Non-convex matrix sensing: Breaking the quadratic rank barrier in the sample complexity

TL;DR

This paper demonstrates that non-convex matrix sensing methods can achieve the same sample complexity as convex methods, breaking the quadratic rank barrier and requiring only linear dependence on the matrix rank.

Contribution

The authors prove that factorized gradient descent with spectral initialization can recover low-rank positive semidefinite matrices with linear sample complexity in rank, improving previous quadratic bounds.

Findings

Non-convex methods match convex sample complexity in matrix sensing.

Factorized gradient descent converges linearly with O(rdκ^2) samples.

Extension to noisy measurements achieves minimax optimal error.

Abstract

For the problem of reconstructing a low-rank matrix from a few linear measurements, two classes of algorithms have been widely studied in the literature: convex approaches based on nuclear norm minimization, and non-convex approaches that use factorized gradient descent. Under certain statistical model assumptions, it is known that nuclear norm minimization recovers the ground truth as soon as the number of samples scales linearly with the number of degrees of freedom of the ground-truth. In contrast, while non-convex approaches are computationally less expensive, existing recovery guarantees assume that the number of samples scales at least quadratically with the rank $r$ of the ground-truth matrix. In this paper, we close this gap by showing that the non-convex approaches can be as efficient as nuclear norm minimization in terms of sample complexity. Namely, we consider the problem of reconstructing a positive semidefinite matrix from a few Gaussian measurements. We show that factorized gradient descent with spectral initialization converges to the ground truth at a linear rate as soon as the number of samples scales with $ \Omega (rd\kappa^2)$, where $d$ is the dimension, and $\kappa$ is the condition number of the ground truth matrix. This improves the previous rank-dependence in the sample complexity of non-convex matrix factorization from quadratic to linear. Furthermore, we extend our theory to the noisy setting, where we show that with noisy measurements, factorized gradient descent with spectral initialization converges to the minimax optimal error up to a factor linear in $\kappa$. Our proof relies on a probabilistic decoupling argument, where we show that the gradient descent iterates are only weakly dependent on the individual entries of the measurement matrices. We expect that our proof technique is of independent interest for other non-convex problems.