  This paper examines a general class of noisy matrix completion tasks where
the goal is to estimate a matrix from observations obtained at a subset of its
entries, each of which is subject to random noise or corruption. Our specific
focus is on settings where the matrix to be estimated is well-approximated by a
product of two (a priori unknown) matrices, one of which is sparse. Such
structural models - referred to here as "sparse factor models" - have been
widely used, for example, in subspace clustering applications, as well as in
contemporary sparse modeling and dictionary learning tasks. Our main
theoretical contributions are estimation error bounds for sparsity-regularized
maximum likelihood estimators for problems of this form, which are applicable
to a number of different observation noise or corruption models. Several
specific implications are examined, including scenarios where observations are
corrupted by additive Gaussian noise or additive heavier-tailed (Laplace)
noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit)
observations. We also propose a simple algorithmic approach based on the
alternating direction method of multipliers for these tasks, and provide
experimental evidence to support our error analyses.
