Reconstruction of a low-rank matrix in the presence of Gaussian noise

This paper addresses the problem of reconstructing a low-rank signal matrix observed with additive Gaussian noise. We first establish that, under mild assumptions, one can restrict attention to orthogonally equivariant reconstruction methods, which act only on the singular values of the observed matrix and do not affect its singular vectors. Using recent results in random matrix theory, we then propose a new reconstruction method that aims to reverse the effect of the noise on the singular value decomposition of the signal matrix. In conjunction with the proposed reconstruction method we also introduce a Kolmogorov–Smirnov based estimator of the noise variance. We show with an extensive simulation study that the proposed method outperforms oracle versions of both soft and hard thresholding methods, and closely matches the performance of the oracle orthogonally equivariant method.