On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms

We introduce a version of the cavity method for diluted mean-field spin models that allows the computation of thermodynamic quantities similar to the Franz-Parisi quenched potential in sparse random graph models. This method is developed in the particular case of partially decimated random constraint satisfaction problems. This allows to develop a theoretical understanding of a class of algorithms for solving constraint satisfaction problems, in which elementary degrees of freedom are sequentially assigned according to the results of a message passing procedure (belief-propagation). We confront this theoretical analysis to the results of extensive numerical simulations.Comment: 32 pages, 24 figure

Improved spectral-norm bounds for clustering

Aiming to unify known results about clustering mixtures of distributions under separation conditions, Kumar and Kannan [KK10] introduced a deterministic condition for clustering datasets. They showed that this single deterministic condition encompasses many previously studied clustering assumptions. More specifically, their proximity condition requires that in the target k-clustering, the projection of a point x onto the line joining its cluster center µ and some other center µ ′ , is a large additive factor closer to µ than to µ ′. This additive factor can be roughly described as k times the spectral norm of the matrix representing the differences between the given (known) dataset and the means of the (unknown) target clustering. Clearly, the proximity condition implies center separation – the distance between any two centers must be as large as the above mentioned bound. In this paper we improve upon the work of Kumar and Kannan [KK10] along several axes. First, we weaken the center separation bound by a factor of √ k, and secondly we weaken the proximity condition by a factor of k (in other words, the revised separation condition is independent of k). Using these weaker bounds we still achieve the same guarantees when al

The number of satisfying assignments of random 2-SAT formulas

We show that throughout the satisfiable phase the normalized number of satisfying assignments of a random 2-SAT formula converges in probability to an expression predicted by the cavity method from statistical physics. The proof is based on showing that the Belief Propagation algorithm renders the correct marginal probability that a variable is set to “true” under a uniformly random satisfying assignment. © 2021 The Authors. Random Structures & Algorithms published by Wiley Periodicals LLC

Information-theoretic thresholds from the cavity method

Vindicating a sophisticated but non-rigorous physics approach called the cavity method, we establish a formula for the mutual information in statistical inference problems induced by random graphs and we show that the mutual information holds the key to understanding certain important phase transitions in random graph models. We work out several concrete applications of these general results. For instance, we pinpoint the exact condensation phase transition in the Potts antiferromagnet on the random graph, thereby improving prior approximate results [Contucci et al.: Communications in Mathematical Physics 2013]. Further, we prove the conjecture from [Krzakala et al.: PNAS 2007] about the condensation phase transition in the random graph coloring problem for any number q ≥ 3 of colors. Moreover , we prove the conjecture on the information-theoretic threshold in the disassortative stochastic block model [Decelle et al.: Phys. Rev. E 2011]. Additionally, our general result implies the conjectured formula for the mutual information in Low-Density Generator Matrix codes [Montanari: IEEE Transactions on Information Theory 2005]

Phase transitions in the qq-coloring of random hypergraphs

31 pages, 7 figuresWe study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes $K$ variables that must be assigned one out of q colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs ($K$ = 2) and bicoloring of random hypergraphs ($q$ = 2), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size q of the domain of the variables and the arity $K$ of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem ($q$ = 2) where we find that for $K$= 3 and $K$ = 4 the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q$ = 2, $K$ ≥ 4. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where $q$ and/or $K$ diverge