# Simple Relative Deviation Bounds for Covariance and Gram Matrices

arXiv: https://arxiv.org/abs/2410.05754 · HTML: https://mlsanity.com/paper/2410.05754/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 41 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 1 of 1)

- [Generalization in Kernel Regression Under Realistic Assumptions](https://mlsanity.com/paper/2312.15995/) (2023)

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

- [Near-optimal Rank Adaptive Inference of High Dimensional Matrices](https://mlsanity.com/paper/2510.08117/) (2025)

## Abstract (arXiv)

We provide non-asymptotic, relative deviation bounds for the eigenvalues of empirical covariance and Gram matrices in general settings. Unlike typical uniform bounds, which may fail to capture the behavior of smaller eigenvalues, our results provide sharper control across the spectrum. Our analysis is based on a general-purpose theorem that allows one to convert existing uniform bounds into relative ones. The theorems and techniques emphasize simplicity and should be applicable across various settings.