Nonconvex Matrix Factorization From Rank-One Measurements

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 factor by minimizing a nonconvex least-squares loss function via vanilla gradient descent, following tailored spectral initialization. When true rank bounded constant, this algorithm guaranteed converge ground truth (up global ambiguity) with near-optimal sample complexity computational complexity. To best our knowledge, first guarantee that achieves near-optimality in both metrics. In particular, key enabler guarantees an implicit regularization phenomenon: without explicit regularization, initialization descent iterates automatically stay within region incoherent measurement vectors. This feature allows one employ much more aggressive step sizes compared ones suggested prior literature, need splitting.