Shrinkage Estimation of Functions of Large Noisy Symmetric Matrices

We study the problem of estimating functions of a large symmetric matrix $A_n$ when we only have access to a noisy estimate $\hat{A}_n=A_n+\sigma Z_n/\sqrt{n}.$ We are interested in the case that $Z_n$ is a Wigner ensemble and suggest an algorithm based on nonlinear shrinkage of the eigenvalues of $\hat{A}_n.$ As an intermediate step we explain how recovery of the spectrum of $A_n$ is possible using only the spectrum of $\hat{A}_n$. Our algorithm has important applications, for example, in solving high-dimensional noisy systems of equations or symmetric matrix denoising. Throughout our analysis we rely on tools from random matrix theory.