Inference from heterogeneous pairwise data

High-dimensional inference problems involving heterogeneous pairwise observations arise in a variety of applications, including covariance estimation, clustering, and community detection. In this talk I will present a unified approach for the analysis of these problems that yields exact formulas for both the fundamental and algorithmic limits. The high-level idea is to model the observations using a linear Gaussian channel whose input is the tensor product of the latent variables. The limits of this general model are then described by a finite-dimensional variational formula, which provides a decoupling between the prior information about the latent variables (usually a product measure) and the specific structure of the observations. I will also discuss some recent results on computationally efficient methods based on approximate message passing.