Nonconvex Deterministic Matrix Completion by Projected Gradient Descent Methods

Hang Xu; Song Li; Junhong Lin

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arXiv · 2401.06592

Nonconvex Deterministic Matrix Completion by Projected Gradient Descent Methods

Hang Xu·Song Li·Junhong Lin

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Abstract

We study deterministic matrix completion problem, i.e., recovering a low-rank matrix from a few observed entries where the sampling set is chosen as the edge set of a Ramanujan graph. We first investigate projected gradient descent (PGD) applied to a Burer-Monteiro least-squares problem and show that it converges linearly to the incoherent ground-truth with respect to the condition number \k{appa} of ground-truth under a benign initialization and large samples. We next apply the scaled variant of PGD to deal with the ill-conditioned case when \k{appa} is large, and we show the algorithm converges at a linear rate independent of the condition number \k{appa} under similar conditions. Finally, we provide numerical experiments to corroborate our results.

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BibTeX RIS

Hang Xu, Song Li, Junhong Lin. 2024-01-12. Nonconvex Deterministic Matrix Completion by Projected Gradient Descent Methods. https://arxiv.org/abs/2401.06592

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