containing at least two valid tracks in which at least three notes are played.\nLikewise in Nlakh-single, we randomly selected instruments for rendering the multi-track MIDI excerpt within the corresponding instrument family.\nThe Nlakh-multi has 100,000 samples for the training set and 10,000 samples for the validation set.\nThe overall process of making the dataset is illustrated in Fig.~\\ref{fig:data}.\n\nAmong other multi-track music datasets that contains audio data, to the best of our knowledge, Nlakh has the largest number of instruments and the largest amount of data at the same time (Table~\\ref{tab:dataset}).\nIn addition to the rendered audio dataset, we also provide a codebase to generate our dataset, so one can use it to render more samples.\n\n\n\n\\begin{table*}[t]\n\\centering\n\\parbox{11cm}{\\caption{Performance of the Multi-Instrument Encoder. Small\/Large indicates the size of the model. Nlakh\/Random indicates which dataset is used for training.}\n\\label{tab:g_eval}\n}\n\\begin{tabular}{@{\\extracolsep{4pt}}lcccccc@{}}\n\\Xhline{2\\arrayrulewidth}\n\\multirow{3}{*}{\\textbf{Model}} & \\multicolumn{2}{c}{\\textbf{Family}} & \\multicolumn{4}{c}{\\textbf{Instrument}} \\\\\n\\cline{2-3} \\cline{4-7}\n& \\multicolumn{2}{c}{\\textbf{F1}} & \\multicolumn{2}{c}{\\textbf{F1}} & \\multicolumn{2}{c}{\\textbf{mAP}} \\\\ \\cline{2-3} \\cline{4-5} \\cline{6-7}\n& macro & weighted & macro & weighted & macro & weighted \\\\ \\hline\nChance & 0.343 & 0.437 & 0.065 & 0.077 & - & -\\\\ \\hline\nSmall-Nlakh & 0.626 & 0.723 & 0.482 & 0.524 & 0.553 & 0.597 \\\\\nLarge-Nlakh & 0.640 & 0.728 & 0.533 & 0.578 & 0.635 & 0.666 \\\\ \nSmall-Random & 0.691 & 0.697 & 0.528 & 0.543 & 0.598 & 0.615 \\\\ \nLarge-Random & \\textbf{0.814} & \\textbf{0.817} & \\textbf{0.694} & \\textbf{0.712} & \\textbf{0.752} & \\textbf{0.760} \\\\ \n\n\\Xhline{2\\arrayrulewidth}\n\\end{tabular}\n\\end{table*}\n\n\\section{Experiments}\n\n\n\\subsection{Single-Instrument Encoder}\n\nWe used the convolutional neural network architecture that was used in~\\cite{han2016deep} for the instrument recognition task as the backbone network of the Single-Instrument Encoder, using mel-spectrogram of the audio as the input.\nWe used Adam optimizer with a learning rate of 0.001, and set batch size as 32.\n\nTo evaluate the Single-Instrument Encoder, we adopted the method proposed by ~\\cite{shi2022use}, which used automatic speaker verification evaluation methodologies for evaluating the instrument embeddings.\nWe first extract the embeddings of five different samples of the target instrument by using the trained Single-Instrument Encoder. \nThe average of those embeddings is used as enrollment embedding.\nWe also make a comparison set that contains 20 embeddings from the target instrument and 20 embeddings from the other instruments.\nThen we compare each embedding in the comparison set with the enrollment embedding in terms of cosine similarity.\nVerifying whether the embeddings in the comparison set correspond to the enrollment embedding or not, we compute the false reject rate and false accept rate for each instrument.\nWe computed the average value of equal error rate (EER), which describes the point where the false reject rate and false accept rate are equal.\n\nThe average EER of our model on Nlakh-single was 0.026 while the previous work's EER on the NSynth dataset was 0.031.\nNote that the samples of the NSynth dataset contain only a single note, while the samples of Nlakh-single contain multiple notes.\nWe also visualized the instrument embeddings of training set and validation set using t-distributed stochastic neighbor embedding (t-SNE)~\\cite{van2008visualizing} in Figure~\\ref{fig:tsne_single}.\nThe results show that the Single-Instrument Encoder could cluster the instrument embeddings robustly even for the unseen instruments in the validation set.\n\n\\begin{figure}[t]\n \\centering\n \\begin{subfigure}[b]{0.49\\columnwidth}\n \\centering\n \\includegraphics[width=\\columnwidth]{Figure4a.png}\n \\caption{}\n \\label{fig:data1}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.49\\columnwidth}\n \\centering\n \\includegraphics[width=\\columnwidth]{Figure4b.png}\n \\caption{}\n \\label{fig:data2}\n \\end{subfigure}\n \\caption{The t-SNE results of Single-Instrument Encoder on Nlakh-single (a) training and (b) validation dataset.}\n \\label{fig:tsne_single}\n\\end{figure}\n\n\n\\subsection{Multi-Instrument Encoder}\n\nThe Multi-Instrument Encoder is trained to extract the embedding of each instrument in the mixture audio.\nIn this experiment, the Multi-Instrument Encoder extracts nine embeddings, which is the maximum number of instruments composing a single mixture in Nlakh-multi. \nFor the network architecture of the Multi-Instrument Encoder, we tried two different network architectures.\nThe first is the same architecture~\\cite{han2016deep} as the Single-Instrument Encoder.\nWe also tried a larger convolutional neural network~\\cite{liu2022convnet} since the task for the Multi-Instrument Encoder is more difficult than that of the Single-Instrument Encoder.\nWe used Adam optimizer with a learning rate of 0.001 and set batch size as 128 for all cases.\n\nDuring the experiment, we noticed an imbalance of the instrument distribution in Nlakh-multi, which may harm the performance of the trained network.\nTo solve this issue, we also trained the network with randomly-mixed audio.\nWe randomly selected a number of musical instruments between two and nine, and then randomly picked the audio samples of selected instruments from Nlakh-single.\nThose samples were used to mix the randomly-mixed audio.\nRather than rendering the finite set of samples for the randomly-mixed dataset, we mixed the audio on-the-fly during training.\n\nGiven mixture audio query, we retrieved the instruments as described in Section~\\ref{section:inference} and computed F1 score.\nWe also calculated the F1 score with the instrument family as the basis of the evaluation.\nThe instrument family is a coarsely categorized class of instruments, which is predefined in NSynth dataset.\nTo calculate the mean Average Precision (mAP), we used the highest cosine similarity between the output embeddings and each embeddings in the embedding library as the similarity score.\n\nTable \\ref{tab:g_eval} shows the evaluation results of the Multi-Instrument Encoder. We had three main observations from the evaluation.\nFirst, every trained network performed significantly better than the chance level in all measurements.\nSecond, the network trained with randomly-mixed audio showed less overfitting than the network trained with Nlakh-multi.\nThird, the network using the larger convolutional neural network showed better performance.\nThe larger convolutional neural network learns more general information and therefore can better handle the extraction of the embedding from an input mixture audio.\n\n\\section{Conclusion}\nIn this work, we proposed a novel method for musical instrument retrieval that employs the Single-Instrument Encoder and the Multi-Instrument Encoder to extract the instrument embeddings.\nTo train and evaluate the proposed model, we suggested the Nlakh dataset, which contains single-track audio and mixture audio from a large number of different musical instruments.\nThe evaluation result showed that the Single-Instrument Encoder was able to learn the mapping from the audio signal of unseen instruments to the instrument embedding space, and the Multi-Instrument Encoder was able to extract multiple embeddings from the mixture audio and retrieve the desired instruments successfully.\nIn the future, we plan to improve the robustness of our method by elaborating our dataset with appliance of various audio effects and expansion of the instrument classes.\n\n\n\n\n\n\n\n\\vfill\\pagebreak\n\\bibliographystyle{IEEEbib}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} Extremal K\\\"ahler metrics were first introduced and studied by E.~Calabi in\n\\cite{cal,cal-2}. Let $X$ denote a connected compact complex manifold of complex dimension $n$. A K\\\"ahler metric $g$ on $X$, with\nK\\\"ahler form $\\omega_g$, is {\\it extremal} if it is a\ncritical point of the functional $g \\mapsto \\int _X s _g ^2 \\, \\frac{\\omega\n_g ^n}{n!}$, where $g$ runs over the set of all K\\\"ahler metrics on $X$\nwithin a fixed K\\\"ahler class $\\Omega = [\\omega]$, and $s_g$ denotes the\nscalar curvature of $g$. As shown in \\cite{cal}, $g$ is extremal if and\nonly if the symplectic gradient $K := {\\rm grad} _{\\omega} s _g = J \\, {\\rm\ngrad} _g s _g$ of $s _g$ is a Killing vector field (i.e. $\\mathcal{L} _K g\n= 0$) or, equivalently, a (real) holomorphic vector field. Extremal K\\\"ahler metrics include K\\\"ahler\nmetrics of constant scalar curvature --- CSC K\\\"ahler metrics for short ---\nand in particular K\\\"ahler--Einstein metrics. Clearly, if the identity component ${\\rm Aut} _0 (X)$ of the automorphism group of $X$ is\nreduced to $\\{1\\}$, i.e. if $X$ has no non-trivial holomorphic vector fields, any extremal K\\\"ahler metric is CSC, whereas a CSC K\\\"ahler metric is K\\\"ahler--Einstein if and only if $\\Omega$ is a multiple of the (real) first Chern class $c _1 (X)$. \n\nIt is conceivable to think about an extremal K\\\"ahler metric $g$ in $\\Omega$ as a {\\it canonical} representative of the K\\\"ahler metrics in the K\\\"ahler class $\\Omega$. One would then expect that the extremal K\\\"ahler metrics in $\\Omega$ reflect most of the holomorphic invariants of the pair $(X, \\Omega)$. In this vein, the goal of this note is to discuss how that the following natural splitting problem fits in with some recent progress in the field.\n\n\\begin{prob}Let $X_i,$ be compact projective manifolds polarized by ample holomorphic line bundles $L_i$ and $X= \\prod_{i=1}^r X_i$ their product endowed with the polarisation $L= \\bigotimes_{i=1}^r L_i$, where $L_i$ is seen as a holomorphic line bundle over $X$ via the natural pull-back. Does any extremal K\\\"ahler metric $g$ in the K\\\"ahler class $\\Omega = 2\\pi c_1(L)$ on $X$ is the Riemannian product of extremal K\\\"ahler metrics $g_i$ in the K\\\"ahler classes $\\Omega_i=2\\pi c_1(L_i)$ on the factors $X_i$?\\end{prob}\n\n\nSeveral remarks are in order.\n\n\\smallskip\nFirst of all, it is well-known (see e.g. \\cite[Thm.~2.1]{yau}) that the answer is positive if we suppose that $g$ is a K\\\"ahler--Einstein metric on $X$. It then follows from a standard Bochner argument (see e.g. \\cite{gauduchon-0,kob}) for the holomorphic projectors $P_j : TX = \\bigoplus_{i=1}^r TX_i\\to TX_i$, where $TX$ (resp. $TX_i$) denotes the holomorphic tangent bundle of $X$ (resp. $X_i$). This is the case when each $X_i$ is either a Calabi--Yau manifold (i.e. $c_1(X_i)=0$) or has ample canonical line bundle $K_{X_i}$ and $L_i= K_{X_i}$, or is a Fano manifold with vanishing Futaki invariant and $L_i= K^{-1}_{X_i}$.\n\n\n\\smallskip\nSecond, it is now known that the extremal K\\\"ahler metrics in a K\\\"ahler class $\\Omega$ are all isometric under the action of the {\\it reduced automorphism group}\\footnote{$\\widetilde{\\mathrm{Aut}}_0 (X)$ is the unique connected {\\it linear\nalgebraic subgroup} of ${\\mathrm{Aut}}_0 (X)$ such that the quotient ${\\rm\nAut} _0 (X)\/\\widetilde{{\\mathrm{Aut}}}_0 (X)$ is the Albanese torus of $X$~\\cite{fujiki-0}; its Lie algebra is the space of\n(real) holomorphic vector fields whose zero-set is non-empty\n~\\cite{fujiki-0,kobayashi,Le-Sim,gauduchon-book}.} $\\widetilde{\\mathrm{Aut}}_0 (X)$~\\cite{BM,CT,Do-one,M4}. Thus, the main difficulty in proving the splitting property is to show that if the polarized projective manifold $(X,L)= \\prod_{i=1}^r(X_i,L_i)$ admits an extremal K\\\"ahler metric, then each factor $(X_i,L_i)$ does also admit extremal K\\\"ahler metric. It was suggested by S.-T.~Yau \\cite{yau-1} that a complete obstruction to the\nexistence of extremal K\\\"ahler metrics in the K\\\"ahler class $\\Omega = 2\\pi c _1\n(L)$ on a projective manifold $X$ polarized by an ample\nholomorphic line bundle $L$ should be expressed in terms of {\\it stability}\nof the pair $(X, L)$. The currently accepted notion of stability is the\n$K$-({\\it poly}){\\it stability} introduced by G.~Tian~\\cite{Tian2} and S.~K.~Donaldson~\\cite{Do2}. The {\\it Yau--Tian--Donaldson conjecture} can\nthen be stated as follows. {\\it A polarized projective manifold $(X, L)$\nadmits a CSC K\\\"ahler metric if and only if it is $K$-polystable.} The implication `CSC $\\Rightarrow$\n{K-polystable}' in the conjecture is now well-established, thanks to work\nby S.~K.~Donaldson \\cite{Do4}, X.~X.~ Chen--G.~Tian~\\cite{CT}, J.~Stoppa\n\\cite{stoppa}, and T.~Mabuchi \\cite{mab-three,mab-three1} but the other direction is still open. In order to\naccount for extremal K\\\"ahler metrics of non-constant scalar curvature,\nG.~Szekelyhidi introduced in \\cite{Sz, gabor} the notion of {\\it\nrelative} $K$-(poly)stability with respect to a maximal torus of the\nconnected component ${\\rm Aut}_0(X,L)$ of the automorphism group ${\\rm Aut}(X,L)$ of the pair $(X, L)$~\\footnote{Recall that ${\\rm Aut}(X,L)$ consist of the automorphisms of $X$ which come from automorphisms of $L$. It is well-known (see e.g. \\cite{kob-0,gauduchon-book}) that ${\\rm Aut}_0(X,L)= \\widetilde{{\\rm Aut}}_0(X)$.} and the\nsimilar implication `extremal $\\Rightarrow$ {relative K-polystable}' was\nobtained in~\\cite{gabor-stoppa}. While it is not hard to see that in the product case (relative) $K$-(poly)stability of $(X,L)$ implies (relative) $K$-(poly)stability of each factor $(X_i, L_i)$, examples from~\\cite{ACGT} suggest that the notion of relative $K$-(poly)stability must be further strengthened in order to establish the other direction in the Yau--Tian--Donaldson correspondence. \n\n\\smallskip Our third observation is that if we start with a product K\\\"ahler metric in the class $2\\pi c_1(L)$, invariant under a maximal connected compact subgroup $K$ of $\\mathrm{Aut}_0(X)= \\prod_{i=1}^r\\mathrm{Aut}_0(X_i)$, then the $K$-relative Calabi flow (a gradient flow for the $K$-relative Mabuchi energy) preserves the Riemannian product structure. On the other hand, it is expected that this flow should converge to an extremal K\\\"ahler metric when it exists (see e.g.~\\cite{D0}). Although this conjecture is very far from being solved, a partial evidence for it is given in~\\cite{chen-hu,huang-zheng, tosatti-weinkove}. Note also that this approach has the advantage to apply to the more general case of a product of compact K\\\"ahler manifolds endowed with a product K\\\"ahler class.\n\n\\smallskip Thus motivated, we prove the splitting property under two additional hypotheses. \n\n\\begin{thm}\\label{main} Let $X_i$ be compact projective manifolds polarized by ample holomorphic line bundles $L_i$ and $X= \\prod_{i=1}^r X_i$ their product endowed with the polarisation $L= \\bigotimes_{i=1}^r L_i$, where $L_i$ is seen as a holomorphic line bundle over $X$ via the natural pull-back. Then, any extremal K\\\"ahler metric $g$ in the K\\\"ahler class $\\Omega = 2\\pi c_1(L)$ on $X$ is the Riemannian product of extremal K\\\"ahler metrics $g_i$ in the K\\\"ahler classes $\\Omega_i=2\\pi c_1(L_i)$ on the factors $X_i$, provided that at least one of the following hypotheses is satisfied. \n\\begin{enumerate} \n\\item[\\rm (i)] The integral Futaki invariants of $(X,L)$ introdced in \\cite{futaki-chow} all vanish.\n\\item[\\rm (ii)] For at most one factor $(X_i,L_i)$, the group ${\\mathrm{Aut}}_0(X_i,L_i)$ has a center of positive dimension. \n\\end{enumerate} \n\\end{thm}\n\nThe hypothesis in (i) automatically holds if ${\\rm Aut}_0(X,L)=\\{ {\\rm Id} \\}$. However, it is known that the hypothesis in (i) is a restrictive condition in the case when ${\\rm Aut}_0(X,L)$ is non-trivial (see e.g. \\cite{OSY}). Also by the results in \\cite{futaki-chow,M2}, in the case when $2\\pi c_1(L)$ admits an extremal K\\\"ahler metric, $(X,L)$ is asymptotically Chow stable if and only if the integral Futaki invariants of $(X,L)$ introdced in \\cite{futaki-chow} all vanish. More generally, the existence of an extremal K\\\"ahler metric in $2\\pi c_1(L)$ is expected to imply that $(X,L)$ is asymptotically Chow stable with respect to a maximal torus $T \\subset {\\rm Aut}_0(X,L)$: we give a precise formulation in Conjecture 1 below and discuss it in the light of the work of T.~Mabuchi~\\cite{M1,M2,M3,mab-three, mab-three1}. We then show how the conjectured correspondence would solve (via Lemma 2 and Theorem 7) the splitting of the extremal K\\\"ahler metrics in the general polarized case.\n\n\\smallskip\nWe now outline the proof of Theorem~\\ref{main}. It uses an idea going back to G.~Tian~\\cite{tian-0} (se also \\cite{yau0}) who proved that any K\\\"ahler metric $\\omega$ in $2\\pi c_1(L)$ can be approximated with induced Fubini--Study metrics from the projective embeddings of the polarized variety $(X, L)$. More precisely, let $h$ be a hermitian metric on $L$ whose curvature is $\\omega$. The induced hermitian metric on each tensor power $L^k$ is still denoted by $h$, and using $h$ and $\\omega$, consider the $L_2$ hermitian inner product on each vector space $H^0(X,L^k)$. Fixing an orthonormal basis for each $H^0(X,L^k)$, define a sequence of embeddings $\\Phi_k : X \\hookrightarrow {\\mathbb C} P^{N_k}$ and induced K\\\"ahler metrics $\\frac{1}{k} \\Phi_k^*(\\omega_{\\rm FS})$ in $\\Omega=2\\pi c_1(L)$. Tian showed that $\\frac{1}{k} \\Phi_k^*(\\omega_{\\rm FS})$ converge to $\\omega$ in $\\cC^2$ as $k \\to \\infty$ while the $\\cC^{\\infty}$ convergence follows from subsequent work by W.