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

Inaccurate records of inventory occur frequently, and by some measures cost retailers approximately 4% in annual sales. Detecting inventory inaccuracies manually is cost-prohibitive, and existing algorithmic solutions rely almost exclusively on learning from longitudinal data, which is insufficient in the dynamic environment induced by modern retail operations. Instead, we propose a solution based on cross-sectional data over stores and SKUs, observing that detecting inventory inaccuracies can be viewed as a problem of identifying anomalies in a (low-rank) Poisson matrix. State-of-the-art approaches to anomaly detection in low-rank matrices apparently fall short. Specifically, from a theoretical perspective, recovery guarantees for these approaches require that non-anomalous 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 entry-wise approach to anomaly detection in low-rank Poisson matrices. Our approach accommodates a general class of probabilistic anomaly models. We show that the cost incurred by our algorithm approaches that of an optimal algorithm at a min-max optimal rate. Using synthetic data and real data from a consumer goods retailer, we show that our approach provides up to a 10× cost reduction over incumbent approaches to anomaly detection. Along the way, we build on recent work that seeks entry-wise error guarantees for matrix completion, establishing such guarantees for sub-exponential matrices, a result of independent interest.