Phase Diagram and Approximate Message Passing for Blind Calibration and Dictionary Learning

We consider dictionary learning and blind calibration for signals and matrices created from a random ensemble. We study the mean-squared error in the limit of large signal dimension using the replica method and unveil the appearance of phase transitions delimiting impossible, possible-but-hard and possible inference regions. We also introduce an approximate message passing algorithm that asymptotically matches the theoretical performance, and show through numerical tests that it performs very well, for the calibration problem, for tractable system sizes.Comment: 5 page

Reweighted belief propagation and quiet planting for random K-SAT

We study the random K-satisfiability problem using a partition function where each solution is reweighted according to the number of variables that satisfy every clause. We apply belief propagation and the related cavity method to the reweighted partition function. This allows us to obtain several new results on the properties of random K-satisfiability problem. In particular the reweighting allows to introduce a planted ensemble that generates instances that are, in some region of parameters, equivalent to random instances. We are hence able to generate at the same time a typical random SAT instance and one of its solutions. We study the relation between clustering and belief propagation fixed points and we give a direct evidence for the existence of purely entropic (rather than energetic) barriers between clusters in some region of parameters in the random K-satisfiability problem. We exhibit, in some large planted instances, solutions with a non-trivial whitening core; such solutions were known to exist but were so far never found on very large instances. Finally, we discuss algorithmic hardness of such planted instances and we determine a region of parameters in which planting leads to satisfiable benchmarks that, up to our knowledge, are the hardest known.Comment: 23 pages, 4 figures, revised for readability, stability expression correcte

Non-adaptive pooling strategies for detection of rare faulty items

We study non-adaptive pooling strategies for detection of rare faulty items. Given a binary sparse N-dimensional signal x, how to construct a sparse binary MxN pooling matrix F such that the signal can be reconstructed from the smallest possible number M of measurements y=Fx? We show that a very low number of measurements is possible for random spatially coupled design of pools F. Our design might find application in genetic screening or compressed genotyping. We show that our results are robust with respect to the uncertainty in the matrix F when some elements are mistaken.Comment: 5 page

Clustering from Sparse Pairwise Measurements

We consider the problem of grouping items into clusters based on few random pairwise comparisons between the items. We introduce three closely related algorithms for this task: a belief propagation algorithm approximating the Bayes optimal solution, and two spectral algorithms based on the non-backtracking and Bethe Hessian operators. For the case of two symmetric clusters, we conjecture that these algorithms are asymptotically optimal in that they detect the clusters as soon as it is information theoretically possible to do so. We substantiate this claim for one of the spectral approaches we introduce

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.Comment: 8 pages, 10 figure

Champs Conditionnels Aléatoires pour l'Annotation d'Arbres

National audienceAvec en vue la transformation de documents semi-structurés de type XML, nous nous intéressons au problème de l'annotation de tels documents par apprentissage statistique, à partir d'exemples de documents déjà annotés. Afin de modéliser la probabilité d'une annotation connaissant un document, nous nous plaçons dans le cadre des champs conditionnels aléatoires. Ce modèle a déjà fait ses preuves pour l'annotation de séquences : nous l'adaptons ici aux arbres ordonnés d'arité non bornée. Nous étudions l'expressivité du nouveau modèle ainsi introduit en le comparant aux automates d'arbres stochastiques (ou grammaires régulières probabilistes d'arbres). Nous présentons aussi en détail l'algorithme de recherche de l'annotation la plus probable et l'algorithme d'inférence pour ce modèle. Ces algorithmes sont implantés dans une librairie Tree CRF écrite en JAVA. Ces travaux sont des préliminaires qui nous permettront par la suite d'étudier les applications du modèle pour la transformation de documents