Application of informationpercolation method to reconstruction problems on graphs
Abstract
In this paper we propose a method of proving impossibility results based on applying strong dataprocessing inequalities to estimate mutual information between sets of variables forming certain Markov random fields. The end result is that mutual information between two "far away" (as measured by the graph distance) variables is bounded by the probability of the existence of an open path in a bondpercolation problem on the same graph. Furthermore, stronger bounds can be obtained by establishing mutual information comparison results with an erasure model on the same graph, with erasure probabilities given by the contraction coefficients. As applications, we show that our method gives sharp threshold for partially recovering a rankone perturbation of a random Gaussian matrix (spiked Wigner model), yields the best known upper bound on the noise level for group synchronization (obtained concurrently by Abbe and Boix), and establishes new impossibility result for community detection on the stochastic block model with $k$ communities.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.04195
 Bibcode:
 2018arXiv180604195P
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Probability;
 Mathematics  Statistics Theory
 EPrint:
 Math. Stat. Learn., vol. 2, no. 1, pp. 124, 2019