# Bootstraps Regularize Singular Correlation Matrices

arXiv: https://arxiv.org/abs/2004.03165 · HTML: https://mlsanity.com/paper/2004.03165/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Apr 2020.

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

- [End-to-End Neural Shrinkage of Indefinite Pairwise Correlation Matrices for Small-Cap-Inclusive Portfolios](https://mlsanity.com/paper/2608.30446/) (2026)

## Abstract (arXiv)

I show analytically that the average of k bootstrapped correlation matrices rapidly becomes positive-definite as k increases, which provides a simple approach to regularize singular Pearson correlation matrices. If n is the number of objects and t the number of features, the averaged correlation matrix is almost surely positive-definite if k> fracee-1fracntsimeq 1.58fracnt in the limit of large t and n. The probability of obtaining a positive-definite correlation matrix with k bootstraps is also derived for finite n and t. Finally, I demonstrate that the number of required bootstraps is always smaller than n. This method is particularly relevant in fields where n is orders of magnitude larger than the size of data points t, e.g., in finance, genetics, social science, or image processing.