DOI

1

Statistical physics of inference: thresholds and algorithms - Zdeborova, Lenka and Krzakala, Florent

ADVANCES IN PHYSICS 65, 453-552 (2016)

Abstract : Many questions of fundamental interest in today's science can be formulated as inference problems: some partial, or noisy, observations are performed over a set of variables and the goal is to recover, or infer, the values of the variables based on the indirect information contained in the measurements. For such problems, the central scientific questions are: Under what conditions is the information contained in the measurements sufficient for a satisfactory inference to be possible? What are the most efficient algorithms for this task? A growing body of work has shown that often we can understand and locate these fundamental barriers by thinking of them as phase transitions in the sense of statistical physics. Moreover, it turned out that we can use the gained physical insight to develop new promising algorithms. The connection between inference and statistical physics is currently witnessing an impressive renaissance and we review here the current state-of-the-art, with a pedagogical focus on the Ising model which, formulated as an inference problem, we call the planted spin glass. In terms of applications we review two classes of problems: (i) inference of clusters on graphs and networks, with community detection as a special case and (ii) estimating a signal from its noisy linear measurements, with compressed sensing as a case of sparse estimation. Our goal is to provide a pedagogical review for researchers in physics and other fields interested in this fascinating topic.

ADVANCES IN PHYSICS 65, 453-552 (2016)

LPS

Abstract : Many questions of fundamental interest in today's science can be formulated as inference problems: some partial, or noisy, observations are performed over a set of variables and the goal is to recover, or infer, the values of the variables based on the indirect information contained in the measurements. For such problems, the central scientific questions are: Under what conditions is the information contained in the measurements sufficient for a satisfactory inference to be possible? What are the most efficient algorithms for this task? A growing body of work has shown that often we can understand and locate these fundamental barriers by thinking of them as phase transitions in the sense of statistical physics. Moreover, it turned out that we can use the gained physical insight to develop new promising algorithms. The connection between inference and statistical physics is currently witnessing an impressive renaissance and we review here the current state-of-the-art, with a pedagogical focus on the Ising model which, formulated as an inference problem, we call the planted spin glass. In terms of applications we review two classes of problems: (i) inference of clusters on graphs and networks, with community detection as a special case and (ii) estimating a signal from its noisy linear measurements, with compressed sensing as a case of sparse estimation. Our goal is to provide a pedagogical review for researchers in physics and other fields interested in this fascinating topic.

DOI

2

Approximate message passing with restricted Boltzmann machine priors - Tramel, Eric W. and Dremeau, Angelique and Krzakala, Florent

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2016)

Abstract : Approximate message passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problems. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernoulli prior which utilizes a restricted Boltzmann machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple i.i.d. priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2016)

LPS

Abstract : Approximate message passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problems. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernoulli prior which utilizes a restricted Boltzmann machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple i.i.d. priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.

DOI

3

Phase Transitions and Sample Complexity in Bayes-Optimal Matrix Factorization - Kabashima, Yoshiyuki and Krzakala, Florent and Mezard, Marc and Sakata, Ayaka and Zdeborova, Lenka

IEEE TRANSACTIONS ON INFORMATION THEORY 62, 4228-4265 (2016)

Abstract : We analyze the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications, such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis, or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics-the cavity and replica methods-to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random-independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable, in principle, in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.

IEEE TRANSACTIONS ON INFORMATION THEORY 62, 4228-4265 (2016)

LPS

Abstract : We analyze the matrix factorization problem. Given a noisy measurement of a product of two matrices, the problem is to estimate back the original matrices. It arises in many applications, such as dictionary learning, blind matrix calibration, sparse principal component analysis, blind source separation, low rank matrix completion, robust principal component analysis, or factor analysis. It is also important in machine learning: unsupervised representation learning can often be studied through matrix factorization. We use the tools of statistical mechanics-the cavity and replica methods-to analyze the achievability and computational tractability of the inference problems in the setting of Bayes-optimal inference, which amounts to assuming that the two matrices have random-independent elements generated from some known distribution, and this information is available to the inference algorithm. In this setting, we compute the minimal mean-squared-error achievable, in principle, in any computational time, and the error that can be achieved by an efficient approximate message passing algorithm. The computation is based on the asymptotic state-evolution analysis of the algorithm. The performance that our analysis predicts, both in terms of the achieved mean-squared-error and in terms of sample complexity, is extremely promising and motivating for a further development of the algorithm.

