arXiv subjects

Guilhem Semerjian

Publications and source records attributed to Guilhem Semerjian.

At least 19 recordsLinked to original sources

arXiv2025

Some observations on the ambivalent role of symmetries in Bayesian inference problems

We collect in this note some observations on the role of symmetries in Bayesian inference problems, that can be useful or detrimental depending on the way they act on the signal and on the observations. We emphasize in particular the need to gauge away unobservable invariances in the definition of a distance between a signal and its estimator, and the consequences this implies for the statistical mechanics treatment of such models, taking as a motivating example the extensive rank matrix factorization problem.

cond-mat.dis-nn↗

arXiv2024

Matrix denoising: Bayes-optimal estimators via low-degree polynomials

We consider the additive version of the matrix denoising problem, where a random symmetric matrix $S$ of size $n$ has to be inferred from the observation of $Y=S+Z$, with $Z$ an independent random matrix modeling a noise. For prior distributions of $S$ and $Z$ that are invariant under conjugation by orthogonal matrices we determine, using results from first and second order free probability theory, the Bayes-optimal (in terms of the mean square error) polynomial estimators of degree at most $D$, asymptotically in $n$, and show that as $D$ increases they converge towards the estimator introduced by Bun, Allez, Bouchaud and Potters in [IEEE Transactions on Information Theory 62, 7475 (2016)]. We conjecture that this optimality holds beyond strictly orthogonally invariant priors, and provide partial evidences of this universality phenomenon when $S$ is an arbitrary Wishart matrix and $Z$ is drawn from the Gaussian Orthogonal Ensemble, a case motivated by the related extensive rank matrix factorization problem.

cond-mat.dis-nn↗

arXiv2024

Faster algorithms for the alignment of sparse correlated Erdös-Rényi random graphs

The correlated Erdös-Rényi random graph ensemble is a probability law on pairs of graphs with $n$ vertices, parametrized by their average degree $λ$ and their correlation coefficient $s$. It can be used as a benchmark for the graph alignment problem, in which the labels of the vertices of one of the graphs are reshuffled by an unknown permutation; the goal is to infer this permutation and thus properly match the pairs of vertices in both graphs. A series of recent works has unveiled the role of Otter's constant $α$ (that controls the exponential rate of growth of the number of unlabeled rooted trees as a function of their sizes) in this problem: for $s>\sqrtα$ and $λ$ large enough it is possible to recover in a time polynomial in $n$ a positive fraction of the hidden permutation. The exponent of this polynomial growth is however quite large and depends on the other parameters, which limits the range of applications of the algorithm. In this work we present a family of faster algorithms for this task, show through numerical simulations that their accuracy is only slightly reduced with respect to the original one, and conjecture that they undergo, in the large $λ$ limit, phase transitions at modified Otter's thresholds $\sqrt{\widehatα}>\sqrtα$, with $\widehatα$ related to the enumeration of a restricted family of trees.

cond-mat.dis-nn↗

arXiv2023

Statistical limits of correlation detection in trees

In this paper we address the problem of testing whether two observed trees $(t,t')$ are sampled either independently or from a joint distribution under which they are correlated. This problem, which we refer to as correlation detection in trees, plays a key role in the study of graph alignment for two correlated random graphs. Motivated by graph alignment, we investigate the conditions of existence of one-sided tests, i.e. tests which have vanishing type I error and non-vanishing power in the limit of large tree depth. For the correlated Galton-Watson model with Poisson offspring of mean $λ>0$ and correlation parameter $s \in (0,1)$, we identify a phase transition in the limit of large degrees at $s = \sqrtα$, where $α\sim 0.3383$ is Otter's constant. Namely, we prove that no such test exists for $s \leq \sqrtα$, and that such a test exists whenever $s > \sqrtα$, for $λ$ large enough. This result sheds new light on the graph alignment problem in the sparse regime (with $O(1)$ average node degrees) and on the performance of the MPAlign method studied in Ganassali et al. (2021), Piccioli et al. (2021), proving in particular the conjecture of Piccioli et al. (2021) that MPAlign succeeds in the partial recovery task for correlation parameter $s>\sqrtα$ provided the average node degree $λ$ is large enough. As a byproduct, we identify a new family of orthogonal polynomials for the Poisson-Galton-Watson measure which enjoy remarkable properties. These polynomials may be of independent interest for a variety of problems involving graphs, trees or branching processes, beyond the scope of graph alignment.

math.ST↗

arXiv2022

The cavity method: from exact solutions to algorithms

The goal of this chapter is to review the main ideas that underlie the cavity method for disordered models defined on random graphs, as well as present some of its outcomes, focusing on the random constraint satisfaction problems for which it provided both a better understanding of the phase transitions they undergo, and suggestions for the development of algorithms to solve them.

