arXiv subjects

Jean Barbier

Publications and source records attributed to Jean Barbier.

65 records · Page 4Linked to original sources

arXiv2016

Mutual information for symmetric rank-one matrix estimation: A proof of the replica formula

Factorizing low-rank matrices has many applications in machine learning and statistics. For probabilistic models in the Bayes optimal setting, a general expression for the mutual information has been proposed using heuristic statistical physics computations, and proven in few specific cases. Here, we show how to rigorously prove the conjectured formula for the symmetric rank-one case. This allows to express the minimal mean-square-error and to characterize the detectability phase transitions in a large set of estimation problems ranging from community detection to sparse PCA. We also show that for a large set of parameters, an iterative algorithm called approximate message-passing is Bayes optimal. There exists, however, a gap between what currently known polynomial algorithms can do and what is expected information theoretically. Additionally, the proof technique has an interest of its own and exploits three essential ingredients: the interpolation method introduced in statistical physics by Guerra, the analysis of the approximate message-passing algorithm and the theory of spatial coupling and threshold saturation in coding. Our approach is generic and applicable to other open problems in statistical estimation where heuristic statistical physics predictions are available.

cs.IT↗

arXiv2016

Threshold Saturation of Spatially Coupled Sparse Superposition Codes for All Memoryless Channels

We recently proved threshold saturation for spatially coupled sparse superposition codes on the additive white Gaussian noise channel. Here we generalize our analysis to a much broader setting. We show for any memoryless channel that spatial coupling allows generalized approximate message-passing (GAMP) decoding to reach the potential (or Bayes optimal) threshold of the code ensemble. Moreover in the large input alphabet size limit: i) the GAMP algorithmic threshold of the underlying (or uncoupled) code ensemble is simply expressed as a Fisher information; ii) the potential threshold tends to Shannon's capacity. Although we focus on coding for sake of coherence with our previous results, the framework and methods are very general and hold for a wide class of generalized estimation problems with random linear mixing.

cs.IT↗

arXiv2016

Proof of Threshold Saturation for Spatially Coupled Sparse Superposition Codes

Recently, a new class of codes, called sparse superposition or sparse regression codes, has been proposed for communication over the AWGN channel. It has been proven that they achieve capacity using power allocation and various forms of iterative decoding. Empirical evidence has also strongly suggested that the codes achieve capacity when spatial coupling and approximate message passing decoding are used, without need of power allocation. In this note we prove that state evolution (which tracks message passing) indeed saturates the potential threshold of the underlying code ensemble, which approaches in a proper limit the optimal threshold. Our proof uses ideas developed in the theory of low-density parity-check codes and compressive sensing.

cs.IT↗

arXiv2015

Scampi: a robust approximate message-passing framework for compressive imaging

Reconstruction of images from noisy linear measurements is a core problem in image processing, for which convex optimization methods based on total variation (TV) minimization have been the long-standing state-of-the-art. We present an alternative probabilistic reconstruction procedure based on approximate message-passing, Scampi, which operates in the compressive regime, where the inverse imaging problem is underdetermined. While the proposed method is related to the recently proposed GrAMPA algorithm of Borgerding, Schniter, and Rangan, we further develop the probabilistic approach to compressive imaging by introducing an expectation-maximizaiton learning of model parameters, making the Scampi robust to model uncertainties. Additionally, our numerical experiments indicate that Scampi can provide reconstruction performance superior to both GrAMPA as well as convex approaches to TV reconstruction. Finally, through exhaustive best-case experiments, we show that in many cases the maximal performance of both Scampi and convex TV can be quite close, even though the approaches are a prori distinct. The theoretical reasons for this correspondence remain an open question. Nevertheless, the proposed algorithm remains more practical, as it requires far less parameter tuning to perform optimally.

cs.IT↗

arXiv2015

Statistical physics and approximate message-passing algorithms for sparse linear estimation problems in signal processing and coding theory

