On the Loss Landscape Geometry of Regularized Deep Matrix Factorization: Uniqueness and Sharpness
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| Autores principales: | , |
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| Formato: | Preprint |
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2026
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| _version_ | 1866917365581611008 |
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| author | Kamber, Anil Parhi, Rahul |
| author_facet | Kamber, Anil Parhi, Rahul |
| contents | Weight decay is ubiquitous in training deep neural network architectures. Its empirical success is often attributed to capacity control; nonetheless, our theoretical understanding of its effect on the loss landscape and the set of minimizers remains limited. In this paper, we show that $\ell^2$-regularized deep matrix factorization/deep linear network training problems with squared-error loss admit a unique end-to-end minimizer for all target matrices subject to factorization, except for a set of Lebesgue measure zero formed by the depth and the regularization parameter. This observation reveals fundamental properties of the loss landscape of regularized deep matrix factorization problems: the Hessian spectrum is constant across all minimizers of the regularized deep scalar factorization problem with squared-error loss. Moreover, we show that, in regularized deep matrix factorization problems with squared-error loss, if the target matrix does not belong to the Lebesgue measure-zero set, then the Frobenius norm of each layer is constant across all minimizers. This, in turn, yields a global lower bound on the trace of the Hessian evaluated at any minimizer of the regularized deep matrix factorization problem. Furthermore, we establish a critical threshold for the regularization parameter above which the unique end-to-end minimizer collapses to zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27072 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Loss Landscape Geometry of Regularized Deep Matrix Factorization: Uniqueness and Sharpness Kamber, Anil Parhi, Rahul Machine Learning Weight decay is ubiquitous in training deep neural network architectures. Its empirical success is often attributed to capacity control; nonetheless, our theoretical understanding of its effect on the loss landscape and the set of minimizers remains limited. In this paper, we show that $\ell^2$-regularized deep matrix factorization/deep linear network training problems with squared-error loss admit a unique end-to-end minimizer for all target matrices subject to factorization, except for a set of Lebesgue measure zero formed by the depth and the regularization parameter. This observation reveals fundamental properties of the loss landscape of regularized deep matrix factorization problems: the Hessian spectrum is constant across all minimizers of the regularized deep scalar factorization problem with squared-error loss. Moreover, we show that, in regularized deep matrix factorization problems with squared-error loss, if the target matrix does not belong to the Lebesgue measure-zero set, then the Frobenius norm of each layer is constant across all minimizers. This, in turn, yields a global lower bound on the trace of the Hessian evaluated at any minimizer of the regularized deep matrix factorization problem. Furthermore, we establish a critical threshold for the regularization parameter above which the unique end-to-end minimizer collapses to zero. |
| title | On the Loss Landscape Geometry of Regularized Deep Matrix Factorization: Uniqueness and Sharpness |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2603.27072 |