~Ruan~\\cite{ruan}. For each $k$, let $\\{ s_0, \\cdots, s_{N_k} \\}$ of $H^0(X,L^k)$ with respect to the $L_2$ hermitian inner product defined by $h_k=h^{\\otimes k}$ and $\\omega$, we denote the corresponding Bergman kernel $\\rho_k$ as \n$$\\rho_k = \\sum_{i=0}^{N_k} h_k(s_i,s_i).\n$$\nThe expansion of Bergman kernel was established by D.~Catlin~\\cite{catlin} and S.~Zeldich~\\cite{zeldich}. The coefficients of the expansion were calculated by Z. Lu \\cite{Lu}. An important ramification of this basic idea, relevant to the problem of existence of CSC metric in $2\\pi c_1(L)$, was given by S.~K.~Donaldson~\\cite{Do-one} who proved that when ${\\mathrm{Aut}}_0(X)=\\{ 1 \\}$, a CSC K\\\"ahler metric $\\omega$ in $2\\pi c_1(L)$ can be approximated in $\\cC^{\\infty}$ by using special projective embeddings called {\\it balanced}, a notion previously introduced and studied by H.~Luo~\\cite{L} and S.~Zhang~\\cite{Z} (see also \\cite{BLY}): a hermitian metric $h_k$ on $L^k$ is called balanced if the corresponding Bergman kernel $\\rho_k$ is a constant function on $X$, or equivalently, if the curvature $\\omega_k$ of $h_k$ satisfies $\\omega_k= \\frac{1}{k} \\Phi_k^*(\\omega_{\\rm FS})$. Thus, S.~K.~Donaldson's theorem states that if ${\\mathrm{Aut}}_0(X,L)=\\{ 1 \\}$ and $\\omega$ is a CSC K\\\"ahler metric in $2\\pi c_1(L)$, then for $k \\gg 1$, there exists a balanced hermitian metric $h_k$ on $L^k$ with curvature $\\omega_k$ and, moreover, $\\omega_k$ converges to $\\omega$ in $\\cC^{\\infty}$ as $k \\to \\infty.$ T.~Mabuchi~\\cite{M1,M2,M3} extended Donaldson's result to the case when ${\\mathrm{Aut}}_0(X,L)$ is non-trivial and $\\omega$ is an extremal K\\\"ahler metric: in this case, $\\omega$ can be approximated in $\\cC^{\\infty}$ by the normalized curvatures $\\omega_k$ of hermitian metrics $h_k$ on $L^k$ which are {\\it balanced relative to a torus} in ${\\rm Aut} _0 (X,L)$: this theory is reviewed in Section~\\ref{s:relative balanced}. For simplicity, we shall momentarily refer to such $h_k$'s as {\\it relative} balanced metrics on $L^k$. In the case when $(X,L)= \\prod_{i=1}^r (X_i,L_i)$, Grauert's direct image theorem for coherent sheaves implies that $H^0(X,L^k) = \\bigotimes_{i=1}^r H^0(X_i,L_i^k)$. It is then easily seen that if each $(X_i, L_i^k)$ admits a relative balanced hermitian metric, then the tensor product metric on $(X,L^k)$ is relative balanced and has curvature compatible with the product structure. Conversely, we show in Section~\\ref{s:proof} (see Theorem~\\ref{reduced}) that if $L^k$ admits some relative balanced metric then each $L_i^k$ does. We achieve this by studying in Section~\\ref{s:functional} the Kempf--Ness function ${\\mathbb D}$ introduced by H.~Luo~\\cite{L} and S.~K.~ Donaldson~\\cite{D1} (it is the function denoted $D$ in \\cite{L} and $\\tilde Z$ in \\cite{D1} and is essentially the $\\log$ of the Chow norm introduced in \\cite{Z}). This observation, together with Mabuchi's approximation result alluded to above, reduces our problem to showing the uniqueness of relative balanced metric on $L$ modulo the action of ${\\rm Aut}_0(X,L)$. This is not automatic in the setting of \\cite{M1,M2,M3} but holds under the assumptions (i) or (ii) of Theorem~\\ref{main}. We thus propose in Section~\\ref{s:relative balanced} a stronger notion of relative balanced metrics (which also appears in the recent work \\cite{M5}) and point out that for such (strongly) relative balanced metrics the uniqueness modulo ${\\rm Aut}_0(X,L)$ automatically holds (Lemma~\\ref{gabor}). \n\n\n\n\n\\section{Hermitian metrics balanced relative to a torus and relative Chow stability}\\label{s:relative balanced}\nIn this section we briefly review some material taken from the works of S.~K.~Donaldson~\\cite{Do-one,D1}, H.~Luo~\\cite{L}, T.~Mabuchi~\\cite{M1,M2,M3,M4,M5} and S.~Zhang~\\cite{Z} that we shall need in the sequel.\n\nLet $X$ be a compact complex projective manifold of complex dimension $n$, polarized by a very ample line bundle $L$, and $N+1$ be the dimension of $V=H^0(X,L)$. Let $\\kappa : X \\hookrightarrow P(V^*)$ denotes the Kodaira embedding with $L= \\kappa^*(\\cO(1))$. For any basis ${\\bf s}=\\{s_0, \\cdots, s_N\\}$ of $V$ we denote \n$$\n\\Phi_{\\bf s} : X \\hookrightarrow \\mathbb{C} {P}^N \n$$\nthe composition of $\\kappa$ with the identification ${\\bf s} : P(V^*) \\cong {\\mathbb C} P^{N}$.\n\n\n\n\n\nThe reduced automorphisms group $\\widetilde{\\mathrm{Aut}}_0(X)$ is the closed connected subgroup of ${\\rm Aut}_0(X)$ whose Lie algebra $\\mathfrak{h}_0$ is the ideal of holomorphic vector fields with zeros on $X$, see \\cite{fujiki-0,kobayashi,Le-Sim,gauduchon-book}. It is well-known (see e.g.~\\cite{gauduchon-book,kob-0}) that $\\widetilde{\\mathrm{Aut}}_0(X)$ coincides with the connected component ${\\mathrm{Aut}}_0(X,L)$ of the group of automorphisms of the pair $(X,L)$ and we obtain a group representation $\\rho : {\\mathrm{Aut}}_0(X,L) \\to {\\rm PGL}(V)$. One can think of ${\\mathrm{Aut}}_0(X,L)$ as the connected group generated by restrictions to $\\kappa (X)$ of elements of ${\\rm PGL}(V)$ which preserve $\\kappa(X) \\subset P(V^*)$; replacing $L$ by the tensor power $L^{N+1}$, we can further lift the action of $\\widetilde{\\mathrm{Aut}}_0(X)={\\rm Aut}_0(X,L)$ on $X$ to an action on the bundle $L$ (see e.g. \\cite{kob-0}), and find a group representation\n\\begin{equation}\\label{representation}\n\\rho : \\mathrm{Aut}_0(X,L) \\rightarrow {\\rm SL}(V).\n\\end{equation}\nIn conclusion, by replacing $L$ with a sufficiently big tensor power if necessarily, we can assume that the reduced automorphisms group $\\widetilde{\\mathrm{Aut}}_0(X) = {\\mathrm{Aut}}_0(X,L)$ of $X$ lifts to act on $L$, and identify the action of $\\widetilde{\\mathrm{Aut}}_0(X) = {\\mathrm{Aut}}_0(X,L)$ on $X$ with the induced action on $\\kappa(X)$ of the connected subgroup ${\\rm SL}_0(V, X)$ of elements ${\\rm SL} (V)$ which preserve $\\kappa(X) \\subset P(V^*)$; furthermore, we shall also assume $N>n$.\n\nFrom now on, we shall fix a real torus $T \\subset \\widetilde{\\mathrm{Aut}}_0(X,J)$ and consider hermitian metrics $h$ on $L$ which are $T$-invariant and whose curvature $\\omega$ defines a $T$-invariant K\\\"ahler form in $2\\pi c_1(L)$. Note also that, by the Calabi theorem~\\cite{cal}, if the K\\\"ahler class $\\Omega= 2\\pi c_1(L)$ admits an extremal K\\\"ahler metric, it will also admits one which is $T$-invariant. Thus, following \\cite{M1}, we are now in position to introduce the notion of a ($T$-invariant) hermitian metric $h$ on $L$ which is balanced relative to $T$. Denote by $T^{c}$ the complexified action of $T$ and consider the lifted linear $T^{c}$-action on $V$ via $\\rho$. Then, for every character $\\chi \\in \\mathrm{Hom}(T^{c}, \\mathbb{C}^*)$, we set\n$$\nV(\\chi) := \\{ s \\in V; \\rho(t)\\cdot s = \\chi(t)\\ s \\ \\mbox{ for all } t \\in T^{c} \\},\n$$\nand obtain the splitting with respect to the mutually distinct characters $\\chi_1, \\ldots, \\chi_{\\nu} \\in \\mathrm{Hom}(T^{c}, \\mathbb{C}^*)$ \n\\begin{equation}\\label{e:split}\nV = \\bigoplus_{k=1}^{\\nu} V(\\chi_k),\n\\end{equation}\nwith $\\prod_{k=1}^{\\nu} \\chi_k^{n_k}=1$ where $n_k= {\\rm dim}_{{\\mathbb C}}(V(\\chi_k))$.\n\n\n\n\n\\begin{defn} Let $m(\\cdot, \\cdot)$ be a hermitian inner product on $V$.\nWe say that $\\{s_0, s_1, \\ldots, s_N \\}$ is an $admissible\\ normal\\ basis$ of $(V,m)$ if it is compatible with the decomposition \\eqref{e:split} and provides a normal basis of $m$ on each factor $V(\\chi_k)$, i.e. if there exist positive real constants $b_k$ ($k=1, \\ldots, \\nu)$, with $\\sum_{k=1}^\\nu n_k b_k = N+1$, and a sub-basis $\\{s_{k,i}, \\ k=1, \\ldots, \\nu, \\ i=1, \\ldots, n_k \\}$ for $V(\\chi_k)$, such that\n\\begin{enumerate}\n\\item[$\\bullet$] $m(s_{k,i}, s_{l,j})=0$ if $l \\neq k$ or $i\\neq j$;\n\\item[$\\bullet$] $m(s_{k,i}, s_{k,i}) = b_k.$\n\\end{enumerate}\nThe vector $b := (b_1, \\ldots, b_\\nu)$ is called $index$ of the admissible normal basis $\\{s_0, \\ldots, s_N\\}$ for $(V,m)$; in the case when the index is $b=(1, \\ldots, 1)$ we shall call the basis {\\it admissible orthonormal} basis of $(V,m)$.\n\\end{defn}\nNote that a hermitian inner product $m( \\cdot, \\cdot)$ admits an admissible normal basis if and only if $V(\\chi_k) \\perp^m V(\\chi_l)$ for $k\\neq l$, which in turn is equivalent to $m$ being $T$-invariant. For any $T$-invariant hermitian metric $h$ on $L$ whose curvature is a K\\\"ahler form in $2\\pi c_1(L)$, $V(\\chi_k) \\perp V(\\chi_l)$ for $k \\neq l$ with respect to the induced $L^2$ hermitian inner product $m=\\langle \\cdot, \\cdot \\rangle_h$ on $V$, defined by \n$$\n\\langle s_1, s_2 \\rangle_h = \\int_X h(s_1, s_2) \\omega^n,\n$$ \nfor any two holomorphic sections $s_1, s_2 \\in H^0(X, L)$. We then define the smooth function $$E_{h, b} := \\sum_{i=0}^N h(s_i,s_i),$$ which is clearly independent of the choice of an admissible normal basis of index $b$ on $(V, m)$.\n\n\\begin{defn}\\label{d:balanced}\nA $T$-invariant hermitian metric $h$ on $L$ whose curvature $\\omega$ is K\\\"ahler metric on $X$ is called {\\it balanced relative to $T$ of index $b$} if the function $E_{h, b}$ is constant for any admissible normal basis of index $b$. The curvature $\\omega$ of $h$ is called a balanced K\\\"ahler metric of index $b$ relative to $T$.\n\\end{defn}\n\n\nThe definition above has the following useful interpretation in terms of the K\\\"ahler geometry of $X$. Consider the space ${\\mathcal B}^T(V)$ of bases of $V=H^0(M,L)$, which are {\\it compatible} with the splitting \\eqref{e:split}, i.e. which are admissible normal bases for some $T$-invariant hermitian inner product $m$. If ${\\bf s}= \\{ s_0, \\cdots, s_N\\}$ is an element of ${\\mathcal B}^T(V)$ and $h$ is any $T$-invariant hermitian metric on $L$, we put \n\\begin{equation}\\label{hs}\nh_{\\bf s} = \\frac{h}{\\sum_{i=0}^N h(s_i, s_i)},\n\\end{equation}\nwhich is manifestly independent of the auxiliary hermitian metric $h$ on $L$. \n\nAny basis ${\\bf s}=\\{s_0, \\cdots, s_N\\}$ in ${\\mathcal B}^T(V)$ determines a $T$-invariant hermitian inner product $m_{\\bf s}$ on $V$ (and $V^*$) such that $\\bf s$ (resp. the dual basis ${\\bf s}^*$) is admissible and orthonormal. The identification ${\\bf s}^* : P(V^*) \\cong {\\mathbb C} P^{N}$ determines a Fubini--Study metric $\\omega_{\\rm FS, {\\bf s}}$ on $P(V^*)$, representing $2\\pi \\cO(1)$; we denote by $\\omega_{X,{\\bf s}}= \\kappa^* (\\omega_{\\rm FS, {\\bf s}})$ the induced K\\\"ahler form on $X$ via the Kodaira embedding $\\kappa$. Note that $\\omega_{X,{\\bf s}}$ is the curvature of the hermitian metric $h_{\\bf s}$ on $L$ defined by \\eqref{hs} and if $\\omega$ is the curvature of $h$, it is easily seen that \n\\begin{equation}\\label{potential}\n\\omega_{X,{\\bf s}} = \\omega + \\frac{1}{2} dd^c \\log (\\sum_{i=0}^N h(s_i,s_i)).\n\\end{equation}\nOne therefore obatins\n\\begin{lemma}\\label{characterization}\nA $T$-invariant hermitian metric $h$ on $L$ is balanced relative to $T$ of index $b$ if and only if with respect to any admissible orthonormal basis ${\\bf s}$ of the hermitian inner product $m_{h,b}$ on $V$, defined by rescaling $\\langle \\cdot, \\cdot \\rangle_h$ on each space $V(\\chi_k)$ by a factor $1\/b_k^2$, $h_{\\bf s}= \\lambda h$ for some positive constant $\\lambda$. \n\\end{lemma}\n\n\n\n\n\n\n\\smallskip\nIn order to give further motivation for the above notions, we now briefly recall the (finite dimensional) momentum map interpretation given by S.~K.~Donaldson~\\cite{Do-one,D1}, and subsequently studied in \\cite{PS,W}. \n\n\n\nOn the space ${\\mathcal B}^T(V)$ the following groups act naturally:\n\\begin{enumerate}\n\\item[$\\bullet$] ${\\mathbb C}^{\\times}$ by scalar multiplications;\n\\item[$\\bullet$] $\\rho(Z_{{\\rm Aut}_0(X,L)}(T))$ where $Z_{{\\rm Aut}_0(X,L)}(T)$ is the connected component of identity of the centralizer of $T$ in ${\\rm Aut}_0(X,L)$;\n\\item[$\\bullet$] $G_T={\\rm S}(\\prod_{k=1}^{\\nu} {\\rm U}(n_k))$, which is also a connected component of the centralizer of $\\rho(T)$ in ${\\rm SU}(N+1)$.\n\\end{enumerate}\nAs the actions of ${\\mathbb C}^{\\times}$ and $\\rho(Z_{{\\rm Aut}_0(X,L)}(T))$ commute with the action of $G_T$, we can consider the quotient space ${\\mathcal Z}^T(V) = {\\mathcal B}^T(V) \/ \\big({\\mathbb C}^{\\times} \\times \\rho(Z_{{\\rm Aut}_0(X,L)}(T))\\big)$ on which $G_T$ acts with stabilizer (of every point) $G_T \\cap \\rho(Z_{{\\rm Aut}_0(X,L)}(T))$; in our setting $\\rho(T) \\subset G_T \\cap \\rho(Z_{{\\rm Aut}_0(X,L)}(T))$. Following \\cite{Do-one,PS}, there is a K\\\"ahler structure on ${\\mathcal Z}^T(V)$, whose definition uses the fact that any point ${\\bf s}= \\{s_{k,i}, \\ k=1,\\ldots \\nu, \\ i=1, \\ldots, n_k\\}$ of ${\\mathcal B}^T(V)$ defines an embedding of $\\Phi_{\\bf s} : X \\hookrightarrow {\\mathbb C}P^N$. With respect to this K\\\"ahler structure $G_T$ acts isometrically with momentum map given (up to a non-zero multiplicative constant) by\n$$\\mu_{G_{T}}({\\bf s}) = i \\ \\Big( \\bigoplus_{k=1}^{\\nu} ( \\langle s_{k,i}, s_{k,j} \\rangle_{h_{\\bf s}}) \\Big)_0, $$\nwhere the $( \\cdot )_0$ denotes the traceless part of the matrix (the Lie algebra ${\\rm su}(N+1)$ being identified with its dual using the positive definite Killing form), and with complexification $G_T^c= {\\rm S}(\\prod_{k=1}^{\\nu}{\\rm GL}(n_k, {\\mathbb C}))$. It follows that ${\\bf s}$ is a zero of the momentum map $\\mu_{G_T}$ if and only if $h_s$ is a balanced metric of index $b=(1, \\ldots, 1)$ relative to $T$; such a metric is also balanced with respect to the trivial torus $T=\\{ {\\rm Id} \\}$ (of index $1$). This is the classic notion of balanced embedding studied in \\cite{L,Z}. It follows from these works that the existence of a balanced basis ${\\bf s}$ is equivalent to the Chow polystability of the variety $(X,L)$, which we briefly recall: Let $d$ be the degree of the image $\\kappa (X) \\subset P(V^*)$ under the Kodaira embedding. Any element $h=(h_0, \\cdots, h_{n})$ of $P(V) \\times \\cdots \\times P(V)$ ($(n+1)$-times) is seen as $(n+1)$ hyper-planes in $P(V^*)$, and\n$$\\{ h \\in P(V) \\times \\cdots \\times P(V) : h_0 \\cap \\cdots \\cap h_n \\cap \\kappa(X) \\neq 0\\}$$\nbecomes a divisor in $P(V) \\times \\cdots \\times P(V)$ defined by a polynomial $\\hat X \\in W=\\big({\\rm Sym}^d(V^*)\\big)^{\\otimes (n+1)}$, called a {\\it Chow line} and determined up to a non-zero scale; the corresponding element $[\\hat X] \\in P(W)$ is the {\\it Chow point} associated to $(X,L)$.\n\\begin{defn}\\label{chow} The polarized variety $(X,L)$ is called {\\it Chow polystable} if the orbit of $\\hat X$ in $W=\\big({\\rm Sym}^d(V^*)\\big)^{\\otimes (n+1)}$ under the natural action of ${\\rm SL}(V)$ is closed. $(X,L)$ is called {\\it asymptotically Chow polystable} if $(X,L^k)$ is Chow polystable for any $k\\gg1$.\n\\end{defn}\nThe result of H.~Luo~\\cite{L} and S.~Zhang~\\cite{Z} (see also \\cite[Theorem~A]{M1}) then states\n\\begin{thm}\\label{luo-zhang} A compact polarized projective complex manifold $(X,L)$ is Chow polystable if and only if $L$ admits a balanced hermitian metric $h$ {\\rm (}of index $1$ relative to $T=\\{ {\\rm Id} \\}${\\rm )}.\n\\end{thm}\n\nThe relevance of balanced metrics to our work comes from the following central result in the theory, proved by S.~K.~Donaldson~\\cite{Do-one} in the case when ${\\mathrm{Aut}}_0(X,L)$ is trivial, and extended by T.~Mabuchi~\\cite{M2} to the general case.\n\\begin{thm}\\label{do-mabuchi} Let $(X,L)$ be an asymptotically Chow polystable compact polarized projective manifold and $\\omega \\in 2\\pi c_1(L)$ a K\\\"ahler metric metric of constant scalar curvature~\\footnote{A. Futaki~\\cite{futaki-chow} showed that the extremal K\\\"ahler metrics in the K\\\"ahler class of an asymptotically Chow stable polarization must be CSC.}. Then, there exist sequences of integers $m_k \\to \\infty$ and hermitian metrics $h_k$ on $L^{m_k}$, with curvatures $\\omega_k$, which are balanced {\\rm (}relative to $T=\\{ 1 \\}$ of index $b=1${\\rm )}, and such that $\\frac{1}{m_k}\\omega_k$ converge in $\\cC^{\\infty}$ to $\\omega$ as $k \\to \\infty$. \\end{thm}\nNote that when ${\\rm Aut}_0(X,L)$ is trivial, S.~K.~Donaldson also shows in \\cite{Do-one} that the existence of a CSC K\\\"ahler metric in $2\\pi c_1(L)$ implies that $(M,L)$ is asymptotically Chow polystable while it is known that the latter condition is restrictive in the case when ${\\rm Aut}_0(X,L)$ is non-trivial (see e.g. \\cite{OSY}). \n\n\\smallskip\nOne therefore needs to further relax the condition on balanced metrics in order to find similar approximations of extremal K\\\"ahler metrics on asymptotically Chow unstable varieties, and this is where the choice of indices $b$ will come to play. Using the momentum map picture described above, a natural approach developed in \\cite{gabor, Sz} would be, instead of zeroes of $\\mu_{G_T}$, to study the critical points of the squared norm $||\\mu_{G_T} ||^2$ (with respect to the positive definite Killing inner product of $\\mathfrak{su}(N+1)$). It follows from the moment map picture that a basis ${\\bf s} \\in {\\mathcal B}^T(V)$ is a critical point of $||\\mu_{G_T}||^2$ if and only if $\\mu_{G_T}({\\bf s})$ is a matrix which belongs to the Lie algebra of the stabilizer of the projection of ${\\bf s}$ to ${\\mathcal Z}^{T}$ for the action of $G_T$. In order to simplify the discussion, and with the application in mind, let us assume that $T$ is a maximal torus in ${\\rm Aut}_0(X,L)$. This implies that $\\rho(Z_{{\\rm Aut}_0(X,L)}(T)) \\cap G = \\rho(T)$ i.e. the stabilizer of any point of ${\\mathcal Z}^T(V)$ is $\\rho(T)$. Therefore, a basis ${\\bf s}$ is a critical point for $||\\mu_{G_T}||^2$ if and only if $\\mu_{G_T}({\\bf s})$ is a diagonal matrix $$ i \\ {\\rm diag}(a_1, \\ldots, a_1, a_2, \\ldots, a_2, \\cdots, a_{\\nu}, \\ldots a_{\\nu})$$ which belongs to the Lie algebra of $\\rho(T)$. In other words, ${\\bf s}$ defines a critical point of $||\\mu_{G_T}||^2$ on ${\\mathcal Z}^{T}$ if and only if the induced hermitian metric $h_{\\bf s}$ on $L$ is balanced relative to $T$ of index $b=(b_1, \\cdots, b_{\\nu})$ with \n\\begin{equation}\\label{constraint}\nb_k= \\frac{1 + \\log |\\chi_k(t)|}{1 + \\sum_{\\ell=1}^{\\nu} \\frac{n_{\\ell} \\log |\\chi_{\\ell}(t)|}{N+1}}, \\ k=1, \\ldots, \\nu\n\\end{equation} \nfor some $t \\in T^c$. \nThe corresponding interpretation in terms of Chow stability has been worked out by T.