Nonconvex Matrix Factorization from Rank-One Measurements

Yuanxin Li, Cong Ma, Yuxin Chen, Yuejie Chi

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

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.

Original languageEnglish (US)
Article number9317779
Pages (from-to)1928-1950
Number of pages23
JournalIEEE Transactions on Information Theory
Volume67
Issue number3
DOIs
StatePublished - Mar 2021

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences

Keywords

  • Matrix factorization
  • gradient descent
  • nonconvex optimization
  • rank-one measurements

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