This thesis is interested in the application of statistical physics methods and inference to sparse linear estimation problems. The main tools are the graphical models and approximate message-passing algorithm together with the cavity method. We will also use the replica method of statistical physics of disordered systems which allows to associate to the studied problems a cost function referred as the potential of free entropy in physics. It allows to predict the different phases of typical complexity of the problem as a function of external parameters such as the noise level or the number of measurements one has about the signal: the inference can be typically easy, hard or impossible. We will see that the hard phase corresponds to a regime of coexistence of the actual solution together with another unwanted solution of the message passing equations. In this phase, it represents a metastable state which is not the true equilibrium solution. This phenomenon can be linked to supercooled water blocked in the liquid state below its freezing critical temperature. We will use a method that allows to overcome the metastability mimicing the strategy adopted by nature itself for supercooled water: the nucleation and spatial coupling. In supercooled water, a weak localized perturbation is enough to create a crystal nucleus that will propagate in all the medium thanks to the physical couplings between closeby atoms. The same process will help the algorithm to find the signal, thanks to the introduction of a nucleus containing local information about the signal. It will then spread as a "reconstruction wave" similar to the crystal in the water. After an introduction to statistical inference and sparse linear estimation, we will introduce the necessary tools. Then we will move to applications of these notions to signal processing and coding theory problems.

cs.IT↗

arXiv2015

Approximate message-passing with spatially coupled structured operators, with applications to compressed sensing and sparse superposition codes

We study the behavior of Approximate Message-Passing, 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 Approximate Message-Passing decoder to perform at large rates for moderate block length.

cs.IT↗

arXiv2014

Error correcting codes and spatial coupling

These are notes from the lecture of Rüdiger Urbanke given at the autumn school "Statistical Physics, Optimization, Inference, and Message-Passing Algorithms", that took place in Les Houches, France from Monday September 30th, 2013, till Friday October 11th, 2013. The school was organized by Florent Krzakala from UPMC and ENS Paris, Federico Ricci-Tersenghi from La Sapienza Roma, Lenka Zdeborovà from CEA Saclay and CNRS, and Riccardo Zecchina from Politecnico Torino. The first three sections cover the basics of polar codes and low density parity check codes. In the last three sections, we see how the spatial coupling helps belief propagation decoding.

cs.IT↗

arXiv2014

Replica Analysis and Approximate Message Passing Decoder for Superposition Codes

Superposition codes are efficient for the Additive White Gaussian Noise channel. We provide here a replica analysis of the performances of these codes for large signals. We also consider a Bayesian Approximate Message Passing decoder based on a belief-propagation approach, and discuss its performance using the density evolution technic. Our main findings are 1) for the sizes we can access, the message-passing decoder outperforms other decoders studied in the literature 2) its performance is limited by a sharp phase transition and 3) while these codes reach capacity as $B$ (a crucial parameter in the code) increases, the performance of the message passing decoder worsen as the phase transition goes to lower rates.

cs.IT↗

arXiv2013

The hard-core model on random graphs revisited

We revisit the classical hard-core model, also known as independent set and dual to vertex cover problem, where one puts particles with a first-neighbor hard-core repulsion on the vertices of a random graph. Although the case of random graphs with small and very large average degrees respectively are quite well understood, they yield qualitatively different results and our aim here is to reconciliate these two cases. We revisit results that can be obtained using the (heuristic) cavity method and show that it provides a closed-form conjecture for the exact density of the densest packing on random regular graphs with degree K>=20, and that for K>16 the nature of the phase transition is the same as for large K. This also shows that the hard-code model is the simplest mean-field lattice model for structural glasses and jamming.

cond-mat.dis-nn↗

arXiv2013

Robust error correction for real-valued signals via message-passing decoding and spatial coupling

We revisit the error correction scheme of real-valued signals when the codeword is corrupted by gross errors on a fraction of entries and a small noise on all the entries. Combining the recent developments of approximate message passing and the spatially-coupled measurement matrix in compressed sensing we show that the error correction and its robustness towards noise can be enhanced considerably. We discuss the performance in the large signal limit using previous results on state evolution, as well as for finite size signals through numerical simulations. Even for relatively small sizes, the approach proposed here outperforms convex-relaxation-based decoders.

cs.IT↗

arXiv2012

Compressed Sensing of Approximately-Sparse Signals: Phase Transitions and Optimal Reconstruction

Compressed sensing is designed to measure sparse signals directly in a compressed form. However, most signals of interest are only "approximately sparse", i.e. even though the signal contains only a small fraction of relevant (large) components the other components are not strictly equal to zero, but are only close to zero. In this paper we model the approximately sparse signal with a Gaussian distribution of small components, and we study its compressed sensing with dense random matrices. We use replica calculations to determine the mean-squared error of the Bayes-optimal reconstruction for such signals, as a function of the variance of the small components, the density of large components and the measurement rate. We then use the G-AMP algorithm and we quantify the region of parameters for which this algorithm achieves optimality (for large systems). Finally, we show that in the region where the GAMP for the homogeneous measurement matrices is not optimal, a special "seeding" design of a spatially-coupled measurement matrix allows to restore optimality.

cs.IT↗