Nonconvex Matrix Factorization from Rank-One Measurements

TL;DR

This paper demonstrates that simple gradient descent with spectral initialization can efficiently recover low-rank matrices from rank-one measurements, achieving near-optimal sample and computational complexity without explicit regularization.

Contribution

It provides the first theoretical guarantee of near-optimal recovery for low-rank matrices from rank-one measurements using nonconvex optimization with implicit regularization.

Findings

Gradient descent converges to the true matrix with high probability.

Implicit regularization keeps iterates incoherent, enabling aggressive step sizes.

Achieves near-optimal sample and computational complexity.

Abstract

We consider the problem of recovering low-rank matrices from random rank-one measurements, which spans numerous applications including covariance sketching, phase retrieval, quantum state tomography, and learning shallow polynomial neural networks, among others. Our approach is to directly estimate the low-rank factor by minimizing a nonconvex quadratic loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is small, this algorithm is guaranteed to converge to the ground truth (up to global ambiguity) with near-optimal sample complexity and computational complexity. To the best of our knowledge, this is the first guarantee that achieves near-optimality in both metrics. In particular, the key enabler of near-optimal computational guarantees is an implicit regularization phenomenon: without explicit regularization, both spectral initialization and the gradient descent iterates automatically stay within a region incoherent with the measurement vectors. This feature allows one to employ much more aggressive step sizes compared with the ones suggested in prior literature, without the need of sample splitting.