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Matrix completion is often applied to data with entries missing not at random(MNAR). For example, consider a recommendation system where users tend to onlyreveal ratings for items they like. In this case, a matrix completion methodthat relies on entries being revealed at uniformly sampled row and columnindices can yield overly optimistic predictions of unseen user ratings.Recently, various papers have shown that we can reduce this bias in MNAR matrixcompletion if we know the probabilities of different matrix entries beingmissing. These probabilities are typically modeled using logistic regression ornaive Bayes, which make strong assumptions and lack guarantees on the accuracyof the estimated probabilities. In this paper, we suggest a simple approach toestimating these probabilities that avoids these shortcomings. Our approachfollows from the observation that missingness patterns in real data oftenexhibit low nuclear norm structure. We can then estimate the missingnessprobabilities by feeding the (always fully-observed) binary matrix specifyingwhich entries are revealed or missing to an existing nuclear-norm-constrainedmatrix completion algorithm by Davenport et al. [2014]. Thus, we tackle MNARmatrix completion by solving a different matrix completion problem first thatrecovers missingness probabilities. We establish finite-sample error bounds forhow accurate these probability estimates are and how well these estimatesdebias standard matrix completion losses for the original matrix to becompleted. Our experiments show that the proposed debiasing strategy canimprove a variety of existing matrix completion algorithms, and achievesdownstream matrix completion accuracy at least as good as logistic regressionand naive Bayes debiasing baselines that require additional auxiliaryinformation.

Missing Not at Random in Matrix Completion: The Effectiveness of Estimating Missingness Probabilities Under a Low Nuclear Norm Assumption