Nuclear norm penalized LAD estimator for low rank matrix recovery

Abstract

In the thesis we propose a novel method for low rank matrix recovery. We study the framework using absolute deviation loss function and nuclear penalty. While nuclear norm penalty is widely utilized heuristic method for shrinkage to low rank solution, the absolute deviation loss function is rarely studied. We establish an near oracle optimal recovery bound and gave a proof using E-net covering argument under certain restricted isometry and restricted eigenvalue assumptions. The estimator is able to recover the underlying matrix with high probability with limited observations that the number of observation is more than the degree of freedom but less than a power of dimension. Our estimator has two advantages. First the theoretical tuning parameter does not depends on the knowledge of the noise level, and the bound can be derived even when noises have fatter tails than normal distribution. The second advantage is that absolute deviation loss function is robust compared with the popular square loss function.