~Mabuchi~\\cite{M5} and is expressed by the closeness in $W$ of the Chow line $\\hat X$ under the natural action of the group\n$$G_{T^{\\perp}} ^c (V)= \\{ {\\rm diag} (A_1, \\cdots, A_{\\nu}) \\in \\prod_{k=1}^{\\nu} {\\rm GL}(V({\\chi_k})) : \\prod_{k=1}^{\\nu} {\\rm det} (A_k)^{1+\\log |\\chi_k(t)|}=1 \\ \\forall t \\in T^c \\}.$$\n\\begin{defn}\\label{stability} We call $(X,L)$ {\\it Chow polystable stable relative to $T$} if the Chow line $\\hat X$ associated to $(X,L)$ has a closed orbit with respect to $G_{T^{\\perp}}^c(V)$; $(X,L)$ is called {\\it asymptotically Chow stable relative to $T$} if $(X, L^k)$ is Chow polystable stable relative to $T$ for all $k\\gg 1$. \n\\end{defn}\n\\noindent We then have (cf. \\cite[Theorem A]{M1} and \\cite[Theorem C]{M5}) \n\\begin{thm}\\label{chow-stability}\nA polarized compact projective complex manifold $(X,L)$ is Chow polystable relative to $T$ if and only if $L$ admits a hermitian metric balanced relative to $T$ for some index $b$ satisfying \\eqref{constraint}. \n\\end{thm}\n\nA generalization of the Kempf--Ness theorem (see \\cite[Theorem 3.5]{gabor} or \\cite[Theorem 1.3.4]{Sz}) in the momentum map set up provides us with the following useful result.\n\\begin{lemma}\\label{gabor} Let $(X,L)$ be a compact polarized projective complex manifold and $T$ a maximal torus in ${\\rm Aut}_0(X,L)$. Then $L$ admits a hermitian metric balanced relative to $T$ of some index $b$ satisfying \\eqref{constraint} if and only if the orbit of some {\\rm (}and hence each{\\rm )} point of ${\\mathcal B}^T(V)$ under the action of the group $$G^c_{T^\\perp}=\\{ {\\rm diag} (A_1, \\cdots, A_{\\nu}) \\in \\prod_{k=1}^{\\nu} {\\rm GL}(n_k, {\\mathbb C})) : \\prod_{k=1}^{\\nu} {\\rm det} (A_k)^{1+\\log |\\chi_k(t)|}=1 \\ \\forall t \\in T^c \\}$$ contains a basis ${\\bf s}$ such that $h_{\\bf s}$ is balanced with respect to $T$ of index satisfying \\eqref{constraint}. Furthermore, any two balanced hermitian metrics relative to $T$ with indices satisfying \\eqref{constraint} are homothetic under the action of ${\\rm Aut}_0(X,L)$. \\end{lemma}\nIt is not difficult to give a direct prove of Lemma~\\ref{gabor}, once one knows the relevant identities to use. The uniqueness part follows from the fact that the $T^c$ action generates balanced metrics relative to $T$ (of some index $b$ satisfying \\eqref{constraint}) and the corresponding admissible bases of index $b$ (see Lemma~\\ref{characterization}) exhaust the $G^c_{T^\\perp}$ orbits of ${\\mathcal Z}^T(V)$; one can then apply Proposition~\\ref{convex} in Section~\\ref{s:functional}. In particular, the index $b$ in Lemma~\\ref{gabor} is uniquely determined.\n\n\n\\smallskip\nIn view of the discussion above, the following provides a natural scope of a generalization of Theorem~\\ref{do-mabuchi}.\n\\begin{conj}\\label{relative-stability} Let $(X,L)$ be a compact polarized projective manifold and $\\omega \\in 2\\pi c_1(L)$ an extremal K\\\"ahler metric which, without loss, is invariant under a maximal torus $T\\subset {\\rm Aut}_0(X,L)$. Then $(M,L)$ is asymptotically Chow polystable relative to $T$, and there exists a sequence of integers $m_k \\to \\infty$ and $T$-invariant hermitian metrics $h_k$ on $L^{m_k}$ with curvatures $\\omega_k$, which are balanced relative to $T$ of indices $b_k$ satisfying \\eqref{constraint}, such that the corresponding relative balanced K\\\"ahler metrics $\\frac{1}{m_k}\\omega_k$ on $X$ converge in $\\cC^{\\infty}$ to $\\omega$ as $k \\to \\infty$.\n\\end{conj}\n\n\n\n\\smallskip\nIn a series of work \\cite{M1,M2,M3}, T.~Mabuchi has established a weaker version of Conjecture~\\ref{relative-stability}. The main idea is to consider instead of the group $G_T$, the smaller group $G = \\prod_{k=1}^{\\nu} {\\rm SU}(n_k)$ which acts on ${\\mathcal Z}^T(V)$ with momentum map\n$$\\mu_{G}({\\bf s})= i \\ \\bigoplus_{k=1}^{\\nu} \\Big( \\langle s_{k,i}, s_{k,j} \\rangle_{h_{\\bf s}}\\Big)_0, $$\nso that the zeroes of $\\mu_G$ correspond to bases ${\\bf s}$ in ${\\mathcal B}^T(V)$ for which the hermitian metrics $h_{\\bf s}$ on $L$ which are balanced relative to $T$ of some index $b$ (not necessarily satisfying \\eqref{constraint}). The corresponding notion of Chow stability is then\n\\begin{defn}\\label{weak-stability} $(X,L)$ is {\\it weakly Chow polystable stable relative to $T$} if the Chow norm $\\hat X$ associated to $(X,L)$ has a closed orbit under the action of $G^c(V)= \\prod_{k=1}^{\\nu} {\\rm SL}(V(\\chi_k))$. We call $(X,L)$ {\\it asymptotically weakly Chow polystable stable relative to $T$} if $(M,L^k)$ is weakly Chow polystable stable relative to $T$ for all $k\\gg 1$. \n\\end{defn}\n\nThe following result is extracted from \\cite{M1,M2,M3}.\n\\begin{thm}\\label{mabuchi} Let $(X,L)$ be a compact polarized projective manifold and $\\omega \\in 2\\pi c_1(L)$ an extremal K\\\"ahler metric which, without loss, is invariant under a maximal compact connected subgroup $K\\subset {\\mathrm{Aut}}_0(X,L)$. Let $T\\subset K$ be any torus in the connected component of the identity of the centre of $K$. Then, $(M,L)$ is asymptotically weakly Chow polystable relative to $T$ and there exists a sequence of integers $m_k \\to \\infty$ and $T$-invariant hermitian metrics $h_k$ on $L^{m_k}$ with curvatures $\\omega_k$, which are balanced relative to $T$, such that $\\frac{1}{m_k}\\omega_k$ on $X$ converge in $\\cC^{\\infty}$ to $\\omega$ as $k \\to \\infty$. \n\\end{thm}\nThe above statement is implicitly established in \\cite{M3} in the course of proof that the existence of an extremal K\\\"ahler metric in $2\\pi c_1(L)$ implies that $(X,L)$ is asymptotically weakly Chow polystable relative to $T$; the choice of $T$ is specified by \\cite[Theorem~I]{M3}. More precisely, \\cite[Theorem~B]{M1} shows that any $T$-invariant extremal K\\\"ahler metric in $2\\pi c_1(L)$ can be approximated by a sequence of {\\it almost critical} metrics; then, combining S.~K.~Donaldson's idea in \\cite{Do-one} and D.~Phong--J.~Sturm's estimates in \\cite{PS}, a perturbation technique is elaborated in \\cite{M2} and applied in \\cite{M3} in order to perturb the almost critical metrics to balanced metrics relative to $T$, in a way that their curvatures converge to $\\omega$.\n\nThe limitation of Theorem~\\ref{mabuchi} to complete the proof of the splitting properly (Theorem~\\ref{main} in the introduction) in full generality is in the lack of analogue of Lemma~\\ref{gabor}, which guarantees that any two balanced metrics relative to $T$ on $L$ are homothetic. We show that this is true under the hypothesis (i) and (ii) of Theorem~\\ref{main}.\n\n\\section{The Kempf--Ness function ${\\mathbb D}$}\\label{s:functional}\nIn this section we are going to apply the well-known `Kempf--Ness' principle related to the problems of studying zeroes of momentum maps. For simplicity, we discuss the existence of hermitian metrics on $L$ which are balanced relative to a fixed torus $T\\subset {\\rm Aut}_0(X,L)$ of some index, but the discussion and all of the results can be easily adapted to the case of indices satisfying \\eqref{constraint} simply by changing the group $G$ to $G_{T^{\\perp}}$. We have seen in Section~\\ref{s:relative balanced} that the problem of finding a basis ${\\bf s}\\in {\\mathcal B}^T(V)$ for which $h_{\\bf s}$ is balanced with respect to $T$ is equivalent to finding zeroes of the momentum map $\\mu_G$ in a given orbit $G^c \\cdot [{\\bf s}_0 ]\\subset {\\mathcal Z}^T(V)$. As $\\mu_G$ is $G$-equivariant, this becomes a problem on the symmetric space $G^c\/G$. On that space we are going to consider a function $F_{{\\bf s}_0} : G^c\/G \\to {\\mathbb R}$, called {\\it Kempf--Ness} function, whose behaviour determines whether or not there exists a zero of $\\mu_G$ on $G^c \\cdot {\\bf s}_0$. This function is geodesically convex and its derivative is essentially $\\mu_G$; hence $\\mu_G$ admits a zero on $G^c \\cdot {\\bf s}_0$ if and only if $F_{{\\bf s}_0}$ attains a minimum on $G^c\/G$.\n\n\n \n\nOn the space ${\\mathcal H}$ of all hermitian inner products $m$ on $V$ such that $V(\\chi_k) \\perp^{m} V(\\chi_l), \\ l \\neq k$ (equivalently, which admit admissible normal bases of some index) the group $G^c(V)$ acts with stabilizer $G(V,m)=G^c(V) \\cap {\\rm U}(V,m)$; thus, for each $m_0 \\in {\\mathcal H}$, by introducing an admissible orthonormal basis ${\\bf s}_0$, we can identify the corresponding orbit ${\\mathcal M}_{m_0}= G^c(V) \\cdot m_0$ with the symmetric space $G^c\/G$ (which is known to be reducible of non-positive sectional curvature). The underlying riemannian metric is explicitly given by (see e.g.~\\cite{helgason})\n\\begin{equation}\\label{metric}\n(M_1, M_2)_m = {\\rm Tr}(M_1 \\cdot m^{-1} \\cdot M_2 \\cdot m^{-1}),\n\\end{equation}\nwhere the hermitian inner product $m$ is identified with a positive-definite hermitian endomorphism of $V$ via $m_0$, and $M_1,M_2 \\in T_m({\\mathcal M}_{m_0})$ with hermitian skew-symmetric endomorphisms of $(V, m_0)$. Another well-known fact (see e.g. \\cite{helgason}) is that geodesics correspond to $1$-parameter subgroups of $G^c(V)$, so the geodesic $m(t)$ joining two points $m_1, m_2 \\in {\\mathcal M}_{m_0}$ is generated by the family of admissible normal bases ${\\bf s}(t)= \\{e^{t\\gamma_0}s_0, \\cdots, e^{t\\gamma_N}s_N\\}$, where ${\\bf s}=\\{s_0, \\cdots, s_N\\}$ is an admissible orthonormal basis for $m_1$ which diagonalizes $m_2$, and $m_2(s_i,s_i)= e^{-2\\gamma_i}$ (with $\\sum_{i=1}^{n_k} \\gamma_{i,k}=0$) and $m(t)$ is the unique hermitian inner product for which $s(t)$ is an admissible orthonormal basis.\n\nDenote by $\\mathcal{K}_\\omega$ be the set of all K\\\"ahler metrics in the K\\\"ahler class $[\\omega]$, i.e.\n$$\n\\mathcal{K}_\\omega = \\{ \\omega_\\varphi \\ | \\ \\omega_\\varphi = \\omega + dd^c \\varphi > 0, \\varphi \\in C^\\infty (X) \\}\n$$\nWe can define a map $\\mathcal{FS} : \\mathcal{H} \\mapsto \\mathcal{K}_\\omega$ as follows: For any $m \\in {\\mathcal H}$ let ${\\bf s}=\\{s_0, \\cdots, s_N\\}$ be an admissible orthonormal basis of $V$ and $\\omega_{FS,{\\bf s}}$ the Fubini--Study it defines on $P(V^*)$. Consider the pull-back $\\omega_{X,{\\bf s}}= \\kappa^* (\\omega_{\\rm FS, {\\bf s}})$ under the Kodaira embedding (satisfying \\eqref{potential}), which is the curvature of the hermitian metric $h_{\\bf s}$ on $L$, given by \\eqref{hs}. Put\n\\begin{eqnarray}\n\\label{w}\n\\mathcal{FS}(m) : = \\omega_{X, {\\bf s}}, \\ \\ h_{m}:= h_{\\bf s}, \n\\end{eqnarray}\nnoting that for a fixed $m$ the right hand sides of \\eqref{hs} and \\eqref{potential} are independent of the choice of orthonormal basis ${\\bf s}$. \n\n\nMany authors have considered (see e.g. \\cite{do,gauduchon-book}) the functional ${\\mathbb I} : \\mathcal{K}_{\\omega} \\to {\\mathbb R}$, defined up to an additive constant by requiring that its derivative $\\delta {\\mathbb I}$ is given by\n$$\n(\\delta {\\mathbb I}) (\\dot{\\varphi}) = \\int_X \\dot{\\varphi} \\ \\omega_\\varphi^n,\n$$\nwhere $\\dot{\\varphi} \\in T_{\\omega_{\\varphi}} (K_{\\omega}) = C^{\\infty}(X)$. Following H.~Luo~\\cite{L} and S.~K.~Donaldson~\\cite{D1}, we introduce ${\\mathbb D} : \\mathcal{H} \\to {\\mathbb R}$ by \n\\begin{equation} \\label{d:d}\n{\\mathbb D} (m) : = - {\\mathbb I} (\\mathcal{FS} (m)).\n\\end{equation}\nThe restriction of ${\\mathbb D}$ to the $G^c(V)$ orbit ${\\mathcal M}_{m_0}$ defines a function on $G^c(V)\/G(V,m_0)$ which, by introducing an admissible orthonormal basis ${\\bf s}_0$ of $m_0$, will be the Kempf--Ness function $F_{{\\bf s}_0} : G^c\/G \\to {\\mathbb R}$ referred to earlier.\n\n\n\\smallskip\nThe following results in this section are essentially proved in \\cite{D1} and \\cite{L}. The way we treat the reduced automorphism group is inspired by X.~X.~Chen's work \\cite{C1}. \n\nWe start by characterizing the critical points of ${\\mathbb D}$.\n\n\\begin{prop} \\label{cha}\nA hermitian inner product $m$ is a critical point of ${\\mathbb D} : G^c(V)\\cdot m_0 \\mapsto {\\mathbb R}$ if and only if the induced hermitian metric $h_m$ defined by \\eqref{hs} and \\eqref{w} is balanced {\\rm (}of some index $b${\\rm )} with respect to $T$, i.e. if and only if any admissible orthonormal basis ${\\bf s}$ of $m$ is a zero of the momentum map $\\mu_G$. Likewise, $m$ is a critical point of ${\\mathbb D} : G^c_{T^{\\perp}}(V)\\cdot m_0 \\mapsto {\\mathbb R}$ if and only if $h_m$ is balanced of index satisfying \\eqref{constraint}.\n\\end{prop}\n\n\\begin{proof}\nFor ${\\mathcal M}_{m_0} = G^c(V) \\cdot m_0$, we will prove that $m$ is a critical point of ${\\mathbb D}: {\\mathcal M}_{m_0} \\mapsto {\\mathbb R}$ if and only if there exist real numbers $(b_1, \\ldots, b_{\\nu})$ such that for some (and hence any) admissible orthonormal basis ${\\bf s}= \\{ s_{k,i}, 1 \\leq k \\leq \\nu, 1 \\leq i \\leq n_k \\}$ of $m$\n\\begin{equation}\\label{conditions}\n\\begin{split}\n\\int_X h_{m} (s_{k,i}, s_{k,i}) \\ {\\mathcal FS}(m)^n & = b_k, \\ i=1, \\ldots, n_k, \\ k=1, \\ldots, \\nu \\\\\n\\int_X h_{m} (s_{k,i}, s_{l,j}) \\ {\\mathcal FS}(m)^n & = 0, \\ {\\rm if} \\ k\\neq l \\ {\\rm or} \\ i \\neq j.\n\\end{split}\n\\end{equation}\nUsing the hermitian metric $h_m$ in the definition \\eqref{hs}, we see that the conditions \\eqref{conditions} are equivalent to $h_m$ being balanced relative to $T$ of index $b=(b_1, \\ldots, b_{\\nu})$. The case of a $G^c_{T\ufffd{\\perp}}(V)$ orbit will follow with obvious modifications of the arguments below. \n\n\n\n\n\\smallskip\n($\\Rightarrow$) Let ${\\bf s}= \\{s_{k,i}, 1 \\leq k \\leq \\nu, 1 \\leq i \\leq n_k\\}$ be an admissible orthonormal basis of $m$. For any choice of $\\gamma_{l,j}$ with \n$$\n\\sum_{i=1}^{n_k} \\gamma_{k,i} = 0.\n$$\nthe basis ${\\bf s}_t=\\{ e^{\\gamma_{l,j}t} s_{l,j} \\}$ defines a hermitian inner product $m(t)$ on $V$ (such that ${\\bf s}_t$ is an admissible orthonormal bases for $m(t)$) and, as we have noticed, $m(t)$ is a geodesic. Put ${\\mathbb D}(t)={\\mathbb D}(m(t))$. Using \\eqref{potential}, \\eqref{w} and \\eqref{d:d}, we obtain for the derivative ${\\mathbb D}'(t)$\n\\begin{equation}\\label{D'}\n\\begin{split}\n{\\mathbb D}'(t) &= \\int_X \\frac{\\sum_{j=0}^N 2\\gamma_j e^{2t\\gamma_j}|s_j|_h^2}{\\sum_{j=0}^N e^{2t\\gamma_j}|s_j|_h^2} \\Big({\\mathcal FS}(m(t))\\Big)^n \\\\\n&= \\int_X Q(t) \\Big( {\\mathcal FS}(m(t))\\Big)^n,\n\\end{split}\n\\end{equation}\nwith \n\\begin{equation}\\label{Q}\nQ(t) = \\frac{\\sum_{j=0}^N 2\\gamma_j e^{2t\\gamma_j}|s_j|_h^2}{\\sum_{j=0}^N e^{2t\\gamma_j}|s_j|_h^2}.\n\\end{equation} \nThen, the fact that $m$ is a critical point of ${\\mathbb D}$ implies \n\\begin{equation}\\label{calcul}\n0 ={\\mathbb D}'(0) = 2 \\sum_{k=1}^{\\nu}\\sum_{i=1}^{n_k} \\gamma_{k,i} \\int_X h_{m} (s_{k,i}, s_{k,i}) \\ {\\mathcal FS}(m)^n,\n\\end{equation}\nwhere we have used the fact that \\eqref{Q} is independent of the choice of a hermitian metric $h$ on $L$. For the latter equality to hold for any choice of real numbers $\\gamma_{k,i}$ as above, $\\int_X h_{m} (s_{k,i}, s_{k,i}) \\ {\\mathcal FS}(m)^n$ must be \nindependent of $i$; since ${\\mathcal FS}(m)$ and $h_{m}$ are $T$ invariant, $\\int_X h_{m}(s_{k,i}, s_{l,j}) \\ {\\mathcal FS}(m)^n = 0$ for $k \\neq l$. Elementary linear algebra shows that if $\\int_X h_{m} (s_{k,i}, s_{k,i}) \\ {\\mathcal FS}(m)^n$ is independent of $i$ for {\\it any} choice of an admissible orthonormal basis ${\\bf s}$ for $m$, then one must have $\\int_X h_{m} (s_{k,i}, s_{k,j}) \\ {\\mathcal FS}(m)^n = 0$ for $ i \\neq j$.\n\n\n($\\Leftarrow$) The conditions \\eqref{conditions} mean that some admissible orthonormal basis ${\\bf s}$ of $m$ is an admissible normal basis (of index ${b}=(b_1, \\ldots, b_{\\nu})$) for the induced $L_2$ hermitian inner product $\\langle \\cdot , \\cdot \\rangle_{h_{m}}$; this is clearly independent of the choice of a particular admissible orthonormal basis of $m$. It is therefore enough to pick one admissible orthonormal basis ${\\bf s}$, and show that if \\eqref{conditions} is satisfied, then $m$ must be a critical point of ${\\mathbb D}$, or equivalently, ${\\mathbb D}'(0)=0$ along any geodesic $m(t)$ issued at $m$. The computation \\eqref{calcul} shows this. \\end{proof}\n\n\n\\begin{rem}\\label{chow-norm} If we consider ${\\mathbb D}$ as a function on $G^c(V)\/G(V,m_0)$ (or $G^c_{T^{\\perp}}(V)\/G_{T^{\\perp}}(V, m_0)$), the computation \\eqref{D'} (compared to a similar result in \\cite{Z}) shows that ${\\mathbb D}$ coincides, up to a positive scale and an additive constant, with the function $\\log || \\cdot ||_{{\\rm CH},m_0}$ defined on $G^c(V)\/G(V,m_0)$ (resp. $G^c_{T^{\\perp}}(V)\/G_{T^{\\perp}}(V, m_0)$), where $|| \\cdot ||_{{\\rm CH},m_0}$ is the $U(V,m_0)$-invariant {\\it Chow norm} on the space $W= {\\rm Sym}^d(V^*)^{\\otimes (n+1)},$ introduced in \\cite{Z}. This, together with Proposition~\\ref{cha}, explains Theorems~\\ref{luo-zhang} and \\ref{chow-stability}, once one proves (as in \\cite{Z} and \\cite{M1}) that $\\log || \\cdot ||_{{\\rm CH},m_0}$ has a critical point on the $G^c(V)$ (resp. $G^c_{T^{\\perp}}(V)$) orbit of $\\hat X$ if and only the orbit is closed (i.e. $(X,L)$ is (weakly) relative Chow stable). Note that the latter condition is independent of the choice of $m_0$, showing that the existence of critical points of ${\\mathbb D}$ is independent of the choice of a $G^c(V)$ orbit in ${\\mathcal H}$.