Implicit Regularization in Perturbed Deep Matrix Factorization: Spectral Conditions and Stability
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917540603625472 |
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| author | Wang, Jingzhe Chou, Hung-Hsu |
| author_facet | Wang, Jingzhe Chou, Hung-Hsu |
| contents | This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent exhibits a low-rank phase in the noiseless setting. These conditions show how the target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. We then analyze the perturbed gradient descent dynamics, proving convergence guarantees and quantifying how the perturbation affects iteration complexity and eigenvalue recovery. Finally, we show that the low-rank phase persists under perturbation, with explicit dependence on the perturbation size. Numerical experiments support the theoretical findings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_28613 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Implicit Regularization in Perturbed Deep Matrix Factorization: Spectral Conditions and Stability Wang, Jingzhe Chou, Hung-Hsu Optimization and Control Machine Learning This paper studies the stability of low-rank implicit regularization in perturbed deep matrix factorization, where the target matrix is corrupted by a noise matrix. We first derive sufficient spectral conditions under which gradient descent exhibits a low-rank phase in the noiseless setting. These conditions show how the target spectrum, initialization, and step size jointly determine the existence of a nonempty low-rank interval. We then analyze the perturbed gradient descent dynamics, proving convergence guarantees and quantifying how the perturbation affects iteration complexity and eigenvalue recovery. Finally, we show that the low-rank phase persists under perturbation, with explicit dependence on the perturbation size. Numerical experiments support the theoretical findings. |
| title | Implicit Regularization in Perturbed Deep Matrix Factorization: Spectral Conditions and Stability |
| topic | Optimization and Control Machine Learning |
| url | https://arxiv.org/abs/2605.28613 |