  Matrix completion is the study of recovering an underlying matrix from a
sparse subset of noisy observations. Traditionally, it is assumed that the
entries of the matrix are "missing completely at random" (MCAR), i.e., each
entry is revealed at random, independent of everything else, with uniform
probability. This is likely unrealistic due to the presence of "latent
confounders", i.e., unobserved factors that determine both the entries of the
underlying matrix and the missingness pattern in the observed matrix. For
example, in the context of movie recommender systems -- a canonical application
for matrix completion -- a user who vehemently dislikes horror films is
unlikely to ever watch horror films. In general, these confounders yield
"missing not at random" (MNAR) data, which can severely impact any inference
procedure that does not correct for this bias. We develop a formal causal model
for matrix completion through the language of potential outcomes, and provide
novel identification arguments for a variety of causal estimands of interest.
We design a procedure, which we call "synthetic nearest neighbors" (SNN), to
estimate these causal estimands. We prove finite-sample consistency and
asymptotic normality of our estimator. Our analysis also leads to new
theoretical results for the matrix completion literature. In particular, we
establish entry-wise, i.e., max-norm, finite-sample consistency and asymptotic
normality results for matrix completion with MNAR data. As a special case, this
also provides entry-wise bounds for matrix completion with MCAR data. Across
simulated and real data, we demonstrate the efficacy of our proposed estimator.
