Work

Nonconvex Matrix Factorization From Rank-One Measurements

Year: 2021

Type: article

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 least-squares loss function via vanilla gradient descent, following a tailored spectral initialization. When the true rank is bounded by a constant, 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.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2021.3050427

Authors: Yuanxin Li, Cong Ma, Yuxin Chen, Yuejie Chi

Institutions: Carnegie Mellon University, University of California, Berkeley, Princeton University

Language: English

Topic: Sparse and Compressive Sensing Techniques

Subfield: Computational Mechanics

Field: Engineering

Domain: Physical Sciences

Open Access status: green