Nuclear Norm Regularization for Deep Learning
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914968815796224 |
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| author | Scarvelis, Christopher Solomon, Justin |
| author_facet | Scarvelis, Christopher Solomon, Justin |
| contents | Penalizing the nuclear norm of a function's Jacobian encourages it to locally behave like a low-rank linear map. Such functions vary locally along only a handful of directions, making the Jacobian nuclear norm a natural regularizer for machine learning problems. However, this regularizer is intractable for high-dimensional problems, as it requires computing a large Jacobian matrix and taking its singular value decomposition. We show how to efficiently penalize the Jacobian nuclear norm using techniques tailor-made for deep learning. We prove that for functions parametrized as compositions $f = g \circ h$, one may equivalently penalize the average squared Frobenius norm of $Jg$ and $Jh$. We then propose a denoising-style approximation that avoids the Jacobian computations altogether. Our method is simple, efficient, and accurate, enabling Jacobian nuclear norm regularization to scale to high-dimensional deep learning problems. We complement our theory with an empirical study of our regularizer's performance and investigate applications to denoising and representation learning. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14544 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nuclear Norm Regularization for Deep Learning Scarvelis, Christopher Solomon, Justin Machine Learning Penalizing the nuclear norm of a function's Jacobian encourages it to locally behave like a low-rank linear map. Such functions vary locally along only a handful of directions, making the Jacobian nuclear norm a natural regularizer for machine learning problems. However, this regularizer is intractable for high-dimensional problems, as it requires computing a large Jacobian matrix and taking its singular value decomposition. We show how to efficiently penalize the Jacobian nuclear norm using techniques tailor-made for deep learning. We prove that for functions parametrized as compositions $f = g \circ h$, one may equivalently penalize the average squared Frobenius norm of $Jg$ and $Jh$. We then propose a denoising-style approximation that avoids the Jacobian computations altogether. Our method is simple, efficient, and accurate, enabling Jacobian nuclear norm regularization to scale to high-dimensional deep learning problems. We complement our theory with an empirical study of our regularizer's performance and investigate applications to denoising and representation learning. |
| title | Nuclear Norm Regularization for Deep Learning |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2405.14544 |