DOI

4

Approximate message-passing with spatially coupled structured operators, with applications to compressed sensing and sparse superposition codes - Barbier, Jean and Schuelke, Christophe and Krzakala, Florent

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2015)

Abstract : We study the behavior of approximate message-passing (AMP), a solver for linear sparse estimation problems such as compressed sensing, when the i.i.d matrices-for which it has been specifically designed-are replaced by structured operators, such as Fourier and Hadamard ones. We show empirically that after proper randomization, the structure of the operators does not significantly affect the performances of the solver. Furthermore, for some specially designed spatially coupled operators, this allows a computationally fast and memory efficient reconstruction in compressed sensing up to the information-theoretical limit. We also show how this approach can be applied to sparse superposition codes, allowing the AMP decoder to perform at large rates for moderate block length.

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT , (2015)

LPS

Abstract : We study the behavior of approximate message-passing (AMP), a solver for linear sparse estimation problems such as compressed sensing, when the i.i.d matrices-for which it has been specifically designed-are replaced by structured operators, such as Fourier and Hadamard ones. We show empirically that after proper randomization, the structure of the operators does not significantly affect the performances of the solver. Furthermore, for some specially designed spatially coupled operators, this allows a computationally fast and memory efficient reconstruction in compressed sensing up to the information-theoretical limit. We also show how this approach can be applied to sparse superposition codes, allowing the AMP decoder to perform at large rates for moderate block length.

DOI

5

Belief-propagation-guided Monte-Carlo sampling - Decelle, Aurelien and Krzakala, Florent

PHYSICAL REVIEW B 89, (2014)

Abstract : A Monte-Carlo algorithm for discrete statistical models that combines the full power of the belief-propagation algorithm with the advantages of a heat-bath approach fulfilling the detailed balance is presented. First we extract randomly a subtree inside the interaction graph of the system. Second, given the boundary conditions, belief propagation is used as a perfect sampler to generate a configuration on the tree, and finally, the procedure is iterated. This approach is best adapted for locally treelike graphs and we therefore tested it on random graphs for hard models such as spin glasses, demonstrating its state-of-the art status in those cases.

PHYSICAL REVIEW B 89, (2014)

LPS

Abstract : A Monte-Carlo algorithm for discrete statistical models that combines the full power of the belief-propagation algorithm with the advantages of a heat-bath approach fulfilling the detailed balance is presented. First we extract randomly a subtree inside the interaction graph of the system. Second, given the boundary conditions, belief propagation is used as a perfect sampler to generate a configuration on the tree, and finally, the procedure is iterated. This approach is best adapted for locally treelike graphs and we therefore tested it on random graphs for hard models such as spin glasses, demonstrating its state-of-the art status in those cases.

DOI

6

Spectral density of the non-backtracking operator on random graphs - Saade, A. and Krzakala, F. and Zdeborova, L.

EPL 107, (2014)

Abstract : The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius v., where. is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero spectral density. The fact that this phase transition is absent in the spectral density of other matrices commonly used for spectral clustering provides a physical justification of the performances of the non-backtracking operator in spectral clustering. Copyright (C) EPLA, 2014

EPL 107, (2014)

LPS

Abstract : The non-backtracking operator was recently shown to provide a significant improvement when used for spectral clustering of sparse networks. In this paper we analyze its spectral density on large random sparse graphs using a mapping to the correlation functions of a certain interacting quantum disordered system on the graph. On sparse, tree-like graphs, this can be solved efficiently by the cavity method and a belief propagation algorithm. We show that there exists a paramagnetic phase, leading to zero spectral density, that is stable outside a circle of radius v., where. is the leading eigenvalue of the non-backtracking operator. We observe a second-order phase transition at the edge of this circle, between a zero and a non-zero spectral density. The fact that this phase transition is absent in the spectral density of other matrices commonly used for spectral clustering provides a physical justification of the performances of the non-backtracking operator in spectral clustering. Copyright (C) EPLA, 2014