Near-Optimal Entrywise Anomaly Detection for Low-Rank Matrices with Sub-Exponential Noise

We study the problem of identifying anomalies in a low-rank matrix observed with sub-exponential noise, motivated by applications in retail and inventory management. State of the art approaches to anomaly detection in low-rank matrices apparently fall short, since they require that nonanomalous entries be observed with vanishingly small noise (which is not the case in our problem, and indeed in many applications). So motivated, we propose a conceptually simple entrywise approach to anomaly detection in low-rank matrices. Our approach accommodates a general class of probabilistic anomaly models. We extend recent work on entrywise error guarantees for matrix completion, establishing such guarantees for subexponential matrices, where in addition to missing entries, a fraction of entries are corrupted by (an also unknown) anomaly model. Viewing the anomaly detection as a classification task, to the best of our knowledge, we are the first to achieve the min-max optimal detection rate (up to log factors). Using data from a massive consumer goods retailer, we show that our approach provides significant improvements over incumbent approaches to anomaly detection.