  This paper considers a large class of problems where we seek to recover a low
rank matrix and/or sparse vector from some set of measurements. While methods
based on convex relaxations suffer from a (possibly large) estimator bias, and
other nonconvex methods require the rank or sparsity to be known a priori, we
use nonconvex regularizers to minimize the rank and $l_0$ norm without the
estimator bias from the convex relaxation. We present a novel analysis of the
alternating proximal gradient descent algorithm applied to such problems, and
bound the error between the iterates and the ground truth sparse and low rank
matrices. The algorithm and error bound can be applied to sparse optimization,
matrix completion, and robust principal component analysis as special cases of
our results.
