It has recently been observed that the permanent of a nonnegative square matrix, i.e., of a square matrix containing only nonnegative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper, we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudocodewords and permanent-based kernels.
Besides the already cited papers @cite_65 @cite_83 on Bethe-approximation-based methods to the permanent of a non-negative matrix, some aspects of the Bethe free energy function were analyzed by Watanabe and Chertkov in @cite_11 and by Chertkov, Kroc, Krzakala, Vergassola, and Zdeborov 'a in @cite_23 . (In particular, the paper @cite_11 applied the loop calculus technique by Chertkov and Chernyak @cite_76 .) Very recent work in that line of research is presented in a paper by A. B. Yedidia and Chertkov @cite_48 that studies so-called fractional free energy functionals, and resulting lower and upper bounds on the permanent of a non-negative matrix.
@cite_48 @cite_65 @cite_83 @cite_23 @cite_76 @cite_11
2761017552 1520400698 1801773550 2078168527 2018836315 2963320833 1893146140 1970851820 1628027163 2169415915 2962824725 1985715224 2042757441 1991114201 2963553740 2498247430 2963963365 2132806676 326163211 2019049541
It has recently been observed that the permanent of a nonnegative square matrix, i.e., of a square matrix containing only nonnegative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper, we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudocodewords and permanent-based kernels. We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze and generalize the Belief Propagation (BP) approach to computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. We introduce a fractional free energy functional parameterized by a scalar parameter @math , where @math corresponds to the BP limit and @math corresponds to the exclusion principle Mean-Field (MF) limit, and show monotonicity and continuity of the functional with @math . We observe that the optimal value of @math , where the @math -parameterized functional is equal to the exact free energy (defined as the minus log of the permanent), lies in the @math range, with the low and high values from the range producing provable low and upper bounds for the permanent. Our experimental analysis suggests that the optimal @math varies for different ensembles considered but it always lies in the @math interval. Besides, for all ensembles considered the behavior of the optimal @math is highly distinctive, thus offering a lot of practical potential for estimating permanents of non-negative matrices via the fractional free energy functional approach. We consider computation of the permanent of a positive (N × N) non-negative matrix, P = (Pji|i, j = 1, ..., N), or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph KN, N. The problem is known to be of likely exponential complexity. Stated as the partition function Z of a graphical model, the problem allows for exact loop calculus representation (Chertkov M and Chernyak V 2006 Phys. Rev. E 72 065102) in terms of an interior minimum of the Bethe free energy functional over non-integer doubly stochastic matrix of marginal beliefs, β = (βji|i, j = 1, ..., N), also correspondent to a fixed point of the iterative message-passing algorithm of the belief propagation (BP) type. Our main result is an explicit expression of the exact partition function (permanent) in terms of the matrix of BP marginals, β, as Z = Perm(P) = ZBP Perm(βji(1 − βji)) ∏i, j(1 − βji), where ZBP is the BP expression for the permanent stated explicitly in terms of β. We give two derivations of the formula, a direct one based on the Bethe free energy and an alternative one combining the Ihara graph-ζ function and the loop calculus approaches. Assuming that the matrix β of the BP marginals is calculated, we provide two lower bounds and one upper bound to estimate the multiplicative term. Two complementary lower bounds are based on the Gurvits–van der Waerden theorem and on a relation between the modified permanent and determinant, respectively. We establish an explicit formula for the limiting free energy density (log-partition function divided by the number of vertices) for ferromagnetic Potts models on uniformly sparse graph sequences converging locally to the d-regular tree for d even, covering all temperature regimes. This formula coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation recursion on the d-regular tree, the so-called replica symmetric solution. For uniformly random d-regular graphs we further show that the replica symmetric Bethe formula is an upper bound for the asymptotic free energy for any model with permissive interactions. We consider homogeneous factor models on uniformly sparse graph sequences converging locally to a (unimodular) random tree T, and study the existence of the free energy density ϕ, the limit of the log-partition function divided by the number of vertices n as n tends to infinity. We provide a new interpolation scheme and use it to prove existence of, and to explicitly compute, the quantity ϕ subject to uniqueness of a relevant Gibbs measure for the factor model on T. By way of example we compute ϕ for the independent set (or hard-core) model at low fugacity, for the ferromagnetic Ising model at all parameter values, and for the ferromagnetic Potts model with both weak enough and strong enough interactions. Even beyond uniqueness regimes our interpolation provides useful explicit bounds on ϕ. In the regimes in which we establish existence of the limit, we show that it coincides with the Bethe free energy functional evaluated at a suitable fixed point of the belief propagation (Bethe) recursions on T. In the special case that T has a Galton–Watson law, this formula coincides with the nonrigorous “Bethe prediction” obtained by statistical physicists using the “replica” or “cavity” methods. Thus our work is a rigorous generalization of these heuristic calculations to the broader class of sparse graph sequences converging locally to trees. We also provide a variational characterization for the Bethe prediction in this general setting, which is of independent interest. We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the belief propagation (BP) approach and its fractional belief propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and Conjectures are verified in experiments, and some new theoretical relations, bounds and Conjectures are proposed. The fractional free energy (FFE) function is parameterized by a scalar parameter γ ∈ [-1;1], where γ = -1 corresponds to the BP limit and γ = 1 corresponds to the exclusion principle (but ignoring perfect matching constraints) mean-field (MF) limit. FFE shows monotonicity and continuity with respect to γ. For every non-negative matrix, we define its special value γ* ∈ [-1;0] to be the γ for which the minimum of the γ-parameterized FFE function is equal to the permanent of the matrix, where the lower and upper bounds of the γ-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of γ* varies for different ensembles but γ* always lies within the [-1;-1 2] interval. Moreover, for all ensembles considered, the behavior of γ* is highly distinctive, offering an empirical practical guidance for estimating permanents of non-negative matrices via the FFE approach. This is the second in a series of papers in which we derive a @math -expansion for the two-dimensional non-local Ginzburg-Landau energy with Coulomb repulsion known as the Ohta-Kawasaki model in connection with diblock copolymer systems. In this model, two phases appear, which interact via a nonlocal Coulomb type energy. Here we focus on the sharp interface version of this energy in the regime where one of the phases has very small volume fraction, thus creating small "droplets" of the minority phase in a "sea" of the majority phase. In our previous paper, we computed the @math -limit of the leading order energy, which yields the averaged behavior for almost minimizers, namely that the density of droplets should be uniform. Here we go to the next order and derive a next order @math -limit energy, which is exactly the Coulombian renormalized energy obtained by Sandier and Serfaty as a limiting interaction energy for vortices in the magnetic Ginzburg-Landau model. The derivation is based on the abstract scheme of Sandier-Serfaty that serves to obtain lower bounds for 2-scale energies and express them through some probabilities on patterns via the multiparameter ergodic theorem. Without thus appealing to the Euler-Lagrange equation, we establish for all configurations which have "almost minimal energy" the asymptotic roundness and radius of the droplets, and the fact that they asymptotically shrink to points whose arrangement minimizes the renormalized energy in some averaged sense. Via a kind of @math -equivalence, the obtained results also yield an expansion of the minimal energy for the original Ohta-Kawasaki energy. This leads to expecting to see triangular lattices of droplets as energy minimizers. In this paper we present the derivation details, logic, and motivation for the three loop calculus introduced in Chertkov and Chernyak (2006 Phys. Rev. E 73 065102(R)). Generating functions for each of the three interrelated discrete statistical models are expressed in terms of a finite series. The first term in the series corresponds to the Bethe–Peierls belief–propagation (BP) contribution; the other terms are labelled by loops on the factor graph. All loop contributions are simple rational functions of spin correlation functions calculated within the BP approach. We discuss two alternative derivations of the loop series. One approach implements a set of local auxiliary integrations over continuous fields with the BP contribution corresponding to an integrand saddle-point value. The integrals are replaced by sums in the complementary approach, briefly explained in Chertkov and Chernyak (2006 Phys. Rev. E 73 065102(R)). Local gauge symmetry transformations that clarify an important invariant feature of the BP solution are revealed in both approaches. The individual terms change under the gauge transformation while the partition function remains invariant. The requirement for all individual terms to be nonzero only for closed loops in the factor graph (as opposed to paths with loose ends) is equivalent to fixing the first term in the series to be exactly equal to the BP contribution. Further applications of the loop calculus to problems in statistical physics, computer and information sciences are discussed. This work describes a method of approximating matrix per- manents efficiently using belief propagation. We formulate aprobability distribution whose partition function is exactly the permanent, then use Bethe free energy to approximate this partition function. After deriving some speedups to standard belief propagation, the resulting algorithm requires (n 2 ) time per iteration. Finally, we demonstrate the advantages of using this approximation. Important inference problems in statistical physics, computer vision, error-correcting coding theory, and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs. The belief propagation (BP) algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree, but only approximate when the factor graph has cycles. We show that BP fixed points correspond to the stationary points of the Bethe approximation of the free energy for a factor graph. We explain how to obtain region-based free energy approximations that improve the Bethe approximation, and corresponding generalized belief propagation (GBP) algorithms. We emphasize the conditions a free energy approximation must satisfy in order to be a "valid" or "maxent-normal" approximation. We describe the relationship between four different methods that can be used to generate valid approximations: the "Bethe method", the "junction graph method", the "cluster variation method", and the "region graph method". Finally, we explain how to tell whether a region-based approximation, and its corresponding GBP algorithm, is likely to be accurate, and describe empirical results showing that GBP can significantly outperform BP. Freedman, Kitaev, and Wang (2002), and later Aharonov, Jones, and Landau (2009), established a quantum algorithm to "additively" approximate the Jones polynomial V(L;t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang (2002) established that the approximation is universal for quantum computation at a non- lattice, principal root of unity. We show that any value-distinguishing approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. Given the power to decide whetherjV(L;t)j b for fixed constants 0 1, T(G; x; y) is #P-hard to approximate within a factor of c even for planar graphs G. Along the way, we clarify and generalize both Aaronson's theorem and the Solovay- Kitaev theorem. We consider the variational free energy approach for compressed sensing. We