Nonnegative Low-rank Matrix Recovery Can Have Spurious Local Minima
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866911389910564864 |
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| author | Zhang, Richard Y. |
| author_facet | Zhang, Richard Y. |
| contents | Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant $δ=0$. This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant $δ>0$, and to higher-rank ground truths $r^{\star}>1$, regardless of how much the search rank $r\ge r^{\star}$ is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03717 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonnegative Low-rank Matrix Recovery Can Have Spurious Local Minima Zhang, Richard Y. Optimization and Control Machine Learning Low-rank matrix recovery is well-known to exhibit benign nonconvexity under the restricted isometry property (RIP): every second-order critical point is globally optimal, so local methods provably recover the ground truth. Motivated by the strong empirical performance of projected gradient methods for nonnegative low-rank recovery problems, we investigate whether this benign geometry persists when the factor matrices are constrained to be elementwise nonnegative. In the simple setting of a rank-1 nonnegative ground truth, we confirm that benign nonconvexity holds in the fully-observed case with RIP constant $δ=0$. This benign nonconvexity, however, is unstable. It fails to extend to the partially-observed case with any arbitrarily small RIP constant $δ>0$, and to higher-rank ground truths $r^{\star}>1$, regardless of how much the search rank $r\ge r^{\star}$ is overparameterized. Together, these results undermine the standard stability-based explanation for the empirical success of nonconvex methods and suggest that fundamentally different tools are needed to analyze nonnegative low-rank recovery. |
| title | Nonnegative Low-rank Matrix Recovery Can Have Spurious Local Minima |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2505.03717 |