  We show that when a high-dimensional data matrix is the sum of a low-rank
matrix and a random error matrix with independent entries, the low-rank
component can be consistently estimated by solving a convex minimization
problem. We develop a new theoretical argument to establish consistency without
assuming sparsity or the existence of any moments of the error matrix, so that
fat-tailed continuous random errors such as Cauchy are allowed. The results are
illustrated by simulations.