\n\\end{rem}\n\n\nThe next results hold for ${\\mathcal M}_{m_0}$ being either the $G^c(V)$ or the $G^c_{T^{\\perp}}(V)$ orbit of $m_0$ in ${\\mathcal H}$.\n\n\\begin{prop}\\label{convex} \n${\\mathbb D}$ is convex along geodesics in $\\mathcal{M}_{m_0}$. Furthermore, for any two critical points $m_1, m_2$ {\\rm (}if they exist{\\rm )}, the geodesic $m(t)$ joining $m_1$ and $m_2$ defines a family of balanced hermitian metrics $h_{m(t)}$ relative to $T$ on $L$, which are isometric under the action of ${\\rm Aut}_0(X,L)$ and, therefore, have the same index $b$.\n\\end{prop}\n\\begin{proof} By \\eqref{D'}, the second derivative of ${\\mathbb D}(t)$ is\n\\begin{eqnarray*}\n{\\mathbb D}''(t) &=& \\int_X \\Big(\\frac{\\partial Q}{\\partial t} + n |dQ|^2_{{\\mathcal FS}(m(t))} \\Big) \\mathcal{FS}(m(t))^n.\n\\end{eqnarray*}\nTo show the convexity of ${\\mathbb D}(t)$ along geodesics, we adopt an argument of T.~Mabuchi~\\cite{M1} by constructing the map\n\\begin{eqnarray*}\n\\eta : [0,1] \\times [0, 2\\pi) \\times X &\\rightarrow& \\mathbb{C}P^N,\\\\\n(t, \\theta, x) &\\mapsto& [e^{(t+\\sqrt{-1}\\theta)\\gamma_0} s_0(x), \\ldots, e^{(t+\\sqrt{-1}\\theta)\\gamma_N} s_N(x)].\n\\end{eqnarray*}\nLetting $z=t+\\sqrt{-1}\\theta$, we have\n\\begin{eqnarray*}\n0 & \\leq & \\int_{\\eta([0,1] \\times [0, 2\\pi) \\times X)} \\omega_{{\\rm FS},{\\bf s}}^{n+1}\\\\\n&=&\\int_{[0,1] \\times [0, 2\\pi) \\times X} (\\eta^*\\omega_{{\\rm FS}, {\\bf s}})^{n+1}\\\\\n&=&\\int_{[0,1] \\times [0, 2\\pi) \\times X} ( \\frac{\\sqrt{-1}}{2\\pi} \\partial_X \\bar{\\partial}_X \\log (\\sum_{j=0}^N e^{2t\\gamma_j} |s_j|^2 ) + \\frac{\\sqrt{-1}}{4\\pi} \\partial_X Q \\wedge d \\bar{z}\\\\\n& & + \\frac{\\sqrt{-1}}{4\\pi} d z \\wedge \\bar{\\partial}_X Q + \\frac{\\sqrt{-1}}{8\\pi} \\frac{\\partial Q}{\\partial t} dz \\wedge d\\bar{z})^{n+1} \\\\\n&=& 2\\pi \\int_0^1 \\int_X \\frac{n+1}{4\\pi} \\frac{\\partial Q}{\\partial t} + \\frac{(n+1)n}{4\\pi} |d Q|^2_{{\\mathcal FS}(m(t))} (\\Phi^*_t \\omega_{{\\rm FS}, {\\bf s}(t)})^n dt \\\\\n&=& \\frac{n+1}{2} \\int_0^1 {\\mathbb D}^{''}(t) dt\n\\end{eqnarray*}\nHence ${\\mathbb D}(t)$ is convex. It follows that any critical point of ${\\mathbb D}$ is a global minimizer. Suppose now $m_1, m_2$ are two minimizers of ${\\mathbb D}$. Joining $m_1$ and $m_2$ with a geodesic $m(t)$ as in the proof of \\eqref{D'}, we have ${\\mathbb D}(0) = {\\mathbb D}(1)$ and ${\\mathbb D}'(0) = 0 = {\\mathbb D}'(1)$, so that ${\\mathbb D}''(t) = 0, 0 \\leq t \\leq 1$. It follows from the calculation above that\n$$\n\\int_{\\eta([0,1] \\times [0, 2\\pi) \\times X)} \\omega_{{\\rm FS}, {\\bf s}}^{n+1} = 0.\n$$\nWe conclude that for any point $p \\in \\eta([0,1] \\times [0, 2\\pi) \\times X) \\subset \\mathbb{C}P^N$, the complex dimension of any neighbourhood of $p$ in $\\eta([0,1] \\times [0, 2\\pi) \\times X)$ is $n$. Hence the image $\\Phi_{{\\bf s}(t)}(X)$ is fixed, showing that the one-parameter group ${\\rm diag}(e^{t\\gamma_0}, \\ldots, e^{t\\gamma_N})$ in ${\\rm SL}(N+1, {\\mathbb C})$ induces a one-parameter group in $\\widetilde{\\mathrm{Aut}}_0(X) = {\\mathrm{Aut}}(X,L)$. \\end{proof}\n\n\nFor any $m \\in {\\mathcal M}_{m_0}$, we introduce the group\n$$\\mathrm{Aut}_m (X, {\\mathcal M}_{m_0}, {\\mathbb D}) = \\{ g \\in \\widetilde{\\mathrm{Aut}}_0(X) \\ | \\ \\rho(g) ({\\mathcal M}_{m_0}) = {\\mathcal M}_{m_0}, \\ {\\mathbb D} (\\rho (g) \\cdot m) = {\\mathbb D}(m) \\},$$\nwhere $\\rho$ is the representation \\eqref{representation}. Clearly, ${\\mathrm{Aut}}_m(X, {\\mathcal M}_{m_0}, {\\mathbb D})$ is a closed subgroup of ${\\mathrm{Aut}}(X,L)= \\widetilde{\\mathrm{Aut}}_0(X)$ while $\\rho({\\mathrm{Aut}}_m(X, {\\mathcal M}_{m_0}, {\\mathbb D}))$ is a closed subgroup of ${\\rm SL}(V)$.\n\n\\begin{lemma} For any two points $m_1, m_2 \\in {\\mathcal M}_{m_0}$, \n$ \\mathrm{Aut}_{{m_1}}(X, {\\mathcal M}_{m_0}, {\\mathbb D}) = \\mathrm{Aut}_{m_2}(X, {\\mathcal M}_{m_0}, {\\mathbb D}).$\n\\end{lemma}\n\n\\begin{proof}\nLet $m(t), 0 \\leq t \\leq 1$ be the geodesic connecting $m_1$ and $m_{2}$ and ${\\bf s}$ an admissible orthonormal basis with respect to $m_1$. For any $g \\in \\mathrm{Aut}_{{m_1}}(X, {\\mathcal M}_{m_0}, {\\mathbb D})$, $\\rho(g) \\cdot m(t)$ is the geodesic connecting $\\rho(g) \\cdot m_1$ and $\\rho(g) \\cdot m_{2}$. Using the integral formula \\eqref{D'}, and noting that \\eqref{Q} is independent of the choice of a hermitian metric $h$, we have $\\frac{d}{dt}{\\mathbb D}(m(t)) = \\frac{d}{dt} {\\mathbb D}(\\rho(g) \\cdot m(t))$, and therefore ${\\mathbb D}(m_{2}) = {\\mathbb D}(\\rho(g) \\cdot m_{2})$. Hence $g \\in \\mathrm{Aut}_{m_{2}}(X, {\\mathcal M}_{m_0}, {\\mathbb D})$. Similarly, $\\mathrm{Aut}_{m_{2}} (X, {\\mathcal M}_{m_0}, {\\mathbb D})\\subset \\mathrm{Aut}_{m_1}(X, {\\mathcal M}_{m_0}, {\\mathbb D})$. \\end{proof}\nIn view of the above lemma, we adopt \n\n\\begin{defn} ${\\mathrm{Aut}}_X({\\mathcal M}_{m_0}, {\\mathbb D})$ is the closed subgroup of $\\widetilde{\\mathrm{Aut}}_0(X)$ of elements which preserve ${\\mathcal M}_{m_0}$ and ${\\mathbb D}$.\n\\end{defn}\n\n\\begin{rem} By definition, $\\rho({\\rm Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})) \\subset \\rho(Z_{{\\mathrm{Aut}}_0(X,L)}(T))\\cap G_{T^{\\perp}}^c$. If $T$ is a maximal torus in ${\\mathrm{Aut}}_0(X,L)$ and $X$ admits an extremal K\\\"ahler metric, a result by E.~Calabi~\\cite{cal} implies that $Z_{{\\mathrm{Aut}}_0(X,L)}(T)= T^c$. We conclude that in this case ${\\rm Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})$ is trivial. \n\nFormula \\eqref{D'} shows that any element of $\\rho(Z_{{\\mathrm{Aut}}_0(X,L)}(T))\\cap G_T^c$ (resp. $\\rho(Z_{{\\mathrm{Aut}}_0(X,L)}(T))\\cap G_{T^{\\perp}}^c$) sends a critical point of ${\\mathbb D}$ to another critical point. It then follows from Proposition~\\ref{convex} that when ${\\mathbb D}$ atteins its minimum on ${\\mathcal M}_{m_0}$ (i.e. $(X,L)$ is (weakly) relative Chow stable, see Theorem~\\ref{chow-stability} and Remark~\\ref{chow-norm}), ${\\rm Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})$ is the sub-group of $Z_{{\\mathrm{Aut}}_0(X,L)}(T)$ of elements whose lifts by $\\rho$ belong to $G^c_T$ (resp. $G^c_{T^{\\perp}}$). \n\\end{rem}\n\\begin{lemma}\\label{l:2} Suppose that ${\\mathbb D}$ has a minimum on ${\\mathcal M}_{m_0}$. Then, the set of all minimizers represents an orbit for the induced action $\\rho(\\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D}))$ and\nfor any $m \\in \\mathcal{M}_{m_0}$ there exists a minimizer $m_{\\mathrm{min}}$ of ${\\mathbb D}$ such that \n$$\nd(m,m_{\\mathrm{min}}) = \\min_{ g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})} d(m, \\rho(g) \\cdot m_{\\mathrm{min}}),\n$$\nwhere $d$ is the distance function defined on $\\mathcal{M}_{m_0}$ with respect to the metric \\eqref{metric}. Furthermore, if $m(t), 0 \\leq t \\leq 1$ is the geodesic connecting $m$ and $m_{\\mathrm{min}}$, then \n$$\nd(m(t),m_{\\mathrm{min}}) = \\min_{g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})} d(m(t), \\rho(g) \\cdot m_{\\mathrm{min}}).\n$$\n\\end{lemma}\n\n\\begin{proof}\nThe first part follows from Proposition~\\ref{convex}. For the second claim, suppose $g_k \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})$ is a sequence such that $$\\lim_{k\\to \\infty} d(m, \\rho(g_k)\\cdot m_{\\rm min}) = \\inf_{g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})} d(m, \\rho(g) \\cdot m_{\\rm min}).$$ Let us denote $m_k= \\rho(g_n) \\cdot m_{\\rm min}$ and choose an admissible normal basis ${\\bf s}^k$ of $m$ which diagonalizes $m_k$. As $G(V)$ (resp. $G_{T^{\\perp}}(V)$) is compact, we can assume that ${\\bf s}^k$ converges to an admissible normal basis ${\\bf s}$ of $m$. On the other hand, as in the proof of Proposition~\\ref{convex}, we can express the geodesic between $m$ and $m_k$ by using a one parameter subgroup of $G^c(V)$ (resp. $G^c_{T^{\\perp}}(V)$) generated by ${\\rm diag}(e^{\\gamma^k_0}, \\cdots , e^{\\gamma^k_N})$ and compute\n$d^2(m,m_k)= \\sum_{i=0}^N |\\gamma_i^k|^2,$\nso that, taking a subsequence, ${\\rm diag}(e^{\\gamma^k_0}, \\cdots , e^{\\gamma^k_N})$ converges to a diagonal matrix ${\\rm diag}(e^{\\gamma_0}, \\cdots , e^{\\gamma_N})$; it defines an element $m_{\\infty} \\in {\\mathcal M}_{m_0}$ such that $m_{\\infty}(s_i,s_j)=0$ for $i\\neq j$ and $m_{\\infty}(s_i,s_i)= e^{-2\\gamma_i}m(s_i,s_i)$. The last conclusion holds easily by using the triangle inequality.\\end{proof}\nThe next result establishes the properness of ${\\mathbb D}$, provided it has critical points on ${\\mathcal M}_{m_0}$. A similar result has been originally established by H.~Luo \\cite{L} in the case when ${\\widetilde \\mathrm{Aut}}_0(X)$ is trivial. \n\\begin{prop}\\label{proper} Suppose $m_{\\min}$ is a minimizer of ${\\mathbb D}$ on $\\mathcal{M}_{m_0}$. For every $C > 0$, there exists $C_1$ such that for any $m \\in \\mathcal{M}_{m_1}$ with the property\n$$\n{\\mathbb D}(m) < {\\mathbb D}(m_{\\min}) + C,\n$$\nthere exists a $g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})$ such that\n$$\nd(m, \\rho(g) \\cdot m_{\\min}) < C_1.\n$$\n\\end{prop}\n\n\\begin{proof}\nSuppose for contradiction that there is a constant $C > 0$ and a sequence $m_i \\in \\mathcal{M}_{m_0}$ such that \n\\begin{equation}\\label{contradiction}\n{\\mathbb D}(m_i) < {\\mathbb D}(m_{\\min}) + C\n\\end{equation}\nand\n\\begin{equation}\\label{contradiction1}\nd(m_i, \\rho(g) \\cdot m_{\\min}) > i\n\\end{equation}\nfor any $g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})$. By Lemma~\\ref{l:2}, for any $i$ there exists $g_i \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})$ such that\n$$\nd(m_i, \\rho(g_i) \\cdot m_{\\min}) = \\min_{g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})} d(m_i, \\rho(g) \\cdot m_{\\min})> i.\n$$\nLet $m_i(t), 0 \\leq t \\leq d_i=d(m_i, m_{\\min})$ be the normal geodesic connecting $m_{\\min}$ and $m_i$. Then, using \\eqref{contradiction}, \\eqref{contradiction1} and Proposition~\\ref{convex}, we get\n\\begin{eqnarray*}\nC & > & {\\mathbb D}(m_i) - {\\mathbb D}(m_{\\min}) \\\\\n& = & \\int_0^{d_i} {\\mathbb D}'(m_i(t)) dt\\\\\n& \\geq & \\int_0^i {\\mathbb D}'(m_i(t)) dt\\\\\n& \\geq & i \\int_0^1 {\\mathbb D}'(m_i(t)) dt\\\\\n&=& i({\\mathbb D}(m_i(1)) - {\\mathbb D}(m_{\\min})).\n\\end{eqnarray*}\nLetting ${\\tilde m}_i = m_i(1)$, we have \n$$\n{\\mathbb D}({\\tilde m}_i) < {\\mathbb D}({\\tilde m}_{\\min}) + \\frac{C}{i}\n$$\nwhile, by Lemma~\\ref{l:2}, \n$$\n1 = d({\\tilde m}_i, m_{\\min}) = \\min_{g \\in \\mathrm{Aut}_X({\\mathcal M}_{m_0}, {\\mathbb D})} d({\\tilde m}_i, \\rho(g) \\cdot m_{\\min}).\n$$\nTaking a subsequence of ${\\tilde m}_i$ converging to a minimizer $m_\\infty$ of ${\\mathbb D}$, we obtain a contradiction (see Lemma~\\ref{l:2}). \\end{proof}\n\n\\section{Proof of Theorem~\\ref{main}}~\\label{s:proof}\nIt is enough to consider the case when the polarized projective manifold $(X,L)$ is the product of two factors $(X_1,L_1)$ and $(X_2,L_2)$. Denote the dimensions of $X, X_1, X_2$ by $n, n_1, n_2$ respectively. Letting $p_i : X \\to X_i$ be the canonical projections, we have $L = \\pi_1^*(L_1) \\otimes \\pi_2^*(L_2)$. \n\nThe holomorphic splitting of the tangent bundle $TX = TX_1 \\oplus TX_2$ induces a product structure $\\widetilde{\\mathrm{Aut}}_0(X) = \\widetilde{\\mathrm{Aut}}_0(X_1)\\times \\widetilde {\\mathrm{Aut}}_0(X_2)$, so we can fix a maximal torus $T \\subset \\widetilde {\\mathrm{Aut}}_0(X)$ of the form $T= T_1 \\times T_2$, where $T_i \\subset \\widetilde {\\mathrm{Aut}}_0(X_i)$ are maximal tori. Taking a common tensor power of the $L_i$'s if necessarily, we will suppose that $(X,L)$ and $(X_i,L_i)$ all satisfy the assumptions made in Section~\\ref{s:relative balanced}. Grauert's direct image theorem for coherent sheaves implies that $V = V_1 \\otimes V_2$ where $V=H^0(X,L)$ and $V_i= H^0(X_i, L_i)$. Notice that if $V_i$ splits under $T_i$ as\n$$\nV_i = \\bigoplus_{k=1}^{\\nu_i} V_i(\\chi^i_k),\n$$\nthen\n$$\nV = \\bigoplus_{j,k} V_1 (\\chi^1_j) \\otimes V_2(\\chi^2_k)\n$$\ngives the decomposition \\eqref{e:split} for $V$ with $\\chi_j^1\\otimes \\chi_i^2 = \\chi_k$. \n\n\n\nLet $m^i_{0}$ be $T_i$-invariant hermitian inner products on $V_i$. Simplifying the notation in Section~\\ref{s:functional}, we let\n$\\mathcal{M}_i$ be the $G_i^c$ (resp. $({G_i})^c_{T_i^{\\perp}})$) orbit of $m^i_{0}$. The tensor product (of hermitian inner products and bases) defines a natural map $\\mathcal{M}_1 \\times \\mathcal{M}_2 \\to {\\mathcal M}$ where ${\\mathcal M}$ is the $G^c$ (resp. $G^c_{T^{\\perp}}$) orbit of $m_0=m^1_{0}\\otimes m^2_{0}$. We define the subspace ${\\mathcal M}_{\\rm prod}$ of decomposable elements of $\\mathcal{M}$ \n$$\n\\mathcal{M}_{\\rm prod} = \\{m =m^1 \\otimes m^2, \\ | \\ m^1 \\in \\mathcal{M}_1, m^2 \\in \\mathcal{M}_2 \\}.\n$$\n\n\\begin{lemma} \\label{subspace} ${\\mathcal M}_{\\rm prod}$ is a closed totally geodesic submanifold of $\\mathcal{M}$ which is stable under the action of $\\rho(\\widetilde{\\mathrm{Aut}}_0(X)) \\cap G^c$. Furthermore, for each $m=m^1\\otimes m^2 \\in {\\mathcal M}_{\\rm prod}$ the induced metric ${\\mathcal FS}(m)= {\\mathcal FS}(m^1) + {\\mathcal FS}(m^2)$ on $X=X_1\\times X_2$ is a product metric.\n\\end{lemma}\n\n\\begin{proof} As $\\widetilde{\\mathrm{Aut}}_0(X)= \\widetilde{\\mathrm{Aut}}_0(X_1) \\times \\widetilde{\\mathrm{Aut}}_0(X_2)$ and we have assumed (by taking a tensor power of $L_i$) that each $\\widetilde{\\mathrm{Aut}}_0(X_i)$ acts on $L_i$, it follows that $\\rho(\\widetilde{\\mathrm{Aut}}_0(X)) \\cap G^c$ preserves ${\\mathcal M}_{\\rm prod}$.\n\nFrom the description of the geodesics of ${\\mathcal M}$ (resp. ${\\mathcal M}_i$) in terms of a 1-parameter subgroups of $G^c$ (resp. $G_i^c$) used in the proof of Proposition~\\ref{convex}, it follows that if $m^i(t)$ is a geodesic of ${\\mathcal M}_i$ ($i=1,2$), then $m(t)=m^1(t)\\otimes m^2(t)$ is a geodesic of ${\\mathcal M}$ which belongs to ${\\mathcal M}$. \n\nThus, in order to established the first part of Lemma~\\ref{subspace}, we only need to show that ${\\mathcal M}_{\\rm prod}$ is a closed \nsubset of ${\\mathcal M}$. Consider a sequence $m_k = m^1_k \\otimes m^2_k \\in {\\mathcal M}_{\\rm prod}$ with $m^i_k \\in \\mathcal{M}_i$. The expression of the geodesic joining $m^i_{0}$ and $m_k^i$ in terms of a 1-parameter subgroup ${\\rm diag}(e^{t\\gamma^i_0}, \\cdots e^{t\\gamma^i_{N_i}})$ of $G_i^c$ (see the previous section) allows us to compute the distance functions $d$ and $d_i$\n\\begin{equation*}\n\\begin{split}\nd_i(m^i_{0}, m^i_k) ^2& = \\sum_{j=0}^{N_i} (\\gamma^i_j)^2, \\\\\nd(m_0, m_k)^2 & = \\sum_{r=0}^{N_1} \\sum_{j=0}^{N_2} (\\gamma^1_r + \\gamma^2_j)^2= N_2 d_1(m_{h_1}^1,m^1_k)^2 + N_1 d_2(m_{h_2}^2,m^2_k)^2,\n\\end{split}\n\\end{equation*}\nwhere we have used that $\\gamma^i_j$ satisfy $\\sum_{j=0}^{N_i} \\gamma_j^i=0$ for $i=1,2$. This completes the first part of the Lemma.\n\nThe final claim is a direct consequence of \\eqref{potential} and the fact that if we have chosen $h=h_1\\otimes h_2$ where $h_i$ is a $T_i$-invariant hermitian metric on $L_i$, then the curvature is $\\omega= \\omega_1 + \\omega_2$. \\end{proof}\n\n\\begin{prop}\\label{split}\nFor any critical point $m$ of ${\\mathbb D}$ on ${\\mathcal M}$, the induced K\\\"ahler metric on $X=X_1 \\times X_2$ is compatible with the the product structure. \n\\end{prop}\n\\begin{proof} Any critical point of ${\\mathcal M}$ must necessarily be a minimizer by Proposition~\\ref{convex}. Let $m_{\\min} $ be such a minimizer. We pick a sequence $m_k \\in {\\mathcal M}_{\\rm prod}$ such that\n$$\n\\lim_{k \\rightarrow \\infty} \\mathbb{D} (m_k) = \\inf_{m \\in {\\mathcal M}_{\\rm prod}} \\mathbb{D} (m).\n$$\nSince the functional ${\\mathbb D}$ defined on $\\mathcal{M}$ is proper in the sense of Proposition~\\ref{proper}, there exist $g_i \\in \\mathrm{Aut}_X({\\mathcal M}, {\\mathbb D})$ such that \n$$\nd(\\rho(g_i)^{-1} \\cdot m_{\\min}, m_i) = d(m_{\\min}, \\rho(g_k) \\cdot m_k) < C_1\n$$\nfor all $i$. Putting ${\\tilde m}_i = \\rho(g_i) \\cdot m_i$, we know by Lemma~\\ref{subspace} that $ {\\tilde m}_i \\in {\\mathcal M}_{\\rm prod}$. Taking a convergent subsequence of ${\\tilde m}_k$ and using the closeness of ${\\mathcal M}_{\\rm prod}$ (see Lemma~\\ref{subspace}), there exists $m \\in {\\mathcal M}_{\\rm prod}$ such that\n$$\n{\\mathbb D}(m) = \\min_{\\bar{m} \\in {\\mathcal M}_{\\rm prod}} {\\mathbb D}(\\bar{m}).