cond-mat.dis-nn↗

arXiv2022

Aligning random graphs with a sub-tree similarity message-passing algorithm

The problem of aligning Erdös-Rényi random graphs is a noisy, average-case version of the graph isomorphism problem, in which a pair of correlated random graphs is observed through a random permutation of their vertices. We study a polynomial time message-passing algorithm devised to solve the inference problem of partially recovering the hidden permutation, in the sparse regime with constant average degrees. We perform extensive numerical simulations to determine the range of parameters in which this algorithm achieves partial recovery. We also introduce a generalized ensemble of correlated random graphs with prescribed degree distributions, and extend the algorithm to this case.

cs.IT↗

arXiv2020

Biased measures for random Constraint Satisfaction Problems: larger interaction range and asymptotic expansion

We investigate the clustering transition undergone by an exemplary random constraint satisfaction problem, the bicoloring of $k$-uniform random hypergraphs, when its solutions are weighted non-uniformly, with a soft interaction between variables belonging to distinct hyperedges. We show that the threshold $α_{\rm d}(k)$ for the transition can be further increased with respect to a restricted interaction within the hyperedges, and perform an asymptotic expansion of $α_{\rm d}(k)$ in the large $k$ limit. We find that $α_{\rm d}(k) = \frac{2^{k-1}}{k}(\ln k + \ln \ln k + γ_{\rm d} + o(1))$, where the constant $γ_{\rm d}$ is strictly larger than for the uniform measure over solutions.

cond-mat.dis-nn↗

arXiv2020

The asymptotics of the clustering transition for random constraint satisfaction problems

Random Constraint Satisfaction Problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of $k$-uniform hypergraphs with a density $α$ of constraints, and the $q$-coloring of random graphs with average degree $c$. We show that in the large $k,q$ limit the clustering transition occurs for $α= \frac{2^{k-1}}{k} (\ln k + \ln \ln k + γ_{\rm d} + o(1))$, $c= q (\ln q + \ln \ln q + γ_{\rm d}+ o(1))$, where $γ_{\rm d}$ is the same constant for both models. We characterize $γ_{\rm d}$ via a functional equation, solve the latter numerically to estimate $γ_{\rm d} \approx 0.871$, and obtain an analytic lowerbound $γ_{\rm d} \ge 1 + \ln (2 (\sqrt{2}-1)) \approx 0.812$. Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at $γ_{\rm r}=1$.

cond-mat.dis-nn↗

arXiv2020

Recovery thresholds in the sparse planted matching problem

We consider the statistical inference problem of recovering an unknown perfect matching, hidden in a weighted random graph, by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted matching. A recent work has demonstrated the existence of a phase transition, in the large size limit, between a full and a partial recovery phase for a specific form of the weights distribution on fully connected graphs. We generalize and extend this result in two directions: we obtain a criterion for the location of the phase transition for generic weights distributions and possibly sparse graphs, exploiting a technical connection with branching random walk processes, as well as a quantitatively more precise description of the critical regime around the phase transition.

cond-mat.dis-nn↗

arXiv2019

Solving Constraint Satisfaction Problems through Belief Propagation-guided decimation

Message passing algorithms have proved surprisingly successful in solving hard constraint satisfaction problems on sparse random graphs. In such applications, variables are fixed sequentially to satisfy the constraints. Message passing is run after each step. Its outcome provides an heuristic to make choices at next step. This approach has been referred to as `decimation,' with reference to analogous procedures in statistical physics. The behavior of decimation procedures is poorly understood. Here we consider a simple randomized decimation algorithm based on belief propagation (BP), and analyze its behavior on random k-satisfiability formulae. In particular, we propose a tree model for its analysis and we conjecture that it provides asymptotically exact predictions in the limit of large instances. This conjecture is confirmed by numerical simulations.

cs.AI↗

arXiv2019

Typology of phase transitions in Bayesian inference problems

Many inference problems, notably the stochastic block model (SBM) that generates a random graph with a hidden community structure, undergo phase transitions as a function of the signal-to-noise ratio, and can exhibit hard phases in which optimal inference is information-theoretically possible but computationally challenging. In this paper we refine this description by emphasizing the existence of more generic phase diagrams with a hybrid-hard phase in which it is computationally easy to reach a non-trivial inference accuracy, but computationally hard to match the information theoretically optimal one. We support this discussion by quantitative expansions of the functional cavity equations that describe inference problems on sparse graphs. These expansions shed light on the existence of hybrid-hard phases, for a large class of planted constraint satisfaction problems, and on the question of the tightness of the Kesten-Stigum (KS) bound for the associated tree reconstruction problem. Our results show that the instability of the trivial fixed point is not a generic evidence for the Bayes-optimality of the message passing algorithms. We clarify in particular the status of the symmetric SBM with 4 communities and of the tree reconstruction of the associated Potts model: in the assortative (ferromagnetic) case the KS bound is always tight, whereas in the disassortative (antiferromagnetic) case we exhibit an explicit criterion involving the degree distribution that separates a large degree regime where the KS bound is tight and a low degree regime where it is not. We also investigate the SBM with 2 communities of different sizes, a.k.a. the asymmetric Ising model, and describe quantitatively its computational gap as a function of its asymmetry, and a version of the SBM with 2 groups of communities. We complement this study with numerical simulations of the Belief Propagation algorithm.

