Nonconvex Matrix Factorization from Rank-One Measurements
This addresses a foundational problem in machine learning and signal processing with applications like phase retrieval and quantum tomography, offering a novel theoretical guarantee.
The paper tackles the problem of recovering low-rank matrices from random rank-one measurements, achieving near-optimal sample and computational complexity with a nonconvex gradient descent algorithm that converges to the ground truth for small true ranks.
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.