# Deterministic equivalent of the Conjugate Kernel matrix associated to Artificial Neural Networks

arXiv: https://arxiv.org/abs/2306.05850 · HTML: https://mlsanity.com/paper/2306.05850/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 33 references that are in the map.
- First posted on arXiv 9 Jun 2023.

## Builds on (most cited first, 11 of 11)

- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Spectra of the Conjugate Kernel and Neural Tangent Kernel for linear-width neural networks](https://mlsanity.com/paper/2005.11879/) (2020)
- [A Fine-Grained Spectral Perspective on Neural Networks](https://mlsanity.com/paper/1907.10599/) (2019)
- [Global Convergence of Deep Networks with One Wide Layer Followed by Pyramidal Topology](https://mlsanity.com/paper/2002.07867/) (2020)
- [Deterministic equivalent and error universality of deep random features learning](https://mlsanity.com/paper/2302.00401/) (2023)
- [Eigenvalue distribution of nonlinear models of random matrices](https://mlsanity.com/paper/1904.03090/) (2019)
- [Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks](https://mlsanity.com/paper/2109.09304/) (2021)
- [Largest Eigenvalues of the Conjugate Kernel of Single-Layered Neural Networks](https://mlsanity.com/paper/2201.04753/) (2022)
- [Analysis of One-Hidden-Layer Neural Networks via the Resolvent Method](https://mlsanity.com/paper/2105.05115/) (2021)
- [Quantitative deterministic equivalent of sample covariance matrices with a general dependence structure](https://mlsanity.com/paper/2211.13044/) (2022)
- [Resolvent convergence under heterogeneous second-moment profiles and quadratic-form control](https://mlsanity.com/paper/2109.02644/) (2021)

## Built on by (most cited first, 5 of 5)

- [Nonlinear spiked covariance matrices and signal propagation in deep neural networks](https://mlsanity.com/paper/2402.10127/) (2024)
- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [Global law of conjugate kernel random matrices with heavy-tailed weights](https://mlsanity.com/paper/2502.18428/) (2025)
- [Eigenvalue distribution of the Neural Tangent Kernel in the quadratic scaling](https://mlsanity.com/paper/2508.20036/) (2025)
- [Bayesian Inference with Shaped Deep Non-linear MLPs](https://mlsanity.com/paper/2605.30860/) (2026)

## Abstract (arXiv)

We study the Conjugate Kernel associated to a multi-layer linear-width feed-forward neural network with random weights, biases and data. We show that the empirical spectral distribution of the Conjugate Kernel converges to a deterministic limit. More precisely we obtain a deterministic equivalent for its Stieltjes transform and its resolvent, with quantitative bounds involving both the dimension and the spectral parameter. The limiting equivalent objects are described by iterating free convolution of measures and classical matrix operations involving the parameters of the model.