first show that the na "ive mean field approach performs remarkably well when coupled with a noise learning procedure. We also notice that it leads to the same equations as those used for iterative thresholding. We then discuss the Bethe free energy and how it corresponds to the fixed points of the approximate message passing algorithm. In both cases, we test numerically the direct optimization of the free energies as a converging sparse-estimationalgorithm. Approximation of the permanent of a matrix with nonnegative entries is a well studied problem. The most successful approach to date for general matrices uses Markov chains to approximately sample from a distribution on weighted permutations, and Jerrum, Sinclair, and Vigoda developed such a method they proved runs in polynomial time in the input. The current bound on the running time of their method is O(n7(log n)4). Here we present a very different approach using sequential acceptance rejection, and show that for a class of dense problems this method has an O(n4 log n) expected running time. In this paper, we are concerned with the one-species Vlasov–Poisson–Boltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional. According to physics predictions, the free energy of random factor graph models that satisfy a certain "static replica symmetry" condition can be calculated via the Belief Propagation message passing scheme [ PNAS, 2007]. Here we prove this conjecture for a wide class of random factor graph models. Specifically, we show that the messages constructed just as in the case of acyclic factor graphs asymptotically satisfy the Belief Propagation equations and that the free energy density is given by the Bethe free energy formula. We extend the De Giorgi--Nash--Moser theory to a class of kinetic Fokker-Planck equations and deduce new results on the Landau-Coulomb equation. More precisely, we first study the H o lder regularity and establish a Harnack inequality for solutions to a general linear equation of Fokker-Planck type whose coefficients are merely measurable and essentially bounded, i.e. assuming no regularity on the coefficients in order to later derive results for non-linear problems. This general equation has the formal structure of the hypoelliptic equations "of type II" , sometimes also called ultraparabolic equations of Kolmogorov type, but with rough coefficients: it combines a first-order skew-symmetric operator with a second-order elliptic operator involving derivatives along only part of the coordinates and with rough coefficients. These general results are then applied to the non-negative essentially bounded weak solutions of the Landau equation with inverse-power law @math @math [--d, 1] whose mass, energy and entropy density are bounded and mass is bounded away from 0, and we deduce the H o lder regularity of these solutions. Abstract This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group G L n ( R [ t , t − 1 ] ) , and for the formal loop group G L n ( R ( ( t ) ) ) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeros of the Edrei–Thoma theorem classifying totally positive functions (corresponding to our case n = 1 ). We give a solution to the “factorization problem” using limits of ratios of minors. This is in a similar spirit to the Berenstein–Fomin–Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi–Yamada’s study of discrete Painleve dynamical systems and Berenstein–Kazhdan’s study of geometric crystals. Sudderth, Wainwright, and Willsky have conjectured that the Bethe approximation corresponding to any fixed point of the belief propagation algorithm over an attractive, pairwise binary graphical model provides a lower bound on the true partition function. In this work, we resolve this conjecture in the affirmative by demonstrating that, for any graphical model with binary variables whose potential functions (not necessarily pairwise) are all log-supermodular, the Bethe partition function always lower bounds the true partition function. The proof of this result follows from a new variant of the "four functions" theorem that may be of independent interest. We develop a definitive physical-space scattering theory for the scalar wave equation on Kerr exterior backgrounds in the general subextremal case |a|<M. In particular, we prove results corresponding to "existence and uniqueness of scattering states" and "asymptotic completeness" and we show moreover that the resulting "scattering matrix" mapping radiation fields on the past horizon and past null infinity to radiation fields on the future horizon and future null infinity is a bounded operator. The latter allows us to give a time-domain theory of superradiant reflection. The boundedness of the scattering matrix shows in particular that the maximal amplification of solutions associated to ingoing finite-energy wave packets on past null infinity is bounded. On the frequency side, this corresponds to the novel statement that the suitably normalised reflection and transmission coefficients are uniformly bounded independently of the frequency parameters. We further complement this with a demonstration that superradiant reflection indeed amplifies the energy radiated to future null infinity of suitable wave-packets as above. The results make essential use of a refinement of our recent proof [M. Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: the full subextremal case |a|<M, arXiv:1402.6034] of boundedness and decay for solutions of the Cauchy problem so as to apply in the class of solutions where only a degenerate energy is assumed finite. We show in contrast that the analogous scattering maps cannot be defined for the class of finite non-degenerate energy solutions. This is due to the fact that the celebrated horizon red-shift effect acts as a blue-shift instability when solving the wave equation backwards. Abstract By looking at fractional D p -branes of type IIA on T 4 Z 2 as wrapped branes and by using boundary state techniques we construct the effective low-energy action for the fields generated by fractional branes, build their worldvolume action and find the corresponding classical geometry. The explicit form of the classical background is consistent only outside an enhancon sphere of radius r e , which encloses a naked singularity of repulson-type. The perturbative running of the gauge coupling constant, dictated by the NS–NS twisted field that keeps its one-loop expression at any distance, also fails at r e .