# Chebyshev Moment Regularization (CMR): Condition-Number Control with Moment Shaping

arXiv: https://arxiv.org/abs/2510.21772 · HTML: https://mlsanity.com/paper/2510.21772/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 9 references that are in the map.
- First posted on arXiv 17 Oct 2025.

## Builds on (most cited first, 4 of 4)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)

## Abstract (arXiv)

We introduce 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 ``kappa-stress'' setting (MNIST, 15-layer MLP), compared to vanilla training, CMR reduces mean layer condition numbers by ~!103 (from ≈3.9!×!103 to ≈3.4 in 5 epochs), increases average gradient magnitude, and restores test accuracy ( ≈10%!→!≈86% ). These results support optimization-driven spectral preconditioning: directly steering models toward well-conditioned regimes for stable, accurate learning.