Bootstraps regularize singular correlation matrices

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