\n$$\n\nLet $m=m^1\\otimes m^2$ be a minimizer of ${\\mathbb D}$ on ${\\mathcal M}_{\\rm prod}$. We claim that $m^i$ is a critical point of the corresponding functional ${\\mathbb D}_i$ on $\\mathcal{M}_i$. Without loss of generality, we only check this for $m^1$. Suppose $m^1(t)$ is a geodesic starting from $m^1(0)=m^1$ in $\\mathcal{M}_1$, expressed in terms of a 1-parameter subgroup ${\\rm diag}(e^{t\\gamma_0^1}, \\cdots, e^{t \\gamma_{N_1}^1})$ of $G_1^c$ (resp. $(G_1)^c_{T_1^{\\perp}}$): there exists an admissible orthonormal basis ${\\bf s}^1 =\\{ s^1_i, 0 \\leq i \\leq N_1\\} $ of $m^1$ such that ${\\bf s}^1(t)=\\{e^{t\\gamma_0^1}s^1_1, \\ldots, e^{t\\gamma_{N_1}^1}s_{N_1} \\}$ is an admissible orthonormal basis for $m^1(t)$. Let ${\\bf s}^2 =\\{ s^2_j, 0 \\leq j \\leq N_2\\} $ be an admissible orthonormal basis for $m^2$. Then $m(t) = m^1(t) \\otimes m^2$ is a geodesic in $\\mathcal{M}$ starting from $m$ and ${\\bf s}^1(t) \\otimes {\\bf s}^2$ is an admissible orthonormal basis for $m(t)$. Since $m(t) \\in {\\mathcal M}_{\\rm prod}$ and $m$ is a minimizer of $\\mathbb{D}$ on ${\\mathcal M}_{\\rm prod}$, we have\n\\begin{eqnarray*}\n0 &=& \\mathbb{D}'(0)\\\\\n&=& \\int_X \\frac{\\sum_{i=0}^{N_1} \\sum_{j=0}^{N_2} 2 \\gamma_i^1 |s^1_i|^2_{h_1} |s^2_j|^2_{h_2}}{\\sum_{i=0}^{N_1} \\sum_{j=0}^{N_2} |s^1_i|^2_{h_1} |s^2_j|^2_{h_2}} \\mathcal{FS}(m)^{n_1+n_2}\\\\\n&=& \\big( \\int_{X_1} \\frac{\\sum_{i=0}^{N_1} 2 \\gamma_i^1 |s^1_i|^2_{h_1} }{\\sum_{i=0}^{N_1} |s^1_i|^2_{h_1}} \\mathcal{FS}(m^1)^{n_1}\\Big) \\times \\Big( \\int_{X_2} \\frac{\\sum_{j=0}^{N_2} |s^2_j|^2_{h_2}}{ \\sum_{j=0}^{N_2} |s^2_j|^2_{h_2}} \\mathcal{FS}(m2)^{n_2}\\Big)\\\\\n&=& C \\int_{X_1} \\frac{\\sum_{i=0}^{N_1} 2 \\gamma_i^1 |s^1_i|^2_{h_1} }{\\sum_{i=0}^{N_1} |s^1_i|^2_{h_1}} \\mathcal{FS}(m^1)^{n_1}= C \\ \\mathbb{D}_1'(0),\n\\end{eqnarray*}\nwhere $C$ is a strictly positive constant. We conclude that $m^1$ is a critical point of $\\mathbb{D}_1$ on $\\mathcal{M}_1$ by using Proposition~\\ref{cha}. Conversely, Proposition~\\ref{cha} also shows that $m$ is a critical point of ${\\mathbb D}$ on $\\mathcal{M}$. Now, by Proposition~\\ref{convex}, the induced K\\\"ahler metrics on $X$ by the critical points of ${\\mathbb D}$ are isometric under the action of $\\widetilde{\\mathrm{Aut}}_0(X)= \\widetilde{\\mathrm{Aut}}_0(X_1) \\times \\widetilde{\\mathrm{Aut}}_0(X_2)$ so, in particular, to the induced product K\\\"ahler metric by $m=m^1\\otimes m2$ (see Lemma~\\ref{subspace}), which completes the proof.\\end{proof}\n\nAs the existence of critical points of ${\\mathbb D}$ is independent of the choice of orbits (see Lemma~\\ref{gabor} and Remark~\\ref{chow-norm}), we obtain as an immediate corollary of Proposition~\\ref{split}\n\\begin{thm}\\label{reduced} Suppose $X$ admits a balanced K\\\"ahler metric relative to $T$ in $2\\pi c_1(L)$. Then there exits a balanced K\\\"ahler metric relative to $T$ in $2\\pi c_1(L)$ compatible with the product structure $X=X_1\\times X_2$.\n\\end{thm}\n\n\\noindent{\\it Proof of Theorem~\\ref{main}.} Combining Theorem~\\ref{reduced} with Theorem~\\ref{do-mabuchi} and Propositions 1 and 2 yields the proof of Theorem~\\ref{main}(i). In order to prove Theorem~\\ref{main}(ii), we use Theorem~\\ref{mabuchi} with $T$ being the connected component of the centre of $\\widetilde{{\\rm Aut}}_0(X)$, so that, by the assumption, for one of the factors, $(X_1,L_1)$ say, $T_1=\\{ {\\rm Id} \\}$. It is not hard to see that in this case {\\it each} $G^c$ orbit of admissible hermitian inner products on $V=V_1\\otimes V_2$ contains products $m=m^1\\otimes m^2$. (The latter is not true in general.) We can then apply Proposition~\\ref{split}. $\\Box$\n\n\\begin{rem} The above arguments and the uniqueness established in Lemma 2 would imply the splitting property should Conjecture \\ref{relative-stability} be true.\n\\end{rem}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{The concept of tipping cascades}\n\nHuman--induced impacts on the Earth system increasingly endanger the integrity of the Earth's climate system and some of its most vulnerable components and processes, the so--called tipping elements \\cite{lenton2008tipping}. Lately, it has been argued that the risk of potential tipping events or even cascading transitions up to a global cascade is rising under ongoing anthropogenic global warming \\cite{steffen2018trajectories,lenton2019climate}. While this is the case, there is considerable debate about the nature of tipping cascades within the scientific community itself and cascading tipping dynamics have been described rather roughly in the recent literature \\cite{steffen2018trajectories,lenton2019climate,lenton2020tipping,lenton2013origin,hughes2013multiscale,rocha2015regime,rocha2018cascading,barnosky2012approaching,brook2013does}. \n\nThe term cascade is used in various fields for a certain class of dynamics possibly exhibited by interacting (sub--)systems. It generally describes the sequential occurrence of similar events (event A is followed by event B which is followed by event C etc.). This sequence of events does not necessarily have to be causal opposed to when event A directly causes event B in a domino effect. The notion of a domino effect is sometimes used synonymously to the term cascade. Examples of cascades comprise cascading failures leading to the collapse of power grids as relevant physical infrastructure networks \\cite{watts2002simple,buldyrev2010catastrophic,gao2011robustness,gao2012networks,hu2011percolation}. Such a cascade may occur as an initial failure increases the likelihood of subsequent failures \\cite{watts2002simple}. In contrast, an initial failure may directly lead to the failure of dependent nodes \\cite{buldyrev2010catastrophic}. \n\nAlong these lines, cascading tipping events or regime shifts are increasingly discussed following the rising awareness of a highly interconnected world in the Anthropocene \\cite{helbing2013globally}. Tipping elements possibly undergoing a transition into a qualitatively different state after the crossing of some critical threshold were identified e.g. in ecology and climate system science \\cite{lenton2008tipping,scheffer2003catastrophic,scheffer2001catastrophic} and comprise, among others, shallow lakes transitioning from a clear to a turbid state \\cite{scheffer1989alternative,scheffer1993alternative}, coral reefs \\cite{hughes1994catastrophes}, the Atlantic Meridional Overturning Circulation \\cite{rahmstorf2005thermohaline,stommel1961thermohaline} and the continental ice sheets on Greenland \\cite{robinson2012multistability} and Antarctica \\cite{garbe2020hysteresis}. \n\nIn the climate system, multiple interactions between large--scale tipping elements have been identified \\cite{kriegler2009imprecise,caesar2018observed,rahmstorf2015exceptional,swingedouw2008antarctic,parsons2019influence,duque2019tipping}. For example, the Atlantic Meridional Overturning Circulation may slow down due to increasing meltwater flux originating from the Greenland Ice Sheet \\cite{caesar2018observed,rahmstorf2015exceptional}. Potential drying over the Amazon rainforest basin leading to loss of rainforest resilience may be influenced by the Atlantic Meridional Overturning Circulation \\cite{parsons2019influence} on the one hand and the El--Ni\u00f1o Southern Oscillation on the other hand \\cite{duque2019tipping}. Rocha et al.~\\cite{rocha2018cascading} identified potential links between ecological systems with alternative states such as the interaction of eutrophication and hypoxia or coupled shifts in coral reefs and mangrove systems. \n\nTipping interactions do not only exist across different large--scale systems, but span various spatial scales as exemplified by spatially extended (and heterogeneous) ecosystems \\cite{lenton2020tipping,rocha2018cascading}. On a local scale, confined ecosystems such as a shallow lake, in fact, consist of discrete units connected through dispersion or other exchange processes with each unit potentially exhibiting alternative stable states \\cite{van2005implications,dakos2010spatial,van2015resilience}. Regionally, regime shifts may propagate from one ecosystem entity to the other transmitted, among others, via small streams and rivers \\cite{hilt2011abrupt,scheffer2004ecology,van2017regime}, moisture recycling \\cite{lenton2020tipping,wunderling2020network,zemp2014importance,zemp2017self} or biotic exchange through e.g. larvae \\cite{brook2013does,van2015resilience,scheffer2012anticipating,lundberg2003mobile}. \n\nMotivated by these and further suggested tipping element interactions, cascading effects arising as potential dynamics have been discussed \\cite{steffen2018trajectories,lenton2019climate,lenton2020tipping,lenton2013origin,hughes2013multiscale,rocha2015regime,rocha2018cascading} as a possible mechanism for creating a potential planetary--scale tipping point (of the biosphere) \\cite{lenton2013origin,hughes2013multiscale,barnosky2012approaching,brook2013does}. Lenton et al.~\\cite{lenton2019climate} stated that we may approach a global cascade of tipping points via the progressive activation of tipping point clusters \\cite{schellnhuber2016right} through the increase of global mean temperature. This could potentially lead to undesirable hothouse climate trajectories \\cite{steffen2018trajectories}. However, it remains unclear whether and how cascade--like dynamics within the Earth system is promoted by the direction and strength of the existing feedbacks \\cite{lenton2020tipping,lenton2013origin,kriegler2009imprecise,wunderling2021modelling}. \n\nRecently, first conceptual steps \\cite{brummitt2015coupled,abraham1991computational} have been undertaken to determine whether the network of Earth system tipping elements is capable to produce global tipping cascades \\cite{wunderling2020interacting,gaucherel2017potential}. Using still conceptual, but process--based models, Dekker et al.~\\cite{dekker2018cascading} demonstrated a possible sequence of tipping events in a coupled system of the Atlantic Meridional Overturning Circulation and El--Ni\u00f1o Southern Oscillation. Social costs of future climate damages caused by carbon emissions originating from domino effects of interacting tipping elements were studied using an integrated assessment model \\cite{lemoine2016economics,cai2016risk}. Earlier, the propagation of critical transitions in lake chains as an ecological example was analyzed, coupling established models of shallow lakes by a unidirectional stream or via diffusion processes \\cite{van2005implications,hilt2011abrupt}. The effect of spatial heterogeneity and connectivity of bistable patches on the overall ecosystem response was further studied by the application of simple models for eutrophication and grazing of a (logictically--growing) resource \\cite{van2005implications,dakos2010spatial}. In addition, examples beyond the biogeophysical Earth system possibly giving rise to the propagation of critical transitions were proposed such as coupled subsystems in the fields of economics and finance \\cite{lenton2020tipping,brummitt2015coupled}.\n\n\\subsection{Descriptions of tipping cascades vary across the literature}\nHowever, tipping cascades or, more generally, patterns of multiple tipping dynamics discussed to arise from the interaction of tipping elements are often loosely described suffering a similar fate as the ancestral 'tipping point' concept \\cite{van2016you}. We encountered important differences across the description of tipping cascades in the recent literature. These differences are in particular related to whether causality is a necessary ingredient for a cascade or not. For example, the pattern where tipping of one system causes the tipping of another system is described as domino dynamics or tipping cascade by Lenton et al.~\\cite{lenton2020tipping}. The propagation of regime shifts by an initial critical transition causing a following one is underpinned by generalized tipping element interactions and termed a cascade by Brummitt et al.~\\cite{brummitt2015coupled}. By comparison, the term cascading tipping is used for a sequence of abrupt transitions in Dekker et al.~\\cite{dekker2018cascading} that may not necessarily be causal. This notion of cascading tipping is exemplary applied to the Atlantic Meridional Overturning Circulation and El--Nino Southern Oscillation as climatic tipping elements \\cite{dekker2018cascading}. Furthermore, and not restricted to causal events, an effect of one regime shift on the occurrence of another regime shift is suggested as cascading in Rocha et al.~\\cite{rocha2018cascading} and confirmed to connect ecological regime shifts such as fisheries collapse and transitions of kelp, mangrove and seagrass ecosystems. \n\nHere we systematically identify and characterize patterns of multiple tipping dynamics such as a domino cascade, a two phase cascade and a joint cascade, which arise in a previously studied system of idealized interacting tipping elements \\cite{brummitt2015coupled,abraham1991computational} (section~\\ref{sec:res}). In particular, these patterns of multiple tipping dynamics differ in the way of how the critical transition propagates from one tipping element to another. The domino cascade, the two phase cascade and the joint cascade are related to the varying descriptions of tipping cascades in the literature and examples of multiple tipping events with comparable characteristics in the Earth system are given. Furthermore, we address the potential for intervention and anticipation by common early warning indicators based on critical slowing down (see Supplementary Material for details). Implications of the distinct patterns of multiple tipping for the resilience of the Earth system, limitations of studying idealized interacting tipping elements and necessary future research are discussed (section~\\ref{ref:disc}). \n \n\n\\section{Patterns of multiple tipping in a model of idealized interacting tipping elements}\n\\label{sec:res}\nIn the following, we present distinct patterns of multiple tipping dynamics, which emerge from the linear bidirectional coupling of two idealized tipping elements (figure~\\ref{fig:fig_1}, \\cite{brummitt2015coupled,abraham1991computational}). Each tipping element depends on its control parameter (or driver), the variation of which may induce a critical transition from a normal to an alternative state with the crossing of a critical control parameter threshold. We consider homogeneous tipping elements, i.e. both tipping elements undergo a critical transition at the same control parameter threshold and on the same intrinsic tipping time scales. A linear coupling term captures the interaction of the tipping elements following Wunderling et al.~\\cite{wunderling2020interacting}, where the state of one tipping element is added to the control parameter of another, coupled tipping element. We refer to Wunderling et al.~\\cite{wunderling2020interacting} and Klose et al.~\\cite{klose2020emergence} for a detailed description of the model of idealized interacting tipping elements. \n\nThe patterns of multiple tipping dynamics described below and illustrated in figure~\\ref{fig:fig_2} originate from different pathways through the control parameter space of both tipping elements: The control parameter~$c_2$ of subsystem~$X_2$ as \\textit{following} tipping element is kept constant at distinct levels (figure~\\ref{fig:fig_2}, going from top to bottom). The control parameter~$c_1$ of subsystem~$X_1$ as \\textit{evolving} tipping element is increased (figure~\\ref{fig:fig_2}, going from left to right) sufficiently slowly such that this subsystem can follow its (moving) equilibrium. In other words, by a separation of the intrinsic system time scale and the time scale of the forcing, the system can be regarded as a fast--slow system \\cite{kuehn2011mathematical}, where the change in the forcing of the system is slow compared to the intrinsic system time scale. We observe the following three qualitatively different dynamic patterns of multiple tipping: \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale = 0.4]{figure1a_1b_1c.pdf}\n\t\\caption{(a) \\& (b): Bifurcation diagram of the idealized tipping elements~(TE)~$X_1$ (a) and $X_2$ (b). The respective differential equation is of the form $\\frac{\\rmd x_1}{\\rmd t} =-x_1^3+x_1+c_1+\\frac{1}{2}d_{21}(x_2+1)$ for subsystem~$X_1$ and $\\frac{\\rmd x_2}{\\rmd t}=-x_2^3+x_2+c_2+\\frac{1}{2}d_{12}(x_1+1)$ for subsystem~$X_2$. Note that for determining the bifurcation diagram of the idealized tipping elements~$X_1$ and $X_2$ the coupling term is not taken into account, i.e. the uncoupled case with $d_{21}= 0$ and $d_{12}= 0$ is shown here. Below the critical threshold~$c_{i_{\\rm{crit}}}$, $i = 1,2$, there exist two stable fixed points. As soon as the control parameter~$c_i$ transgresses its critical value~$c_{i_{\\rm{crit}}}$, a fold--bifurcation occurs and the system tips from the lower (normal) state~$x_{i^-}^*$ to the upper (alternative) state~$x_{i^+}^*$. (c) Sketch of the potential landscape of the two subsystems in case they do not interact shown as a ball--and--cup diagram.}\n\t\\label{fig:fig_1} \n\\end{figure}\n\n\\subsection{Two phase cascade (figure~\\ref{fig:fig_2}(a))}\n\nAn increase of the control parameter~$c_1$ across its threshold and the resulting critical transition of subsystem~$X_1$ is not sufficient to directly trigger a critical transition in subsystem~$X_2$. The system converges intermediately to a stable fixed point (as seen in the phase space portraits) and only a further increase of the control parameter~$c_1$ can initiate the critical transition in subsystem~$X_2$ by the loss of the intermediately occupied stable fixed point. Thus, by limiting the further increase in the control parameter~$c_1$ after the first tipping event of subsystem~$X_1$, a full two phase cascade can be mitigated. We can identify the two phase cascade with the cascade described and simulated in Dekker et al.