This article is an extended version of previous work of the authors [40, 41] on low-rank matrix estimation in the presence of constraints on the factors into which the matrix is factorized. Low-rank matrix factorization is one of the basic methods used in data analysis for unsupervised learning of relevant features and other types of dimensionality reduction. We present a framework to study the constrained low-rank matrix estimation for a general prior on the factors, and a general output channel through which the matrix is observed. We draw a paralel with the study of vector-spin glass models - presenting a unifying way to study a number of problems considered previously in separate statistical physics works. We present a number of applications for the problem in data analysis. We derive in detail a general form of the low-rank approximate message passing (Low- RAMP) algorithm, that is known in statistical physics as the TAP equations. We thus unify the derivation of the TAP equations for models as different as the Sherrington-Kirkpatrick model, the restricted Boltzmann machine, the Hopfield model or vector (xy, Heisenberg and other) spin glasses. The state evolution of the Low-RAMP algorithm is also derived, and is equivalent to the replica symmetric solution for the large class of vector-spin glass models. In the section devoted to result we study in detail phase diagrams and phase transitions for the Bayes-optimal inference in low-rank matrix estimation. We present a typology of phase transitions and their relation to performance of algorithms such as the Low-RAMP or commonly used spectral methods.
In physics, message passing equations are always closely linked with the Bethe free energy who's stationary points are the message passing fixed point equations. In the presence of multiple fixed points it is the value of the Bethe free energy that decides which of the fixed points is the correct one. In section and appendix we derive the Bethe free energy on a single instance of the low-rank matrix estimation problem. The form of free energy that we derive has the convenience to be variational in the sense that in order to find the fixed point we are looking for a maximum of the free energy, not for a saddle. Corresponding free energy for the compressed sensing problem was derived in @cite_74 @cite_59 and can be used to guide the iterations of the Low-RAMP algorithm @cite_66 .
@cite_59 @cite_66 @cite_74
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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 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. 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 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.