Phase diagram of matrix compressed sensing

TL;DR

This paper analyzes the phase transitions and performance limits of Bayesian inference in matrix compressed sensing, revealing connections to matrix factorization and providing insights into algorithmic solvability.

Contribution

It introduces a theoretical analysis of phase transitions in matrix compressed sensing using the replica method, linking it to matrix factorization and evaluating P-BiG-AMP performance.

Findings

Identification of different phase transition types affecting solvability

Theoretical performance bounds match empirical results of P-BiG-AMP

Asymptotic equations are equivalent to those in matrix factorization

Abstract

In the problem of matrix compressed sensing we aim to recover a low-rank matrix from few of its element-wise linear projections. In this contribution we analyze the asymptotic performance of a Bayes-optimal inference procedure for a model where the matrix to be recovered is a product of random matrices. The results that we obtain using the replica method describe the state evolution of the recently introduced P-BiG-AMP algorithm. We show the existence of different types of phase transitions, their implications for the solvability of the problem, and we compare the results of the theoretical analysis to the performance reached by P-BiG-AMP. Remarkably the asymptotic replica equations for matrix compressed sensing are the same as those for a related but formally different problem of matrix factorization.