Adaptive Estimation of Noise Variance and Matrix Estimation via USVT Algorithm
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Consider the problem of denoising a large m× n matrix. This problem has received widespread attention in recent years. A common approach is to assume that the denoised matrix is of low rank and then apply a thresholding algorithm. The error bound in this case is a function of the unknown rank. Another approach comes from the empirical observation that a low rank matrix may have small nuclear norm and therefore approximating the mean matrix via penalization of its nuclear norm sounds reasonable. In 2015 Sourav Chatterjee Chatterjee investigated this problem in the case where the observed entries are a noisy version of a small fraction of the original entries. In his work he introduced the USVT algorithm for matrices with bounded entries (or normally distributed entries with known variance). In this work we want to consider the case that the entries do not necessarily lie in a bounded interval and we don't know their common variance, σ^2, although then we have to consider only the case that all the entries have been observed. In this work we consider two estimators for σ and study their mean square error. Then we use a modified version of the USVT algorithm to give an estimation for the mean value matrix.
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