Linear inverse problems on Erdos-Renyi graphs: Information-theoretic limits and efficient recovery

Abstract

This paper considers the inverse problem with observed variables Y = BGX circle plus Z, where B-G is the incidence matrix of a graph G, X is the vector of unknown vertex variables with a uniform prior, and Z is a noise vector with Bernoulli (epsilon) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery of X is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n)/n for Erdos-Renyi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by alpha = np/log(n), it is shown that exact recovery is possible if and only if alpha > 2/(1 - 2 epsilon)(2) + o(1/(1 - 2 epsilon)(2)). In other words, 2/(1 - 2 epsilon)(2) is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. Full version available in [1].