Chebyshev Moment Regularization (CMR): Condition-Number Control with Moment Shaping
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| Format: | Preprint |
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2025
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| _version_ | 1866917041488789504 |
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| author | Baek, Jinwoo |
| author_facet | Baek, Jinwoo |
| contents | We introduce \textbf{Chebyshev Moment Regularization (CMR)}, a simple, architecture-agnostic loss that directly optimizes layer spectra. CMR jointly controls spectral edges via a log-condition proxy and shapes the interior via Chebyshev moments, with a decoupled, capped mixing rule that preserves task gradients. We prove strictly monotone descent for the condition proxy, bounded moment gradients, and orthogonal invariance. In an adversarial ``$κ$-stress'' setting (MNIST, 15-layer MLP), \emph{compared to vanilla training}, CMR reduces mean layer condition numbers by $\sim\!10^3$ (from $\approx3.9\!\times\!10^3$ to $\approx3.4$ in 5 epochs), increases average gradient magnitude, and restores test accuracy ( $\approx10\%\!\to\!\approx86\%$ ). These results support \textbf{optimization-driven spectral preconditioning}: directly steering models toward well-conditioned regimes for stable, accurate learning. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_21772 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chebyshev Moment Regularization (CMR): Condition-Number Control with Moment Shaping Baek, Jinwoo Machine Learning Numerical Analysis We introduce \textbf{Chebyshev Moment Regularization (CMR)}, a simple, architecture-agnostic loss that directly optimizes layer spectra. CMR jointly controls spectral edges via a log-condition proxy and shapes the interior via Chebyshev moments, with a decoupled, capped mixing rule that preserves task gradients. We prove strictly monotone descent for the condition proxy, bounded moment gradients, and orthogonal invariance. In an adversarial ``$κ$-stress'' setting (MNIST, 15-layer MLP), \emph{compared to vanilla training}, CMR reduces mean layer condition numbers by $\sim\!10^3$ (from $\approx3.9\!\times\!10^3$ to $\approx3.4$ in 5 epochs), increases average gradient magnitude, and restores test accuracy ( $\approx10\%\!\to\!\approx86\%$ ). These results support \textbf{optimization-driven spectral preconditioning}: directly steering models toward well-conditioned regimes for stable, accurate learning. |
| title | Chebyshev Moment Regularization (CMR): Condition-Number Control with Moment Shaping |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2510.21772 |