  In this paper we study the problem of reconstruction of a low-rank matrix
observed with additive Gaussian noise. First we show that under mild
assumptions (about the prior distribution of the signal matrix) we can restrict
our attention to reconstruction methods that are based on the singular value
decomposition of the observed matrix and act only on its singular values
(preserving the singular vectors). Then we determine the effect of noise on the
SVD of low-rank matrices by building a connection between matrix reconstruction
problem and spiked population model in random matrix theory. Based on this
knowledge, we propose a new reconstruction method, called RMT, that is designed
to reverse the effect of the noise on the singular values of the signal matrix
and adjust for its effect on the singular vectors. With an extensive simulation
study we show that the proposed method outperform even oracle versions of both
soft and hard thresholding methods and closely matches the performance of a
general oracle scheme.
