Bootstraps Regularize Singular Correlation Matrices

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.