Exploiting random matrix theory to improve noisy low-rank matrix approximation

We consider an estimation and denoising problem where the measurement matrix is modeled as a low-rank signal matrix corrupted by a Gaussian white noise matrix. We exploit recent results from random matrix theory to develop an algorithm for improving the quality of the estimated low-rank signal matrix that explicitly accounts for the noisiness of the estimated signal singular vectors. We explain why we are able to obtain this improvement relative to the Eckart-Young-Mirsky theorem motivated “optimal” approximation that employs the rank-k SVD of the measurement matrix and discuss extensions of the result to settings where the Gaussianity assumption can be dropped.