Global and Local Analyses of Nonlinear Low-Rank Matrix Recovery Problems

Yingjie Bi; Javad Lavaei

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arXiv · 2010.04349

Global and Local Analyses of Nonlinear Low-Rank Matrix Recovery Problems

Yingjie Bi·Javad Lavaei

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Abstract

The restricted isometry property (RIP) is a well-known condition that guarantees the absence of spurious local minima in low-rank matrix recovery problems with linear measurements. In this paper, we introduce a novel property named bound difference property (BDP) to study low-rank matrix recovery problems with nonlinear measurements. Using RIP and BDP jointly, we first focus on the rank-1 matrix recovery problem, for which we propose a new criterion to certify the nonexistence of spurious local minima over the entire space. We then analyze the general case with an arbitrary rank and derive a condition to rule out the possibility of having a spurious solution in a ball around the true solution. The developed conditions lead to much stronger theoretical guarantees than the existing bounds on RIP.

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Yingjie Bi, Javad Lavaei. 2020-10-09. Global and Local Analyses of Nonlinear Low-Rank Matrix Recovery Problems. https://arxiv.org/abs/2010.04349

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