cond-mat.dis-nn↗

arXiv2019

Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.

cond-mat.dis-nn↗

arXiv2018

Generating dense packings of hard spheres by soft interaction design

Packing spheres efficiently in large dimension $d$ is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be effectively constructed by this method, up to a packing fraction close to $7\, d\, 2^{-d}$. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.

cond-mat.dis-nn↗

arXiv2017

Phase transitions in the $q$-coloring of random hypergraphs

We 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 $q=2$, $K \ge 4$. 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.

cond-mat.dis-nn↗

arXiv2017

The large deviations of the whitening process in random constraint satisfaction problems

Random constraint satisfaction problems undergo several phase transitions as the ratio between the number of constraints and the number of variables is varied. When this ratio exceeds the satisfiability threshold no more solutions exist; the satisfiable phase, for less constrained problems, is itself divided in an unclustered regime and a clustered one. In the latter solutions are grouped in clusters of nearby solutions separated in configuration space from solutions of other clusters. In addition the rigidity transition signals the appearance of so-called frozen variables in typical solutions: beyond this threshold most solutions belong to clusters with an extensive number of variables taking the same values in all solutions of the cluster. In this paper we refine the description of this phenomenon by estimating the location of the freezing transition, corresponding to the disappearance of all unfrozen solutions (not only typical ones). We also unveil phase transitions for the existence and uniqueness of locked solutions, in which all variables are frozen. From a technical point of view we characterize atypical solutions with a number of frozen variables different from the typical value via a large deviation study of the dynamics of a stripping process (whitening) that unveils the frozen variables of a solution, building upon recent works on atypical trajectories of the bootstrap percolation dynamics. Our results also bear some relevance from an algorithmic perspective, previous numerical studies having shown that heuristic algorithms of various kinds usually output unfrozen solutions.

cond-mat.dis-nn↗

arXiv2016

Network dismantling

We study the network dismantling problem, which consists in determining a minimal set of vertices whose removal leaves the network broken into connected components of sub-extensive size. For a large class of random graphs, this problem is tightly connected to the decycling problem (the removal of vertices leaving the graph acyclic). Exploiting this connection and recent works on epidemic spreading we present precise predictions for the minimal size of a dismantling set in a large random graph with a prescribed (light-tailed) degree distribution. Building on the statistical mechanics perspective we propose a three-stage Min-Sum algorithm for efficiently dismantling networks, including heavy-tailed ones for which the dismantling and decycling problems are not equivalent. We also provide further insights into the dismantling problem concluding that it is an intrinsically collective problem and that optimal dismantling sets cannot be viewed as a collection of individually well performing nodes.

physics.soc-ph↗

arXiv2015

Thermal, quantum and simulated quantum annealing: analytical comparisons for simple models

We study various annealing dynamics, both classical and quantum, for simple mean-field models and explain how to describe their behavior in the thermodynamic limit in terms of differential equations. In particular we emphasize the differences between quantum annealing (i.e. evolution with Schrödinger equation) and simulated quantum annealing (i.e. annealing of a Quantum Monte Carlo simulation).

cond-mat.stat-mech↗

arXiv2014

Minimal contagious sets in random regular graphs

The bootstrap percolation (or threshold model) is a dynamic process modelling the propagation of an epidemic on a graph, where inactive vertices become active if their number of active neighbours reach some threshold. We study an optimization problem related to it, namely the determination of the minimal number of active sites in an initial configuration that leads to the activation of the whole graph under this dynamics, with and without a constraint on the time needed for the complete activation. This problem encompasses in special cases many extremal characteristics of graphs like their independence, decycling or domination number, and can also be seen as a packing problem of repulsive particles. We use the cavity method (including the effects of replica symmetry breaking), an heuristic technique of statistical mechanics many predictions of which have been confirmed rigorously in the recent years. We have obtained in this way several quantitative conjectures on the size of minimal contagious sets in large random regular graphs, the most striking being that 5-regular random graph with a threshold of activation of 3 (resp. 6-regular with threshold 4) have contagious sets containing a fraction 1/6 (resp. 1/4) of the total number of vertices. Equivalently these numbers are the minimal fraction of vertices that have to be removed from a 5-regular (resp. 6-regular) random graph to destroy its 3-core. We also investigated Survey Propagation like algorithmic procedures for solving this optimization problem on single instances of random regular graphs.

cond-mat.dis-nn↗