# Quantitative deterministic equivalent of sample covariance matrices with a general dependence structure

arXiv: https://arxiv.org/abs/2211.13044 · HTML: https://mlsanity.com/paper/2211.13044/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 26 references that are in the map.
- First posted on arXiv 23 Nov 2022.

## Builds on (most cited first, 2 of 2)

- [Implicit Regularization of Random Feature Models](https://mlsanity.com/paper/2002.08404/) (2020)
- [Resolvent convergence under heterogeneous second-moment profiles and quadratic-form control](https://mlsanity.com/paper/2109.02644/) (2021)

## Built on by (most cited first, 9 of 9)

- [Deterministic equivalent and error universality of deep random features learning](https://mlsanity.com/paper/2302.00401/) (2023)
- [Nonlinear spiked covariance matrices and signal propagation in deep neural networks](https://mlsanity.com/paper/2402.10127/) (2024)
- [Dimension-free deterministic equivalents and scaling laws for random feature regression](https://mlsanity.com/paper/2405.15699/) (2024)
- [A Random Matrix Theory Perspective on the Spectrum of Learned Features and Asymptotic Generalization Capabilities](https://mlsanity.com/paper/2410.18938/) (2024)
- [Asymptotics of Learning with Deep Structured (Random) Features](https://mlsanity.com/paper/2402.13999/) (2024)
- [Deterministic equivalent of the Conjugate Kernel matrix associated to Artificial Neural Networks](https://mlsanity.com/paper/2306.05850/) (2023)
- [Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime](https://mlsanity.com/paper/2608.23938/) (2026)
- [A Random Matrix Theory of Masked Self-Supervised Regression](https://mlsanity.com/paper/2601.23208/) (2026)
- [Characterizing the Generalization Error of Random Feature Regression with Arbitrary Data-Augmentation](https://mlsanity.com/paper/2605.10290/) (2026)

## Abstract (arXiv)

We study sample covariance matrices arising from rectangular random matrices with i.i.d. columns. It was previously known that the resolvent of these matrices admits a deterministic equivalent when the spectral parameter stays bounded away from the real axis. We extend this work by proving quantitative bounds involving both the dimensions and the spectral parameter, in particular allowing it to get closer to the real positive semi-line. As applications, we obtain a new bound for the convergence in Kolmogorov distance of the empirical spectral distributions of these general models. We also apply our framework to the problem of regularization of Random Features models in Machine Learning without Gaussian hypothesis.