Inference and uncertainty quantification for noisy matrix completion

Significance Matrix completion finds numerous applications in data science, ranging from information retrieval to medical imaging. While substantial progress has been made in designing estimation algorithms, it remains unknown how to perform optimal statistical inference on the unknown matrix given the obtained estimates—a task at the core of modern decision making. We propose procedures to debias the popular convex and nonconvex estimators and derive distributional characterizations for the resulting debiased estimators. This distributional theory enables valid inference on the unknown matrix. Our procedures 1) yield optimal construction of confidence intervals for missing entries and 2) achieve optimal estimation accuracy in a sharp manner. Noisy matrix completion aims at estimating a low-rank matrix given only partial and corrupted entries. Despite remarkable progress in designing efficient estimation algorithms, it remains largely unclear how to assess the uncertainty of the obtained estimates and how to perform efficient statistical inference on the unknown matrix (e.g., constructing a valid and short confidence interval for an unseen entry). This paper takes a substantial step toward addressing such tasks. We develop a simple procedure to compensate for the bias of the widely used convex and nonconvex estimators. The resulting debiased estimators admit nearly precise nonasymptotic distributional characterizations, which in turn enable optimal construction of confidence intervals/regions for, say, the missing entries and the low-rank factors. Our inferential procedures do not require sample splitting, thus avoiding unnecessary loss of data efficiency. As a byproduct, we obtain a sharp characterization of the estimation accuracy of our debiased estimators in both rate and constant. Our debiased estimators are tractable algorithms that provably achieve full statistical efficiency.