Tight oracle bounds for low-rank matrix recovery from a minimal number of random measurements

TL;DR

This paper establishes tight theoretical bounds for low-rank matrix recovery from minimal random measurements, demonstrating stable recovery with noise and extending results to full-rank matrices with decaying singular values.

Contribution

It provides the first stable recovery guarantees from few measurements using nuclear-norm minimization and extends error bounds to full-rank matrices with decaying singular values.

Findings

Stable recovery from minimal measurements is possible with nuclear-norm minimization.

Recovery error is within a constant factor of minimax and oracle errors.

Error bounds are extended to full-rank matrices with decaying singular values.

Abstract

This paper presents several novel theoretical results regarding the recovery of a low-rank matrix from just a few measurements consisting of linear combinations of the matrix entries. We show that properly constrained nuclear-norm minimization stably recovers a low-rank matrix from a constant number of noisy measurements per degree of freedom; this seems to be the first result of this nature. Further, the recovery error from noisy data is within a constant of three targets: 1) the minimax risk, 2) an oracle error that would be available if the column space of the matrix were known, and 3) a more adaptive oracle error which would be available with the knowledge of the column space corresponding to the part of the matrix that stands above the noise. Lastly, the error bounds regarding low-rank matrices are extended to provide an error bound when the matrix has full rank with decaying singular values. The analysis in this paper is based on the restricted isometry property (RIP) introduced in [6] for vectors, and in [22] for matrices.