# Eigenvalue distribution of nonlinear models of random matrices

arXiv: https://arxiv.org/abs/1904.03090 · HTML: https://mlsanity.com/paper/1904.03090/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2019.

## Built on by (most cited first, 27 of 27)

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- [A Random Matrix Theory Perspective on the Spectrum of Learned Features and Asymptotic Generalization Capabilities](https://mlsanity.com/paper/2410.18938/) (2024)
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- [High-dimensional ridge regression with random features for non-identically distributed data with a variance profile](https://mlsanity.com/paper/2504.03035/) (2025)
- [Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling](https://mlsanity.com/paper/2508.20036/) (2025)
- [Gaussian Equivalence for Self-Attention: Asymptotic Spectral Analysis of Attention Matrix](https://mlsanity.com/paper/2510.06685/) (2025)
- [A Random Matrix Theory of Masked Self-Supervised Regression](https://mlsanity.com/paper/2601.23208/) (2026)
- [Feature Learning in Linear-Width Two-Layer Networks: Two vs. One Step of Gradient Descent](https://mlsanity.com/paper/2605.17767/) (2026)
- [Conditional KRR: Injecting Unpenalized Features into Kernel Methods with Applications to Kernel Thresholding](https://mlsanity.com/paper/2605.26067/) (2026)
- [Spectral Perturbation of the Empirical Fisher Information Matrix under Weight Quantization](https://mlsanity.com/paper/2606.28432/) (2026)

## Abstract (arXiv)

This paper is concerned with the asymptotic empirical eigenvalue distribution of a non linear random matrix ensemble. More precisely we consider M= frac1m YY^* with Y=f(WX) where W and X are random rectangular matrices with i.i.d. centered entries. The function f is applied pointwise and can be seen as an activation function in (random) neural networks. We compute the asymptotic empirical distribution of this ensemble in the case where W and X have sub-Gaussian tails and f is real analytic. This extends a previous result where the case of Gaussian matrices W and X is considered. We also investigate the same questions in the multi-layer case, regarding neural network applications.