# Why High-rank Neural Networks Generalize?: An Algebraic Framework with RKHSs

arXiv: https://arxiv.org/abs/2509.21895 · HTML: https://mlsanity.com/paper/2509.21895/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 23 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [Stronger generalization bounds for deep nets via a compression approach](https://mlsanity.com/paper/1802.05296/) (2018)
- [Norm-Based Capacity Control in Neural Networks](https://mlsanity.com/paper/1503.00036/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Orthogonal Deep Neural Networks](https://mlsanity.com/paper/1905.05929/) (2019)
- [Data-dependent Sample Complexity of Deep Neural Networks via Lipschitz Augmentation](https://mlsanity.com/paper/1905.03684/) (2019)
- [Improved Sample Complexities for Deep Networks and Robust Classification via an All-Layer Margin](https://mlsanity.com/paper/1910.04284/) (2019)
- [Compression based bound for non-compressed network: unified generalization error analysis of large compressible deep neural network](https://mlsanity.com/paper/1909.11274/) (2019)
- [Robust Fine-Tuning of Deep Neural Networks with Hessian-based Generalization Guarantees](https://mlsanity.com/paper/2206.02659/) (2022)
- [Truth or Backpropaganda? An Empirical Investigation of Deep Learning Theory](https://mlsanity.com/paper/1910.00359/) (2019)
- [Koopman-based generalization bound: New aspect for full-rank weights](https://mlsanity.com/paper/2302.05825/) (2023)
- [Deep Ridgelet Transform and Unified Universality Theorem for Deep and Shallow Joint-Group-Equivariant Machines](https://mlsanity.com/paper/2405.13682/) (2024)

## Built on by (most cited first, 1 of 1)

- [Unified generalization analysis for physics informed neural networks](https://mlsanity.com/paper/2605.13260/) (2026)

## Abstract (arXiv)

We derive a new Rademacher complexity bound for deep neural networks using Koopman operators, group representations, and reproducing kernel Hilbert spaces (RKHSs). The proposed bound describes why the models with high-rank weight matrices generalize well. Although there are existing bounds that attempt to describe this phenomenon, these existing bounds can be applied to limited types of models. We introduce an algebraic representation of neural networks and a kernel function to construct an RKHS to derive a bound for a wider range of realistic models. This work paves the way for the Koopman-based theory for Rademacher complexity bounds to be valid for more practical situations.