# 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 Â_n=A_n+σ Z_n/√(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 Â_n. As an intermediate step we explain how recovery of the spectrum of A_n is possible using only the spectrum of Â_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.

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