We observe that a crucial inventory management problem ('phantom inventory'), that by some measures costs retailers approximately 4% in annual sales 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 extend recent work on entry-wise error guarantees for matrix completion, establishing such guarantees for sub-exponential matrices, where in addition to missing entries, a fraction of entries are corrupted by (an also unknown) anomaly model. We show that for any given budget on the false positive rate (FPR), our approach achieves a true positive rate (TPR) that approaches the TPR of an (unachievable) optimal algorithm at a min-max optimal rate. Using data from a massive consumer goods retailer, we show that our approach provides significant improvements over incumbent approaches to anomaly detection.
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