Improved Algorithms for Matrix Recovery from Rank-One Projections

TL;DR

This paper introduces two fast, non-convex algorithms for low-rank matrix recovery from noisy rank-one projections, with proven linear convergence and sample complexity independent of the matrix's condition number.

Contribution

The paper presents novel non-convex algorithms with theoretical guarantees and computational advantages for matrix recovery from rank-one projections.

Findings

Algorithms achieve linear convergence.

Sample complexity is independent of the condition number.

Numerical experiments demonstrate speed-ups over existing methods.

Abstract

We consider the problem of estimation of a low-rank matrix from a limited number of noisy rank-one projections. In particular, we propose two fast, non-convex \emph{proper} algorithms for matrix recovery and support them with rigorous theoretical analysis. We show that the proposed algorithms enjoy linear convergence and that their sample complexity is independent of the condition number of the unknown true low-rank matrix. By leveraging recent advances in low-rank matrix approximation techniques, we show that our algorithms achieve computational speed-ups over existing methods. Finally, we complement our theory with some numerical experiments.