Adaptive Estimation of Noise Variance and Matrix Estimation via USVT Algorithm

We propose a method for estimating the entries of a large noisy matrix when the variance of the noise, $\sigma^2$, is unknown without putting any assumption on the rank of the matrix. We consider the estimator for $\sigma$ introduced by Gavish and Donoho \cite{Gavish} and give an upper bound on its mean squared error. Then with the estimate of the variance, we use a modified version of the Universal Singular Value Thresholding (USVT) algorithm introduced by Chatterjee \cite{Chatterjee} to estimate the noisy matrix. Finally, we give an upper bound on the mean squared error of the estimated matrix.