Can Learning Be Explained By Local Optimality In Robust Low-rank Matrix Recovery?

• URL: http://arxiv.org/abs/2302.10963v3
• Date: Fri, 04 Apr 2025 15:57:51 GMT
• Title: Can Learning Be Explained By Local Optimality In Robust Low-rank Matrix Recovery?
• Authors: Jianhao Ma, Salar Fattahi,
• Abstract summary: We show that the true solutions corresponding to $Xstar$ do not emerge as local optima, but rather as strict saddle points.<n>Our findings challenge the conventional belief that all strict saddle points are undesirable and should be avoided.
• Score: 18.49274803854387
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We explore the local landscape of low-rank matrix recovery, focusing on reconstructing a $d_1\times d_2$ matrix $X^\star$ with rank $r$ from $m$ linear measurements, some potentially noisy. When the noise is distributed according to an outlier model, minimizing a nonsmooth $\ell_1$-loss with a simple sub-gradient method can often perfectly recover the ground truth matrix $X^\star$. Given this, a natural question is what optimization property (if any) enables such learning behavior. The most plausible answer is that the ground truth $X^\star$ manifests as a local optimum of the loss function. In this paper, we provide a strong negative answer to this question, showing that, under moderate assumptions, the true solutions corresponding to $X^\star$ do not emerge as local optima, but rather as strict saddle points -- critical points with strictly negative curvature in at least one direction. Our findings challenge the conventional belief that all strict saddle points are undesirable and should be avoided.

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