~\\cite{dekker2018cascading}. Within the climate system, a stepwise change in the oxygen isotopic ratio at the Eocene--Oligocene transition may be interpreted as a two phase cascade of the Atlantic Meridional Overturning Circulation as the evolving tipping element and the Antarctic Ice Sheet as the following tipping element in response to a slowly decreasing atmospheric carbon dioxide concentration \\cite{dekker2018cascading,tigchelaar2011new}. \n\nAn increasingly slower recovery from perturbations and thus an increase in common statistical indicators such as autocorrelation and variance are observed for subsystem~$X_1$ on the approach of the two phase cascade in a \\textit{pre--tipping time span} before the critical transition of subsystem~$X_1$ (Supplementary Material, figure~S1--S3). In contrast, for subsystem~$X_2$, an increasingly slower recovery from perturbations as well as increasing autocorrelation and variance can not be detected in the pre--tipping time span prior to the critical transition of subsystem~$X_1$ (Supplementary Material, figure~S1--S3). However, given the intermediate convergence to a stable fixed point after the critical transition of subsystem~$X_1$ and prior to the critical transition of subsystem~$X_2$, an \\textit{intermediate time span} offers the possibility to indicate the upcoming critical transition of subsystem~$X_2$ in the two phase cascade. A step--like change to a relatively higher level of the statistical indicators for subsystem~$X_2$ compared to the respective level in the pre--tipping time span is observed (Supplementary Material, figure~S2--S3, compare also \\cite{dekker2018cascading}), indicating an increased vulnerability of subsystem~$X_2$ to a critical transition. The height of the step--like change in the statistical indicators varies with the magnitude of the constant control parameter~$c_2$ as a consequence of an increasingly slower recovery from perturbations in the intermediate time span with increasing magnitude of the constant control parameter~$c_2$. This observation corresponds to the rotation of the eigenvectors and the change in the eigenvalue magnitude of the system of interacting tipping elements, which determine the magnitude and direction of the recovery to perturbations and hence critical slowing down prior to a bifurcation--induced critical transition (\\cite{boerlijst2013catastrophic,dakos2018identifying}, Supplementary Material). However, no threshold, i.e. a height of the step--like change above which this following tipping occurs, can be observed but it rather is a continuous and relative quantity. In other words, a step--like change of the statistical indicators (though comparably smaller) may also be present after the critical transition of subsystem~$X_1$ even if a critical transition of subsystem~$X_2$ does not follow. Thus, to use this height of the step--like change to clearly indicate an upcoming following transition may be difficult in practice.\n\n\\subsection{Domino cascade (figure~\\ref{fig:fig_2}(b))}\n\nFor a slightly elevated level of the constant control parameter~$c_2$, the increase of the control parameter~$c_1$ across its threshold and the corresponding critical transition of subsystem~$X_1$ towards its alternative state is sufficient to trigger a critical transition of subsystem~$X_2$. Note that, in contrast to the two phase cascade, no further increase of the control parameter~$c_1$ is necessary to observe the domino cascade, but the tipping of one subsystem (the evolving tipping element) directly causes and initiates the tipping of another (the following tipping element). This corresponds to the description of a tipping cascade given in Lenton et al.~\\cite{lenton2020tipping} and Brummitt et al.~\\cite{brummitt2015coupled} and the general notion of a domino effect including causality \\cite{hornby2015dict}. A notable feature is the expected path of the system in the phase space. Even though the intermediately occupied stable fixed point involved in the two phase cascade is absent, it still influences the dynamics (see phase space, figure~\\ref{fig:fig_2}(b)) as a 'ghost' (e.g. \\cite{strogatz1989predicted,sardanyes2006ghosts,sardanyes2009ghosts,duarte2012chaos}). As demonstrated recently in a conceptual model, domino cascades may propagate through tipping elements in the Earth system, such as the large ice sheets on Greenland and West Antarctica and the Atlantic Meridional Overturning Circulation \\cite{wunderling2020interacting, wunderling2020basin}. \n\nA domino cascade may not be preceded clearly by the increase of the common early warning indicators and relying on these indicators may lead to an unexpected following critical transition of the following tipping element. An increasingly slower recovery from perturbations and thus increasing autocorrelation and variance as common statistical indicators are observed for subsystem~$X_1$ on the approach of the domino cascade in the pre--tipping time span (Supplementary Material, figure~S1--S3). The statistical indicators for subsystem~$X_2$ remain constant though on a relatively higher level than for the two phase cascade in the pre--tipping time span (Supplementary Material, figure~S1--S3). However, no clear intermediate time span prior to the critical transition of subsystem~$X_2$ exists allowing for an additional detection of early warning signals as for the two phase cascade. \n\n\\subsection{Joint cascade (figure~\\ref{fig:fig_2}(c))}\n\nSubsystem~$X_1$ and subsystem~$X_2$ may tip jointly with a possible trajectory evolving close to the phase space diagonal for an increase of the control parameter~$c_1$ across its threshold as opposed to the other two multiple tipping patterns. Such a joint cascade is observed with a strongly elevated level of the constant control parameter~$c_2$. The critical transitions of the respective subsystems cannot be clearly distinguished with regard to their order of tipping. Though the case of a joint cascades has not been treated explicitly in the recent literature on interacting tipping elements, a similar behaviour may be observed in spatially extended bistable ecosystems subject to regime shifts \\cite{van2005implications,dakos2010spatial}.\n\nFor both subsystems, a slower recovery from perturbations is expected prior to their joint tipping (Supplementary Material, figure~S1--S2). For subsystem~$X_1$ autocorrelation and variance increase on the approach of the joint cascade with increasing control parameter~$c_1$. Subsystem~$X_2$ exhibits a relatively high constant level of these statistical indicators prior to the joint cascade corresponding to the level of the constant control parameter~$c_2$ and indicating the vulnerability of this subsystem to critical transitions (Supplementary Material, figure~S3). \n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[scale = 0.5]{figure2a_2b_2c.pdf}\n\t\\caption{Three different types of tipping cascades depicted as three different situations. From left to right, the critical parameter~$c_1$ of the evolving tipping element~(TE)~$X_1$ is driven closer to and over its tipping point (compare to figure~\\ref{fig:fig_1}). From top to bottom, the critical parameter~$c_2$ of the following tipping element~(TE)~$X_2$ is also driven closer to, but not across, its tipping point. In this setting, three different patterns of multiple tipping or cascades can occur. (a) Two phase cascade: The first subsystem~$X_1$ tips and is then shifted closer towards subsystem~$X_2$ by an increase of the control parameter~$c_1$. Then subsystem~$X_2$ tips as well. (b) Domino cascade: The subsystems~$X_1$ and $X_2$ are closer together than in the two phase cascade such that a tipping of subsystem~$X_1$ (middle panel) is sufficient to trigger a critical transition in subsystem~$X_2$. (c) Joint cascade: The two subsystems are very close to each other such that the beginning of a tipping event in subsystem~$X_1$ immediately causes the tipping of the second subsystem~$X_2$ and the tipping events cannot be distinguished. The respective stable fixed point attractors and phase diagrams are shown below the domino sketches. Orange dots represent stable fixed points, while unstable fixed points are given by red dots. The background colour indicates the normalized speed $v = \\sqrt{\\dot{x}_{1}^2+\\dot{x}_{2}^2}\/v_{max}$ going from close to zero (purple) to fast (yellow--green).\n}\n\t\\label{fig:fig_2} \n\\end{figure}\n\n\\section{Discussion}\n\\label{ref:disc}\nStudying a system of idealized interacting tipping elements \\cite{brummitt2015coupled,abraham1991computational}, qualitatively different dynamic patterns of multiple tipping were identified and characterized as a two phase cascade, a domino cascade and a joint cascade. \n\nThe various patterns of multiple tipping originating from two idealized interacting tipping elements are related to different, though simplified and specific pathways through the control parameter space. In the end, the control parameter evolution determines the emergence of the specific system behavior, which may be a domino cascade, a two phase cascade or a joint cascade. In other words, the control parameter evolution, i.e., the evolution of the forcing, can therefore determine the characteristics of multiple tipping that are observed. However, other factors such as the strength and the sign of coupling are as well decisive for the emergence of tipping cascades. Moreover, in more complex systems, control parameters can not be treated separately for each tipping element and drivers may be shared \\cite{rocha2018cascading}.\n\nThe different observed patterns of multiple tipping may have implications for the mitigation of tipping by controlling the respective drivers. A limitation of the forcing can prevent the two phase cascade since a critical transition of the evolving tipping element is not sufficient for the spread of a tipping event to a following subsystem. Instead, the critical transition needs to be followed by a further evolution of the respective subsystem's state before a following critical transition is initiated. However, in a domino cascade an initial critical transition of the evolving tipping element is sufficient to trigger a slightly delayed but inevitable following critical transition of another tipping element.\n\nIn addition, the potential success of anticipating the emergence of tipping cascades through early warning indicators based on critical slowing down \\cite{wissel1984universal,scheffer2009critical,lenton2011early} was assessed and demonstrated to differ across the patterns of multiple tipping (see Supplementary Material). Using insights of Boerlijst et al.~\\cite{boerlijst2013catastrophic} and Dakos~\\cite{dakos2018identifying} on critical slowing down in multi--component systems in relation to the eigenvector orientation, it is shown how critical slowing down and common statistical indicators for the anticipation of critical transitions are related to the rotation of eigenvectors and the change in the eigenvalues' magnitude. Thereby, the analysis of statistical properties of the two phase cascade in Dekker et al.~\\cite{dekker2018cascading} is expanded. We find that these common statistical indicators based on critical slowing down may fail for upcoming domino cascades in a system of idealized interacting tipping elements. While increasing autocorrelation and variance are observed for the evolving tipping element on the approach of the domino cascade, constant levels of these statistical indicators were determined for the following tipping element. In the case of a two phase cascade or a joint cascade, the critical slowing down based indicators express some degree of vulnerability (or resilience) in the system of interacting tipping elements. However, their application may be unfeasible in practice. In particular, for the two phase cascade, the critical transition of the evolving tipping element is preceded by increasing autocorrelation and variance of the respective subsystem, while a step--like change towards a relatively higher level of the statistical indicators in the intermediate time span is found for the following tipping element. The joint cascade may be conceivable with a raised but constant level of autocorrelation and variance for the following tipping element accompanied by an increase of statistical indicators for the evolving tipping element. With the slower recovery from perturbations for both tipping elements, correlations between the subsystems' time series comparable to the application of spatial early warning signals \\cite{dakos2010spatial,dakos2011slowing,donangelo2010early,guttal2009spatial,kefi2014early} may unfold. \n\nAs these very specific and simplified scenarios of control parameter evolution demonstrate that an increase of autocorrelation and variance prior to multiple tipping events cannot necessarily be expected, these common early warning indicators should not be relied on as the only way of anticipating cascading critical transitions in systems of interacting tipping elements. Additionally taking into account often referenced limitations, false alarms and false positives in the application of critical slowing down based indicators to individual tipping elements and the anticipation of upcoming critical transitions \\cite{boettiger2013early,dakos2015resilience,ditlevsen2010tipping}, it seems to be necessary to invoke a combination of process-\u2013based modelling accompanied by monitoring the system under investigation resulting in predictions as well as data--driven techniques \\cite{dakos2015resilience,ditlevsen2010tipping,dakos2012methods} to detect upcoming multiple transitions and, in particular, the domino cascade. \n\nNote that the presented discussion is restricted to bifurcation--induced tipping with a relatively weak noise and a sufficiently slow change of the tipping element driver is applied. Hence, our examination of tipping cascades excludes early tipping \\cite{lohmann2021abrupt} and flickering \\cite{dakos2013flickering} due to noise as well as rate--induced effects, which will further influence the presented patterns of multiple tipping, their characteristics such as the intermediate time span of the two phase cascade and hence the potential for anticipation and mitigation. In a related stochastic system, similar patterns were demonstrated as fast and slow domino effects \\cite{ashwin2017fast}. The patterns of multiple tipping are expected to change in response to a fast change of the tipping element driver with respect to the intrinsic response time scales, which cannot be ruled out given the current unprecedented anthropogenic forcing of the biogeophysical Earth system \\cite{joos2008rates,zeebe2015anthropogenic}. In addition, rate--induced transitions may occur \\cite{ashwin2012tipping,wieczorek2011excitability} as suspected based on modelling studies for the Atlantic Meridional Overturning Circulation \\cite{alkhayuon2019basin,stocker1997influence,lohmann2021risk}, predator--prey systems \\cite{o2019tipping,scheffer2008pulse,siteur2016ecosystems} and for the release of soil carbon in the form of the compost--bomb instability \\cite{wieczorek2011excitability,luke2011soil} and may further complicate the early warning of cascading tipping \\cite{lohmann2021abrupt,ritchie2016early}. Heterogeneity across the response of tipping elements to the same control parameter level \\cite{brook2013does,scheffer2012anticipating} and in the intrinsic time scales of tipping \\cite{wunderling2020interacting,ritchie2021overshooting,hughes2013living} was neglected. \n\nFinally, it is assumed that the long--term behaviour of many real\u2013world systems in terms of the system's state such as the overturning strength of the Atlantic Meridional Overturning Circulation \\cite{stommel1961thermohaline,cessi1994simple}, the ice volume of the Greenland Ice Sheet \\cite{levermann2016simple} and the algae density in shallow lakes \\cite{scheffer1989alternative,scheffer1993alternative} can be qualitatively captured by the studied idealized tipping elements featuring a fold bifurcation as tipping mechanism. However, biogeophysical and biogeochemical processes involved in the behaviour of these real\u2013-world systems and included in some more complex climate models may either give rise to further types of cascading tipping or may dampen the overall possibilities of tipping behavior \\cite{wunderling2020interacting}. \n\n\\section{Conclusion}\n\nQualitatively different patterns of multiple tipping dynamics in interacting nonlinear subsystems of the climate and ecosystems have been identified in this work. These multiple tipping patterns may emerge as illustrated in a system of idealized interacting tipping elements and include the cases of joint cascades, domino cascades and two phase cascades. As described in Lenton et al.~\\cite{lenton2020tipping} and Brummitt et al.~\\cite{brummitt2015coupled} as well as corresponding to the general notion of a domino effect \\cite{hornby2015dict}, tipping of one subsystem causes or triggers the tipping of another subsystem in a domino cascade. In addition, we find a two phase cascade corresponding to the tipping pattern presented in Dekker et al.