Robust Low-Rank Matrix Recovery

This paper focuses on robust estimation of the low-rank matrix $\Theta^*$ from the trace regression model $Y=\text{Tr} (\Theta^{*T} X) +\epsilon$. It encompasses four popular problems: sparse linear models, compressive sensing, matrix completion and multi-task regression as its specific examples. Instead of optimizing penalized least-squares, our robust penalized least-squares approach is to replace the quadratic loss by its robust version, which is obtained by the appropriate truncations or shrinkage of the data and hence is very easy to implement. Under only bounded $2+\delta$ moment condition on the noise, we show that the proposed robust penalized trace regression yields an estimator that possesses the same rates as those presented in Negahban and Wainwright (2011, 2012a) with sub-Gaussian error assumption. The rates of convergence are explicitly derived. As a byproduct, we also give a robust covariance matrix estimator called the shrinkage sample covariance and establish its concentration inequality in terms of the spectral norm when the random sample has only bounded fourth moment.