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arXiv · 2601.21333

Certifying optimality in nonconvex robust PCA

Pinxi Gong·Lexiao Lai·Jianhao Ma
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Abstract

Robust principal component analysis seeks to recover a low-rank matrix from fully observed data with sparse corruptions. A scalable approach fits a low-rank factorization by minimizing the sum of entrywise absolute residuals, leading to a nonsmooth and nonconvex objective. Under standard incoherence conditions and a random model for the corruption support, we study factorizations of the ground-truth rank-$r$ matrix with both factors of rank $r$. With high probability, every such factorization is a Clarke critical point. We also characterize the local geometry: when the factorization rank equals $r$, these solutions are sharp local minima; when it exceeds $r$, they are strict saddle points.

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Pinxi Gong, Lexiao Lai, Jianhao Ma. 2026-01-29. Certifying optimality in nonconvex robust PCA. https://arxiv.org/abs/2601.21333

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