Statistical limits of graphical channel models and a semidefinite programming approach

Abstract

Community recovery is a major challenge in data science and computer science. The goal in community recovery is to find the hidden clusters from given relational data, which is often represented as a labeled hyper graph where nodes correspond to items needing to be labeled and edges correspond to observed relations between the items. We investigate the problem of exact recovery in the class of statistical models which can be expressed in terms of graphical channels. In a graphical channel model, we observe noisy measurements of the relations between k nodes while the true labeling is unknown to us, and the goal is to recover the labels correctly. This generalizes both the stochastic block models and spiked tensor models for principal component analysis, which has gained much interest over the last decade. We focus on two aspects of exact recovery: statistical limits and efficient algorithms achieving the statistic limit. For the statistical limits, we show that the achievability of exact recovery is essentially determined by whether we can recover the label of one node given other nodes labels with fairly high probability. This phenomenon was observed by Abbe et al. for generic stochastic block models, and called "local-to-global amplification". We confirm that local-to-global amplification indeed holds for generic graphical channel models, under some regularity assumptions. As a corollary, the threshold for exact recovery is explicitly determined. For algorithmic concerns, we consider two examples of graphical channel models, (i) the spiked tensor model with additive Gaussian noise, and (ii) the generalization of the stochastic block model for k-uniform hypergraphs. We propose a strategy which we call "truncate-and-relax", based on a standard semidefinite relaxation technique. We show that in these two models, the algorithm based on this strategy achieves exact recovery up to a threshold which orderwise matches the statistical threshold. We complement this by showing the limitation of the algorithm.