~\\cite{dekker2018cascading}. While we reveal that it may be possible to find critical slowing down based early warning indicators for the two phase cascade, such indicators can fail in the case of a domino cascade. \n\nHowever, our results are limited by the conceptual nature of the system investigated here. In particular, in more complex and process--detailed models of tipping elements the respective nonlinear properties might be smeared out and the presented characteristics of the emerging multiple tipping patterns might be altered due to processes such as strong noise, interactions to other system components or further biogeophysical processes that are not modelled here.\n\nSince cascading tipping dynamics have been described rather roughly in the recent literature and the presented patterns of multiple tipping dynamics differ in the potential of their mitigation and anticipation, we suggest to be more precise in future discussions on potential dynamics arising from the interaction of tipping elements and, in particular, on tipping cascades. \nIn the future, a quantitative assessment of interacting tipping elements with an ongoing improvement of their representation in complex (climate) models e.g. by including interactive evolving ice sheets into Earth system models \\cite{kreuzer2021coupling} as well as the additional use of paleoclimate data \\cite{thomas2020tipping} may help to reduce uncertainties on the preconditions for the emergence of tipping cascades and possible early warning indicators based on process--understanding. To the end, these insights may contribute to reflections on the boundaries of the safe--operating space for humanity, and to a better understanding of Earth system resilience with respect to anthropogenic perturbations more generally. \n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nWe consider lossy compression of a binary symmetric source\n(BSS) using a low-density generator-matrix (LDGM) code as shown in\nFigure~\\ref{fig:ldgmtanner}. More precisely, let $S \\in \\GF^m$ represent\nthe binary source of length $m$. We have $S=\\{S_1,S_2,\\dots,S_m\\}$,\nwhere the $\\{S_i\\}_{i=1}^{m}$ are iid random variables with\n$\\prob\\{S_i=1\\}=\\frac12$, $i\\in [m]$. Let $\\mathcal{S}$ denote the set\nof all source words.\n\\begin{figure}[htp]\n\\begin{center}\n\\input{ps\/ldgmtanner}\n\\end{center}\n\\caption{\\label{fig:ldgmtanner}\nThe Tanner graph corresponding to a simple LDGM code used for\nlossy compression of a BSS. We have $m=7$, $R=\\frac47$, \nand $L(x)=x^3$.\n}\n\\end{figure}\n\nGiven a source word $s \\in {\\mathcal S}$, we compress it by mapping it to\none of the $2^{m R}$ index words $w \\in {\\mathcal W} = \\GF^{m R}$, where\n$R$ is the {\\em rate}, $R \\in [0, 1]$. \nWe denote this encoding map by\n$\\encoder: s \\mapsto W$ (the map can be random). The reconstruction is\ndone via an LDGM code determined by a sparse binary $m R \\times m$ generator\nmatrix $G$. Let $\\hat{s}$ denote the reconstructed word associated to\n$w$. We have $\\hat{s} = w G$. We denote this decoding map by $\\decoder:\nw \\mapsto \\hat{s}$. Let $\\code$ denote the code, $\\code=\\{\\hat{s}^{(1)}, \\dots,\n\\hat{s}^{(2^{m R})}\\}$, $\\hat{s}^{(i)} \\in \\GF^m$. The codewords are not necessarily distinct.\n\nWe call the components of the index word $w=\\{w_1, \\dots, w_{m R}\\}$\nthe {\\em generators} and the associated nodes in the factor graph\nrepresenting the LDGM code the {\\em generator nodes}. \nWe assume that these generators nodes have a normalized degree distribution \n$L(x)=\\sum_{i} L_i x^i$. This means that $L_i$ represents the fraction \n(out of $m R$) of generator nodes of degree $i$.\n\nWe are interested in the trade-off between rate and distortion which is achievable \nin this setting. Let $\\distortion(\\cdot, \\cdot)$ denote the Hamming distortion function,\n$\\distortion: \\GF^m \\times \\GF^m \\rightarrow \\naturals$.\nThe average distortion is then given by\n\\begin{align*}\n\\frac1{m} \\expectation[d(S, \\decoder(\\encoder(S))].\n\\end{align*}\nWe are interested in the minimum of this average distortion, where the minimum\nis taken over all LDGM codes of a given rate, generator degree distribution $L(x)$, and length,\nas well as over all encoding functions.\n\n\\section{Review}\nGiven the success of sparse graph codes applied to the channel coding problem, it\nis not surprising that there is also interest in the use\nof sparse graph codes for the source coding problem.\nMartinian and Yedidia \\cite{MaYe03} were probably the first to \nwork on lossy compression using sparse graph codes. \nThey considered a memoryless ternary source with erasures and demonstrated a duality result between\ncompression of this source and the transmission problem over\na binary erasure channel (both using iterative encoding\/decoding).\nMezard, Zecchina, and Ciliberti \\cite{CiMe05} considered the lossy compression\nof the BSS using LDGM codes with a Poisson distribution on the generators.\nThey derived the one-step replica symmetry-breaking (1RSB) solution \nand the average\nrate-distortion function. According to this analysis, this ensemble approaches\nthe Shannon rate-distortion curve exponentially fast in the average degree.\nThey observed that the iterative interpretation associated to the 1RSB analysis\ngives rise to an algorithm, which they called {\\em survey propagation}. In\n\\cite{CiMeZe05} the same authors implement an encoder that utilizes a Tanner graph with random non-linear\nfunctions at the check nodes and a {\\em survey propagation} based\ndecimation algorithm for data compression of the BSS. \nIn \\cite{WaM05}, Wainwright and Maneva also considered the lossy compression of a\nBSS using an LDGM code with a given degree distribution. They showed how survey\npropagation can be interpreted as belief propagation algorithm \n(as did Braunstein and Zecchina \\cite{BZ03})\non an enlarged set of assignments and demonstrated\nthat the survey propagation algorithm is a practical and efficient encoding\nscheme.\nRecently, Filler and Friedrich \\cite{FiFr07} demonstrated experimentally that\neven standard belief propagation based decimation algorithms using optimized\ndegree distributions for LDGM codes and a proper initialization of the messages\ncan achieve a rate-distortion trade-off very close to the Shannon bound. \nMartinian and Wainwright \\cite{MaWa06,MaW06a,MaW06b} constructed {\\em compound LDPC\nand LDGM code ensembles} and gave rigorous {\\em upper bounds} on their distortion\nperformance. A standard LDGM code ensemble is a special case of their\nconstruction, hence they also provide {\\em upper bounds} on the rate-distortion\nfunction of LDGM ensembles. By using the first and second moment method they proved\nthat a code chosen randomly from the {\\em compound ensemble} under optimal encoding and decoding achieves the Shannon\nrate-distortion curve with high probability. Finally, they pointed out that such constructions are\nuseful also in a more general context (e.g., the Wyner-Ziv or the Gelfand-Pinsker\nproblem).\nDimakis et al \\cite{DiWaRa07} were the first authors to provide\nrigorous {\\em lower bounds} on the rate-distortion function of LDGM code\nensembles. \n\\begin{theorem}[Dimakis, Wainwright, Ramchandran \\cite{DiWaRa07}]\n\\label{the:dwrbound} Let $\\code$ be a binary code of blocklength $m$ and rate\n$R$ chosen uniformly at random from an ensemble of left Poisson LDGM Codes with check-node degree\n${\\mathtt r}$. Suppose that we perform MAP decoding. With high probability the \nrate-distortion pair ($R,D$) achieved by $\\code$ fulfills\n\\begin{align*} \nR & \\geq \\frac{1-h(D)}{1-e^{-\\frac{(1-D){\\mathtt r}}{R}}} > 1-h(D).\n\\end{align*}\n\\end{theorem}\n\n\\subsection{Outline}\nIn the spirit of Gallager's information theoretic bound for LDPC codes,\nwe are interested in deriving lower bounds on the rate-distortion function \nwhich are valid for {\\em any} LDGM code with a given generator node degree distribution $L(x)$.\nOur approach is very simple.\nPick a parameter $D$, $D \\in [0, \\frac12]$ (think of this parameter as\nthe distortion). Consider the set of ``covered'' sequences \\begin{align}\n\\label{equ:cofd} {\\mathcal C}(D) & = \\bigcup_{\\hat{s} \\in \\code} {\\mathcal\nB}(\\hat{s}, D m), \\end{align} where ${\\mathcal B}(x, i)$, $x \\in \\GF^m$,\n$i \\in [m]$, is the Hamming ball of radius $i$ centered at $x$. In words,\n${\\mathcal C}(D)$ represents the set of all those source sequences that\nare within Hamming distance at most $D m$ from at least one code word.\n\nRecall that for any $s \\in {\\cal S}$, ${\\encoder}(s) \\in {\\mathcal W}$\nrepresents the index word and that ${\\decoder}({\\encoder}(s))$ denotes the\nreconstructed word. We have\n\\begin{align*}\n\\distortion(s, {\\decoder}({\\encoder}(s))) & \\geq \n\\begin{cases}\n0, & s \\in {\\mathcal C}(D), \\\\\nDm, & s \\in \\GF^m \\setminus {\\mathcal C}(D).\n\\end{cases}\n\\end{align*}\nTherefore,\n\\begin{align} \n&\\frac1{m} \\expectation[\\distortion(S, {\\decoder}({\\encoder}(S)))] \\nonumber \\\\\n& = \\frac1{m}\\sum_{s \\in \\GF^m} 2^{-m} \\distortion(s, {\\decoder}({\\encoder}(s)))\n \\geq \\frac{2^{-m}}{m} \\sum_{s \\in \\GF^m \\setminus {\\mathcal C}(D)} \\distortion(s, {\\decoder}({\\encoder}(s))) \\nonumber \\\\\n& \\geq 2^{-m} D |\\GF^m \\setminus {\\mathcal C}(D)| \\geq D \\bigl(1-2^{-m} |{\\mathcal C}(D)| \\bigr).\n\\label{equ:averagedistortion}\n\\end{align}\nIf the codewords are well spread out then we know from Shannon's random coding\nargument that for a choice $D=h^{-1}(1-R)$, $|{\\mathcal C}(D)| \\approx 2^m$, \\cite{CoT91}.\nBut the codewords of an LDGM code are clustered since changing a\nsingle generator symbol only changes a constant number of symbols in\nthe codeword. There is therefore substantial overlap of the balls.\nWe will show that there\nexists a $D$ which is strictly larger than the distortion corresponding\nto Shannon's rate-distortion bound so that $|{\\mathcal C}(D)|$ is exponentially\nsmall compared to $2^m$ regardless of the specific code. From\n(\\ref{equ:averagedistortion}) this implies that the distortion is at\nleast $D$.\n\nTo derive the required upper bound on $|{\\mathcal C}(D)|$ we use two\ndifferent techniques. In Section~\\ref{sec:boundviacounting} we use a\nsimple combinatorial argument. In Section~\\ref{sec:boundviatestchannel},\non the other hand, we employ a probabilistic argument based on the\n``test channel'' which is typically used to show the achievability of\nthe Shannon rate-distortion function.\n\nAlthough both bounds prove that the rate-distortion function is\nstrictly bounded away from the Shannon rate-distortion function for\nthe whole range of rates and any LDGM code, we conjecture that a\nstronger bound is valid. We pose our conjecture as an open problem in\nSection~\\ref{sec:openquestions}.\n\n\\section{Bound Via Counting}\\label{sec:boundviacounting}\n\\begin{theorem}[Bound Via Counting]\\label{the:boundviacounting}\nLet $\\code$ be an LDGM code with blocklength $m$ and with generator\nnode degree distribution $L(x)$ and define $L'=L'(1)$. Let\n\\begin{align*}\nf(x) = \\prod_{i=0}^{d} (1+x^i)^{L_i}, \\;\\;\na(x) = \\prod_{i=0}^{d} i L_i \\frac{x^i}{1+x^i}, \\\\\n\\hat{R}(x) = \\frac{1-h(\\frac{x}{1+x})}{1-\\log \\frac{f(x)}{x^{a(x)}}}, \\;\\;\n\\hat{D}(x) = \\frac{x}{1+x} - a(x) \\hat{R}(x).\n\\end{align*}\nFor $R \\in [\\frac{1}{L'}, 1]$ let $x(R)$ be the unique positive solution of $\\hat{R}(x)=R$.\nDefine the curve $D(R)$ as\n\\begin{align*}\n& \\begin{cases}\n\\frac12 \\Bigl(1-R L' \\bigl(1-2\\bigl( \\frac{x(\\frac{1}{L'})}{1+x(\\frac{1}{L'})} -\n\\frac{a(x(\\frac{1}{L'}))}{{\\mathtt l}}\\bigr)\\bigr)\\Bigr), &\n R \\in [0, \\frac{1}{L'}], \\\\\n\\hat{D}(x(R)), R \\in [\\frac{1}{L'}, 1].\n\\end{cases}\n\\end{align*}\nThen, for any blocklength $m$, the achievable distortion of an LDGM code of rate $R$ and generator degree distribution $L(x)$\nis lower bounded by $D(R)$.\n\\end{theorem}\nDiscussion: \n(i) As stated above, if we are considering a single code of rate $R$ then\nthe lower bound on the distortion is $D(R)$. If, on the other hand we are considering\na family of codes, all with the same generator degree distribution $L(x)$ but with\ndifferent rates $R$, then it is more convenient to plot the lower bound in a parametric\nform. First plot the curve $(\\hat{D}(x), \\hat{R}(x))$ for $x \\in [0, 1]$. Then connect\nthe point $(D=\\frac12, R=0)$ to the point on the $(\\hat{D}(x), \\hat{R}(x))$ curve\nwith $\\hat{R}(x)=\\frac{1}{L'}$ by a straight line. The resulting upper envelope\ngives the stated lower bound for the whole range. This construction is shown in \nFigure~\\ref{fig:rdconstruction}.\n\\begin{figure}[htp]\n\\begin{center}\n\\input{ps\/rdconstruction}\n\\end{center}\n\\caption{\\label{fig:rdconstruction}\nConstruction of the bound for codes with $L(x)=x^2$ so that $L'=2$ (all generator\nnodes have degree $2$). The solid gray curve corresponds to the Shannon rate-distortion curve.\nThe black curve just above, which is partially solid and partially dotted, corresponds\nto the curve $(\\hat{D}(x), \\hat{R}(x))$ for $x \\in [0, 1]$. It starts at the point $(0, 1)$\n(which corresponds to $x=0$) and ends at $(\\frac{L'-1}{2 L'}=\\frac14, \\frac{1}{(L')^2}=\\frac14)$ \nwhich corresponds to $x=1$. The straight line goes from the point $(\\hat{D}(x(\\frac{1}{L'})), \\frac{1}{L'})$ to the point $(\\frac12, 0)$. Any achievable $(R, D)$ pair must lie in the lightly shaded region.\nThis region is strictly bounded away from the Shannon rate-distortion function over the whole range.\n}\n\\end{figure}\n(ii) \nAlthough this is difficult to glance from the expressions, we will\nsee in the proof that for any bounded generator degree distribution\n$L(x)$ the performance is strictly bounded away from the Shannon\nrate-distortion function. From a practical perspective however the gap\nto the rate-distortion bound decreases quickly in the degree.\n\n\\begin{example}[Generator-Regular LDGM Codes]\n\\label{exa:rdgeneratorregular}\nConsider codes with generator degree equal to ${\\mathtt l}$ and\nan arbitrary degree distribution on the check nodes.\nIn this case we have $f(x)=1+x^{\\mathtt l}$ and \n$a(x) = \\frac{{\\mathtt l} x^{\\mathtt l}}{1+x^{\\mathtt l}}$.\nFigure~\\ref{fig:rdgeneratorregular} compares the lower bound to\nthe rate-distortion curve for ${\\mathtt l}=1$, $2$, and $3$. For each case the\nachievable region is strictly bounded away from the Shannon rate-distortion curve.\n\\begin{figure}[htp]\n\\begin{center}\n\\input{ps\/rdgeneratorregular}\n\\end{center}\n\\caption{\\label{fig:rdgeneratorregular}\nBounds for $L(x)=x^{\\mathtt l}$ for ${\\mathtt l}=1$, $2$, and $3$.\nFor ${\\mathtt l}=2$ the $3$ gray dots correspond to the special\ncases $R=\\frac23$, $R=\\frac12$, and $R=\\frac25$ respectively.\nThe corresponding lower bounds on the distortion are\n$D(\\frac23) \\geq 0.0616> 0.0614905$ (rate-distortion bound), \n$D(\\frac12) \\geq0.115 > 0.11$ (rate-distortion bound), and $D(\\frac25) \\geq 0.1924 >0.1461$ (rate-distortion bound).\n}\n\\end{figure}\n\\end{example}\n\n\\begin{example}[$({\\mathtt l}, {\\mathtt r})$-Regular LDGM Codes]\nIn this case we have $R={\\mathtt l}\/{\\mathtt r}$ and $L(x)=x^{{\\mathtt l}}$.\nThe same bound as in Example~\\ref{exa:rdgeneratorregular} applies.\nThe three special cases $({\\mathtt l}=2, {\\mathtt r}=3)$, $({\\mathtt l}=2, {\\mathtt r}=4)$, and $({\\mathtt l}=2, {\\mathtt r}=5)$,\nwhich correspond to $R=\\frac23$, $R=\\frac12$, and $R=\\frac25$ respectively,\nare marked in Figure~\\ref{fig:rdgeneratorregular} as gray dots.\n\\end{example}\n\n\\begin{example}[${\\mathtt r}$-Regular LDGM Codes of Rate $R$]\nAssume that all check nodes have degree ${\\mathtt r}$ and that the \nconnections are chosen uniformly at random with repetitions.\nFor large blocklengths this implies that the degree distribution\non the variable nodes converges to a Poisson distribution, i.e., we have\nin the limit\n\\begin{align*}\nL(x) & = \\sum_{i=1}^{\\infty} L_i x^i = e^{\\frac{{\\mathtt r}}{R} (x-1)}.\n\\end{align*}\nLet us evaluate our bound for this generator degree distribution.\nNote that since the average degree of the {\\em check} nodes is fixed we have a different\ngenerator degree distribution $L(x)$ for each rate $R$.\nFigure~\\ref{fig:rdsourceregular} compares the resulting bound with the Shannon rate-distortion function\nas well as the bound of Theorem~\\ref{the:dwrbound}. The new bound\nis slightly tighter. But more importantly, it applies to {\\em any} LDGM code. \n\\begin{figure}[htp]\n\\begin{center}\n\\input{ps\/rdsourceregular}\n\\end{center}\n\\caption{\\label{fig:rdsourceregular}\nLower bound on achievable $(R, D)$ pairs for ${\\mathtt r}$-regular LDGM codes with a\nPoisson generator degree distribution and ${\\mathtt r}=2, 4$. The dashed curve corresponds to \nthe bound of Theorem~\\ref{the:dwrbound} and the solid black curve represents the bound\nof Theorem~\\ref{the:boundviacounting}.\nThe gray curve is the Shannon rate-distortion tradeoff. \n}\n\\end{figure}\n\n\\end{example}\n\n{\\em Proof of Theorem~\\ref{the:boundviacounting}.}\nFrom the statement in Theorem~\\ref{the:boundviacounting} you see that the\nbound consists of a portion of the curve $(\\hat{D}(x), \\hat{R}(x))$\nand a straight-line portion. The straight-line portion is easily\nexplained. Assume that all generator nodes have degree ${\\mathtt l}$ (for the\ngeneral case replace all mentions of ${\\mathtt l}$ by the average degree\n$L'$). Then the maximum number of check nodes that can depend on the\nchoice of generator nodes is $n {\\mathtt l}$. Therefore, if the rate $R$ is\nlower than $\\frac{1}{{\\mathtt l}}$ then at least a fraction $(1-R {\\mathtt l})$ of the\ncheck nodes cannot be connected to any generator node. For those nodes\nthe average distortion is $\\frac12$, whereas for the fraction $R {\\mathtt l}$\nof the check nodes which are (potentially) connected to at least one\ngenerator node the best achievable distortion is the same for any $0 \\leq\nR \\leq \\frac{1}{{\\mathtt l}}$. It suffices therefore to restrict our attention\nto rates in the range $[\\frac{1}{L'}, 1]$ and to prove that their $(R,\nD)$ pairs are lower bounded by the curve $(\\hat{D}(x), \\hat{R}(x))$.