Causal Matrix Completion
Abstract
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 finitesample consistency and asymptotic normality of our estimator. Our analysis also leads to new theoretical results for the matrix completion literature. In particular, we establish entrywise, i.e., maxnorm, finitesample consistency and asymptotic normality results for matrix completion with MNAR data. As a special case, this also provides entrywise bounds for matrix completion with MCAR data. Across simulated and real data, we demonstrate the efficacy of our proposed estimator.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.15154
 Bibcode:
 2021arXiv210915154A
 Keywords:

 Economics  Econometrics;
 Computer Science  Machine Learning;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning