Matrix Completion with Quantified Uncertainty through Low Rank Gaussian Copula

Modern large scale datasets are often plagued with missing entries; indeed, in the context of recommender system, most entries are missing. While a flurry of imputation algorithms are proposed, almost none can estimate the uncertainty of its imputations. This paper proposes a probabilistic and scalable framework for missing value imputation with quantified uncertainty. Our model, the Low Rank Gaussian Copula, augments a standard probabilistic model, Probabilistic Principal Component Analysis, with marginal transformations for each column that allow the model to better match the distribution of the data. It naturally handles Boolean, ordinal, and real-valued observations and quantifies the uncertainty in each imputation. The time required to fit the model scales linearly with the number of rows and the number of columns in the dataset. Empirical results show the method yields state-of-the-art imputation accuracy across a wide range of datasets, including those with high rank. Our uncertainty measure predicts imputation error well: entries with lower uncertainty do have lower imputation error (on average). Boolean and ordinal entries with the lowest uncertainty have almost zero error. Moreover, for real-valued data, the resulting confidence intervals are well-calibrated.