\n\nAs a second simplification note that although the bound is valid for\nall blocklengths $m$ we only need to prove it for the limit of infinite\nblocklengths. To see this, consider a particular code of blocklength\n$m$. Take $k$ identical copies of this code and consider these $k$\ncopies as one code of blocklength $k m$. Clearly, this large code has\nthe same rate $R$, the same generator degree distribution $L(x)$,\nand the same distortion $D$ as each component code. By letting\n$k$ tend to infinity we can construct an arbitrarily large code of\nthe same characteristics and apply the bound to this limit. Since our bound\nbelow is valid for {\\em any} sequence of codes whose blocklength tends to\ninfinity the claim follows.\n\nPick $w \\in \\naturals$ so that $D m + w \\leq \\frac{m}{2}$. Then\n\\begin{align*}\n|{\\mathcal C}(D)| \n& = |\\bigcup_{\\hat{s} \\in \\code} {\\mathcal B}(\\hat{s}, D m)| \\\\\n& \\stackrel{\\text{(i)}}{\\leq} \\frac{1}{A_m(w)} \\sum_{\\hat{s} \\in \\code} |{\\mathcal B}(\\hat{s}, Dm +w)| \\\\\n& \\stackrel{\\text{(ii)}}{\\leq} 2^{-mR \\log \\frac{f(x_{\\omega})}{x_{\\omega}^{\\omega}}+o_m(1)} 2^{m R} 2^{m h(D+w\/m)} \\\\\n& \\stackrel{\\text{(iii)}}{=} 2^{m (-R \\log \\frac{f(x_{\\omega})}{x_{\\omega}^{a(x_{\\omega})}}+ \nR+ h(D+a(x_{\\omega}) R) + o_m(1))}.\n\\end{align*}\n\nTo see (i) note that a ``big'' sphere ${\\mathcal B}(\\hat{s}, Dm +w)$, where $\\hat{s} \\in \\code$, contains \nall ``small'' spheres of the form ${\\mathcal B}(\\hat{s}', D m)$, where $\\hat{s}' \\in \\code$ so that\n$\\distortion(\\hat{s}, \\hat{s}') \\leq w$.\nLet $A_m(w)$ be the number of codewords of Hamming weight at most $w$. \nThen, by symmetry, each small sphere\n${\\mathcal B}(\\hat{s}', Dm)$ is in exactly $A_m(w)$ big spheres ${\\mathcal B}(\\hat{s}, Dm +w)$.\nIt follows that every point in $\\bigcup_{\\hat{s} \\in \\code} {\\mathcal B}(\\hat{s}, D m)$ is counted at least\n$A_m(w)$ times in the expression $\\sum_{\\hat{s} \\in \\code} |{\\mathcal B}(\\hat{s}, Dm +w)|$.\n\nConsider now step (ii). We need a lower bound on $A_m(w)$. Assume at first that\nall generator nodes have degree ${\\mathtt l}$. Assume that exactly $g$ generator nodes\nare set to $1$ and that all other nodes are set to $0$. There are $\\binom{mR}{g}$\nways of doing this. Now note that for each such constellation the weight\nof the resulting codeword is at most $w=g {\\mathtt l}$. It follows that in the generator regular case\nwe have\n\\begin{align} \\label{equ:anofwone}\nA_m(w) \\geq \\sum_{g=0}^{w\/{\\mathtt l}} \\binom{mR}{g}. \n\\end{align}\nWe can rewrite (\\ref{equ:anofwone}) in the form\n\\begin{align} \\label{equ:anofwtwo}\nA_m(w) & \\geq \\sum_{i=0}^{w} \\text{coef}\\{(1+x^{\\mathtt l})^{mR}, x^i\\}, \n\\end{align}\nwhere $\\text{coef}\\{ (1+x^{\\mathtt l})^{mR}, x^i\\}$ indicates the coefficient of the polynomial\n$(1+x^{\\mathtt l})^{mR}$ in front of the monomial $x^i$. \nThe expression (\\ref{equ:anofwtwo}) stays valid also for irregular generator degree distributions $L(x)$\nif we replace $(1+x^{\\mathtt l})^{mR}$ with $f(x)^{mR}$, where $f(x)=\\prod_i(1+x^i)^{L_i}$ as defined in\nthe statement of the theorem. This of course requires that $n$ is chosen in such a way\nthat $n L_i \\in \\naturals$ for all $i$. \n\nDefine $N_m(w) = \\sum_{i=0}^{w} \\text{coef}\\{f(x)^{mR}, x^i\\}$, so that\n(\\ref{equ:anofwtwo}) can be restated as $A_m(w) \\geq N_m(w)$. Step (ii) now\nfollows by using the asymptotic expansion of $N_m(w)$ stated as Theorem~1\n\\cite{MB04}, where we define $\\omega=w\/(mR)$ and where $x_{\\omega}$ is the\nunique positive solution to $a(x)=\\omega$.\n\nFinally, to see (iii) we replace $w$ by $mR a(x_{\\omega})$ and thus\nwe get the claim.\nSince this bound is valid for any $w \\in \\naturals$ so that $D m + w \\leq \\frac{m}{2}$ we get the bound\n\\begin{align*}\n\\lim_{m \\rightarrow \\infty} \\frac{1}{m} \\log |{\\mathcal C}(D)| \\leq g(D, R),\n\\end{align*}\nwhere\n\\begin{align*}\ng(D, R) & = \\inf_{\\stackrel{x \\geq 0}{D+a(x)R \\leq \\frac12}} -R \\log \\frac{f(x)}{x^{a(x)}}+ R+ h(D+a(x) R).\n\\end{align*}\n\n\nNow note that as long as $g(D, R)<1$,\n$|{\\mathcal C}(D)|$ is exponentially small compared to $2^m$.\nTherefore, looking back at (\\ref{equ:averagedistortion}) we see that in this case\nthe average distortion converges to at least $D$ in the limit $m \\rightarrow \\infty$.\nWe get the tightest bound by looking for the condition for equality, i.e. by looking\nat the equation $g(R, D)=1$. \nIf we take the derivative with respect to $x$ and set it to $0$ then we get the condition\n\\begin{align*}\n\\frac{x}{1+x} = D+R a(x).\n\\end{align*}\nRecall that $D + a(x) R \\leq \\frac12$, so that this translates to $x \\leq 1$.\nThis means that $x \\leq 1$. Replace $D+a(x) R$ in the entropy term by $\\frac{x}{1+x}$,\nset the resulting expression for $g(R, x)$ equal to $1$, and solve for $R$.\nThis gives $R$ as a function of $x$ and so we also get $D$ as a function\nof $x$. We have\n\\begin{align*}\nR(x) = \\frac{1-h(\\frac{x}{1+x})}{1-\\log \\frac{f(x)}{x^{a(x)}}}, \\,\\,\nD(x) = \\frac{x}{1+x} - a(x) R(x) .\n\\end{align*}\nA check shows that $x=0$ corresponds to $(D, R)=(0, 1)$ and that $x=1$ corresponds to \n$(D, R)=(\\frac{L'-1}{2 L'}, \\frac{1}{(L')^2})$. Further, $R$ and $D$ are monotone functions of $x$.\nRecall that we are only interested in the bound for $R \\in [\\frac{1}{L'}, 1]$. We get the corresponding\ncurve by letting $x$ take values in $[0, x(\\frac{1}{L'})]$. For smaller values of the rate\nwe get the aforementioned straight-line bound.\n\nLooking at the above expression for $g(D, R)$ one can see why this\nbound is strictly better than the rate-distortion curve for $D \\in (0,\n\\frac12)$. Assume at first that the generator degree distribution is\nregular. Let the degree be ${\\mathtt l}$. In this case a quick check shows\nthat $-R \\log \\frac{f(x)}{x^{a(x)}}$ is equal to $-R h(\\frac{a(x)}{{\\mathtt l}})$. Since\n$a(0)=0$ we get the rate distortion bound if we set $x=0$.\nThe claim follows by observing that $a(x)$ is a continuous strictly\nincreasing function and that $h(x)$ has an infinite derivative at $x=0$\nwhile $h(D+a(x)R)$ has a finite derivative at $x=0$. It follows that\nthere exists a sufficiently small $x$ so that $R h(\\frac{a(x)}{{\\mathtt l}})$ is strictly\nlarger than $h(D+a(x)R)-h(D)$ and so that $D+a(x) R \\leq \\frac12$. Hence, $g(D,\nR)$ is strictly decreasing as a function of $x$ at $x=0$. This bounds\nthe achievable distortion strictly away from the rate-distortion bound.\nThe same argument applies to an irregular generator degree distribution;\nthe simplest way to see this is to replace ${\\mathtt l}$ by the maximum degree\nof $L(x)$.\n\n\n\\section{Bound Via Test Channel}\\label{sec:boundviatestchannel}\nInstead of using a combinatorial approach to bound $|{\\mathcal C}(D)|$\none can also use a probabilistic argument using the ``test channel''\nshown in Figure~\\ref{fig:testchannel}.\n\\begin{figure}[htp]\n\\begin{center}\n\\input{ps\/testchannel}\n\\end{center}\n\\caption{\\label{fig:testchannel} \nThe generator words $W$ are chosen uniformly at random from ${\\mathcal W}$.\nThis generates a codeword $\\hat{S}$ uniformly at random. Each component of\n$\\hat{S}$ is then sent over a binary symmetric channel with transition probability $D'$.\n}\n\\end{figure}\n\nFor the cases we have checked \nthe resulting bound is numerically identical to the bound of\nTheorem~\\ref{the:boundviacounting} (excluding the straight-line portion).\nWe restrict our exposition to the regular case. \nThe generalization to the irregular case is straightforward.\n\\begin{theorem}[Bound Via Test Channel]\\label{the:boundviatestchannel}\nLet $\\code$ be an LDGM code with blocklength $m$, \ngenerator degree distribution $L(x)=x^{{\\mathtt l}}$, and rate $R$.\nThen for any pair $(R, D)$, where $D$ is the average distortion, we have\n\\begin{align*}\nR & \\geq \\sup_{D \\leq D' \\leq \\frac12} \\frac{1-h(D)-\\text{KL}(D \\| D')}{1-\\log_2\\Bigl(1+\\frac{(D')^{\\mathtt l}}{(1-D')^{\\mathtt l}} \\Bigr)} \\\\\n& \\geq \\frac{1-h(D)}{1-\\log_2\\Bigl(1+\\frac{D^{\\mathtt l}}{(1-D)^{\\mathtt l}} \\Bigr)} > 1-h(D),\n\\end{align*}\nwhere $\\text{KL}(D \\| D')=D \\log_2(D\/D')+(1-D) \\log_2((1-D)\/(1-D'))$.\n\\end{theorem}\n{\\em Proof.} \nThe same remark as in the proof of Theorem~\\ref{the:boundviacounting} applies: although\nthe bound is valid for any blocklength it suffices to prove it for the limit of\nblocklengths tending to infinity. Also, for simplicity we have not stated the bound\nin its strengthened form which includes a straight-line portion. But the same technique\nthat was applied in the proof of Theorem~\\ref{the:boundviacounting} applies also to the present case.\n\nAs remarked earlier, the idea of the proof is based on bounding\n$|{\\mathcal C}(D)|$ by using the ``test channel.'' More precisely,\nchoose $W$ uniformly at random from the set of\nall binary sequences of length $m R$. Subsequently compute\n$\\hat{S}$ via $\\hat{S} = W G$, where $G$ is the generator matrix\nof the LDGM code. Finally, let $S=\\hat{S}+Z$, where $Z$ has iid components with\n$\\prob\\{Z_i=1\\}=D'$.\n\nConsider the set of sequences $s \\in {\\mathcal C}(D)$.\nFor each such $s$ we know that there exists an $\\hat{s} \\in \\code$ so\nthat $\\distortion(s, \\hat{s}) \\leq Dm$. \nWe have\n\\begin{align*}\n& \\prob\\{S=s \\mid s \\in {\\mathcal C}(D) \\} \\\\\n& = \\sum_{\\hat{s}' \\in \\code} \\prob\\{S=s, \\hat{S}=\\hat{s}' \\mid s \\in {\\mathcal C}(D) \\} \\\\ \n& = \\sum_{w=0}^{m} \\sum_{\\hat{s}' \\in \\code: \\distortion(\\hat{s}', \\hat{s})=w} \\prob\\{S=s, \\hat{S}=\\hat{s}' \\mid s \\in {\\mathcal C}(D) \\} \\\\ \n& = \\sum_{w=0}^{m} A_{m}(w) \\prob\\{S=s, \\hat{S}=\\hat{s}' \\mid s \\in {\\mathcal C}(D), \\distortion(\\hat{s}', \\hat{s})=w \\} \\\\ \n& = \\sum_{w=0}^{m} A_{m}(w) 2^{-m R} \\Bigl(\\frac{D'}{1-D'}\\Bigr)^{\\distortion(s, \\hat{s}')} (1-D')^m\n\\end{align*}\n\\begin{align*}\n& \\geq \\sum_{w=0}^{m} A_{m}(w) 2^{-m R} \\Bigl(\\frac{D'}{1-D'}\\Bigr)^{\\distortion(s, \\hat{s})+\\distortion(\\hat{s}, \\hat{s}')} (1-D')^m \\\\\n& \\stackrel{ \\distortion(\\hat{s}', \\hat{s})=w}{=} \\sum_{w=0}^{m} A_{m}(w) 2^{-m R} \\Bigl(\\frac{D'}{1-D'}\\Bigr)^{\\distortion(s, \\hat{s})+w} (1-D')^m \\\\\n& \\stackrel{\\distortion(s, \\hat{s}) \\leq D m}{\\geq} \\sum_{w=0}^{m} A_{m}(w) 2^{-m R} \\Bigl(\\frac{D'}{1-D'}\\Bigr)^{D m+w} (1-D')^m \\\\\n& = 2^{-m R -mh(D)-m \\text{KL}(D \\| D')} \\sum_{w=0}^{m} A_m(w) \\Bigl(\\frac{D'}{1-D'}\\Bigr)^{w},\n\\end{align*}\nwhere $A_m(w)$ denotes the number of codewords in $\\code$ of Hamming weight $w$. Due to\nthe linearity of the code this is also the number of codewords in $\\code$ of Hamming\ndistance $w$ from $\\hat{s}$.\nUsing summation by\nparts and setting $c=D'\/(1-D')<1$, we have\n\\begin{align*}\n& \\sum_{w=0}^{m} A_m(w) c^w \\\\\n& = c^{m+1} 2^{mR}+ \\sum_{w=0}^{m}\\Bigl(\\sum_{i=0}^{w-1} A_m(i) \\Bigr) (c^w-c^{w+1}) \\\\\n& \\stackrel{(\\ref{equ:anofwtwo})}{\\geq} c^{m+1} 2^{mR}+ \\sum_{w=0}^{m}\\Bigl(\\sum_{i=0}^{\\lfloor(w-1)\/{\\mathtt l} \\rfloor} \n\\binom{mR}{i} \\Bigr) (c^w-c^{w+1}) \\\\\n& = \\sum_{w=0}^{\\lfloor m\/{\\mathtt l} \\rfloor} \\binom{mR}{w} c^{{\\mathtt l} w} + c^{m+1} \\Bigl(2^{mR} -\n\\sum_{i=0}^{\\lfloor m\/{\\mathtt l} \\rfloor} \\binom{mR}{i} \\Bigr) \\\\\n& \\geq \\sum_{w=0}^{\\lfloor m\/{\\mathtt l} \\rfloor} \\binom{mR}{w} c^{{\\mathtt l} w} \\geq \\frac1m (1+c^{\\mathtt l})^{m R}.\n\\end{align*}\nThe last step is valid as long as $\\frac{R c^{\\mathtt l}}{1+c^{\\mathtt l}} < \\frac{1}{{\\mathtt l}}$. In\nthis case the maximum term (which appears at $\\frac{R c^{\\mathtt l}}{1+c^{\\mathtt l}} m$) is\nincluded in the sum (which goes to $m\/{\\mathtt l}$) and is thus greater than equal to\nthe average of all the terms, which is $\\frac1m (1+c^{\\mathtt l})^{m R} $ . This\ncondition is trivially fulfilled for $R {\\mathtt l} < 1$. Assume for a moment that it\nis also fulfilled for $R {\\mathtt l} \\geq 1$ and the optimum choice of $D'$. It then\nfollows that\n\\begin{align*}\n\\prob\\{S=s \\mid s \\in {\\mathcal C}(D) \\} & \n\\geq \\frac1m 2^{-m(R+h(D)+\\text{KL}(D \\| D')-R \\log_2(1+c^{\\mathtt l}))}.\n\\end{align*}\nSince \n\\begin{align*}\n1 & = \\sum_{s \\in \\GF^m} \\prob\\{S=s\\} \n\\geq \\sum_{s \\in {\\mathcal C}(D)} \\prob\\{S=s\\} \\\\\n& \\geq |{\\mathcal C}(D)| \\frac1m 2^{-m(R+h(D)+\\text{KL}(D \\| D')-R \\log_2(1+c^{\\mathtt l}))},\n\\end{align*}\nwe have\n$|{\\mathcal C}(D)| \\leq m 2^{m(R+h(D)+\\text{KL}(D \\| D')-R \\log_2(1+c^{\\mathtt l}))}$.\nProceeding as in (\\ref{equ:averagedistortion}), we have\n\\begin{align*}\n& \\expectation[\\distortion(S, {\\decoder}({\\encoder}(S)))] \n\\geq D \\bigl(1-2^{-m} |{\\mathcal C}(D)| \\bigr) \\\\\n& \\geq D \\bigl(1-m 2^{m(R+h(D)+\\text{KL}(D \\| D')-R \\log_2(1+c^{\\mathtt l})-1)} \\bigr).\n\\end{align*}\nWe conclude that if for some $D \\leq D' \\leq \\frac12$, \n$R+h(D)+\\text{KL}(D \\| D')-R \\log_2(1+\\frac{(D')^{\\mathtt l}}{(1-D')^{\\mathtt l}})-1<0$\nthen the distortion is at least $D$. All this is still conditioned on\n$\\frac{R {\\mathtt l} c^{\\mathtt l}}{1+c^{\\mathtt l}} < 1$ for the optimum choice of $D'$.\nFor $R {\\mathtt l} <1$ we already checked this. So assume that $R {\\mathtt l} \\geq 1$.\nThe above condition can then equivalently be written\nas $D' < \\frac{1}{1+(R {\\mathtt l} -1)^{\\frac{1}{{\\mathtt l}}}}$.\nOn the other hand, taking the derivative of our final expression on\nthe rate-distortion function with respect to $D'$ we get the condition\nfor the maximum to be\n$D' = \\frac{1}{1+(1+\\frac{R {\\mathtt l}}{D'-D})^{\\frac{1}{{\\mathtt l}}}} < \\frac{1}{1+(R {\\mathtt l} -1)^{\\frac{1}{{\\mathtt l}}}}$.\nWe see therefore that our assumption $\\frac{R {\\mathtt l} c^{\\mathtt l}}{1+c^{\\mathtt l}} < 1$ is also correct\nin the case $R {\\mathtt l} \\geq 1$.\n\nNumerical experiments show that the present bound yields for the regular case\nidentical results as plotting the curve corresponding to $g(D, R)=1$, where\n$g(D, R)$ was defined in the proof of Theorem~\\ref{the:boundviacounting}.\nThis can be interpreted as follows. Choose $D'$ equal to the optimal radius of the Hamming\nball in the proof of Theorem~\\ref{the:boundviacounting}. Then the points $\\hat{s}'$ that\ncontribute most to the probability of $S=s$ must be those that have a distance to $\\hat{s}$ of\n$m(D'-D)$.\n\n\\section{Discussion and Open Questions}\\label{sec:openquestions}\nIn the preceding sections we gave two bounds. Both of them are\nbased on the idea of counting the number of points that are ``covered''\nby spheres centered around the codewords of an LDGM code. In the \nfirst case we derived a bound by double counting this number. In the second\ncase we derived a bound by looking at a probabilistic model using the test channel.\n\nAn interesting open question is to determine the exact relationship of\nthe test channel model to the rate-distortion problem.\nMore precisely, it is tempting to conjecture that a pair $(R, D)$ is only achievable\nif $H(S)=m$ in this test channel model. This would require to show\nthat only elements of the typical set of ${\\mathcal S}$ under the test channel model\nare covered, i.e., have code words within distance $D$. For the test channel model it is \nvery easy to determine a criterion in the spirit of Gallager's original bound.\nWe have\n\\begin{align*}\nH(S) \n& = H(W)+H(S \\mid W)- H(W \\mid S) \\\\\n& = m R + m h(D) - \\sum_{g=1}^{m R} H(W_g \\mid S, W_{1}, \\dots, W_{g-1}) \\\\\n& \\stackrel{\\text{(i)}}{\\leq} m R + m h(D) - \\sum_{g=1}^{m R} H(W_g \\mid S, W_{\\sim g}) \\\\\n& \\stackrel{\\text{(ii)}}{=} m R + m h(D) - \\sum_{g=1}^{m R} H(W_g \\mid S_g, W_{\\sim g}),\n\\end{align*}\nwhere $S_g$ denotes the subset of the components of the $S$ vectors which\nare connected to the generator $g$.\nStep (i) follows since conditioning decreases entropy. Step (ii) follows since\nknowing ($S_g, W_{\\sim g}$), $W_g$ is not dependent on $S_{\\sim g}$. The term\n$H(W_g \\mid S_g, W_{\\sim g}) $ represents the EXIT function of a repetition\ncode when transmitting over BSC($D$) channel. \nIf one could show that $H(S)=m$ is a necessary condition for \nachieving average distortion of $D$ then a quick calculation shows that\nthe resulting bound would read\n\\begin{align*}\nR & \\geq \n\\frac{1 - h(D)}{1-\\sum_{i=0}^{{\\mathtt l}} \\binom{{\\mathtt l}}{i} (1-D)^i \nD^{{\\mathtt l}-i} \\log_2\\Bigl(1+\\bigl(\\frac{D}{1-D}\\bigr)^{2 i -{\\mathtt l}}\\Bigr)}. \n\\end{align*}\nThis ``bound'' is similar in spirit to the original bound given by Gallager, except\nthat in Gallager's original bound for LDPC codes we have a term corresponding to the\nentropy of single-parity check codes, whereas here we have terms that correspond\nto the entropy of repetition codes; this would be quite fitting given the duality of the problems.\n\n\\section*{Acknowledgment} We gratefully acknowledge the support by the\nSwiss National Science Foundation under grant number 200020-113412.\n\n\\bibliographystyle{IEEEtran}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}