MMSE of probabilistic low-rank matrix estimation: Universality with respect to the output channel

TL;DR

This paper analyzes the MMSE in probabilistic low-rank matrix estimation with non-linear measurements, revealing that MMSE depends solely on the Fisher information of the output channel, and explores its implications in community detection.

Contribution

It introduces an AMP algorithm and state evolution for non-linear low-rank matrix estimation, establishing a universality result linking MMSE to Fisher information.

Findings

MMSE depends only on Fisher information of the output channel

Identifies computational and statistical boundaries in community detection

Provides an AMP algorithm with theoretical performance characterization

Abstract

This paper considers probabilistic estimation of a low-rank matrix from non-linear element-wise measurements of its elements. We derive the corresponding approximate message passing (AMP) algorithm and its state evolution. Relying on non-rigorous but standard assumptions motivated by statistical physics, we characterize the minimum mean squared error (MMSE) achievable information theoretically and with the AMP algorithm. Unlike in related problems of linear estimation, in the present setting the MMSE depends on the output channel only trough a single parameter - its Fisher information. We illustrate this striking finding by analysis of submatrix localization, and of detection of communities hidden in a dense stochastic block model. For this example we locate the computational and statistical boundaries that are not equal for rank larger than four.