In Proceedings of the 14th International Conference on Randomization and Computation (RANDOM), volume 6302 of Lecture Notes in Computer Science, pages 434-447. Springer, 2010
In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when lambda < (ln 2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1. This is almost tight as reconstruction is known to hold when lambda> (e+o(1))ln^2(k). We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees.
A key insight from statistical physics about spin systems on random graphs is the central role played by Gibbs measures on trees. We determine the local weak limit of the hardcore model on random regular graphs asymptotically until just below its condensation threshold, showing that it converges in probability locally in a strong sense to the free boundary condition Gibbs measure on the tree. As a consequence we show that the reconstruction threshold on the random graph, indicative of the onset of point to set spatial correlations, is equal to the reconstruction threshold on the $d$-regular tr…
The hardcore model is a model of lattice gas systems which has received much attention in statistical physics, probability theory and theoretical computer science. It is the probability distribution over independent sets $I$ of a graph weighted proportionally to $λ^{|I|}$ with fugacity parameter $λ$. We prove that at the uniqueness threshold of the hardcore model on the $d$-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree $d$. Specifically, we show that unless NP$=$RP there is no polynomial time approximation scheme for the partition f…
In this paper, we consider a broadcasting process in which information is propagated from a given root node on a noisy tree network, and answer the question that whether the symbols at the nth level of the tree contain non-vanishing information of the root as n goes to infinity. Although the reconstruction problem on the tree has been studied in numerous contexts including information theory, mathematical genetics and statistical physics, the existing literatures with rigorous reconstruction thresholds established are very limited. In the remarkable work of Borgs, Chayes, Mossel and Roch (The …
The reconstruction problem on the tree has been studied in numerous contexts including statistical physics, information theory and computational biology. However, rigorous reconstruction thresholds have only been established in a small number of models. We prove the first exact reconstruction threshold in a non-binary model establishing the Kesten-Stigum bound for the 3-state Potts model on regular trees of large degree. We further establish that the Kesten-Stigum bound is not tight for the $q$-state Potts model when $q \geq 5$. Moreover, we determine asymptotics for the reconstruction thresho…
Queensborough Community College, City University of New York（纽约城市大学皇后区社区学院）；University of Washington（华盛顿大学）
The tree reconstruction problem is to collect and analyze massive data at the $n$th level of the tree, to identify whether there is non-vanishing information of the root, as $n$ goes to infinity. Its connection to the clustering problem in the setting of the stochastic block model, which has wide applications in machine learning and data mining, has been well established. For the stochastic block model, an "information-theoretically-solvable-but-computationally-hard" region, or say "hybrid-hard phase", appears whenever the reconstruction bound is not tight of the corresponding reconstruction o…
We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of "broadcasting on a tree"). The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2x2 transition matrix P. We say that "reconstruction is possible" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\to\infty$. Adapting a method of Brightwell and Winkler, we obtain new conditions under whi…
We establish the exact threshold for the reconstruction problem for a binary asymmetric channel on the b-ary tree, provided that the asymmetry is sufficiently small. This is the first exact reconstruction threshold obtained in roughly a decade. We discuss the implications of our result for Glauber dynamics, phylogenetic reconstruction, and so-called ``replica symmetry breaking'' in spin glasses and random satisfiability problems.
We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $λ$, called the activity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $ω$ which is the positive solution to $λ=ω(1+ω)^b$, and vertices are occupied with probability $ω/(1+ω)$ when their parent is uno…
The stochastic block model is one of the oldest and most ubiquitous models for studying clustering and community detection. In an exciting sequence of developments, motivated by deep but non-rigorous ideas from statistical physics, Decelle et al. conjectured a sharp threshold for when community detection is possible in the sparse regime. Mossel, Neeman and Sly and Massoulie proved the conjecture and gave matching algorithms and lower bounds. Here we revisit the stochastic block model from the perspective of semirandom models where we allow an adversary to make `helpful' changes that strengthen…
It is known that the Kesten-Stigum reconstruction bound is tight for roughly symmetric binary channels. In this paper, we will adopt a refined analysis of moment recursion on a weighted version of the magnetization, which is engaged in Sly [2011] to handle the symmetric Potts model, and establish the critical condition of the asymmetric Ising model to make Kesten-Stigum bound the reconstruction threshold on regular $d$-ary trees.
We study the problem of reconstructing a perfect matching $M^*$ hidden in a randomly weighted $n\times n$ bipartite graph. The edge set includes every node pair in $M^*$ and each of the $n(n-1)$ node pairs not in $M^*$ independently with probability $d/n$. The weight of each edge $e$ is independently drawn from the distribution $\mathcal{P}$ if $e \in M^*$ and from $\mathcal{Q}$ if $e \notin M^*$. We show that if $\sqrt{d} B(\mathcal{P},\mathcal{Q}) \le 1$, where $B(\mathcal{P},\mathcal{Q})$ stands for the Bhattacharyya coefficient, the reconstruction error (average fraction of misclassified e…
INRIA（法国国家信息与自动化研究所）；ENS（巴黎高等师范学院）；PSL Research University（PSL研究大学）
We study the fundamental limits for reconstruction in weighted graph (or matrix) database alignment. We consider a model of two graphs where $π^*$ is a planted uniform permutation and all pairs of edge weights $(A_{i,j}, B_{π^*(i),π^*(j)})_{1 \leq i<j \leq n}$ are i.i.d. pairs of Gaussian variables with zero mean, unit variance and correlation parameter $ρ\in [0,1]$. We prove that there is a sharp threshold for exact recovery of $π^*$: if $n ρ^2 \geq (4+ε) \log n + ω(1)$ for some $ε>0$, there is an estimator $\hatπ$ -- namely the MAP estimator -- based on the observation of databases $A,B$ tha…
Genome sequencing is the basis for many modern biological and medicinal studies. With recent technological advances, metagenomics has become a problem of interest. This problem entails the analysis and reconstruction of multiple DNA sequences from different sources. Shotgun genome sequencing works by breaking up long DNA sequences into shorter segments called reads. Given this collection of reads, one would like to reconstruct the original collection of DNA sequences. For experimental design in metagenomics, it is important to understand how the minimal read length necessary for reliable recon…
Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant $α_2$ in the $2$-log-Sobolev inequality ($2$-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., $p$-NLSI, $p\ge1$). As an example, we show $α_1 = 1+\frac{1+o(1)}{\log k}$. By integrating the $1$-NLSI we obtain the new strong data processing inequality (SDPI), which in turn allows us to improve results of Mos…
