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This paper addresses detecting anomalous patterns in images, time-series, andtensor data when the location and scale of the pattern is unknown a priori. Themultiscale scan statistic convolves the proposed pattern with the image atvarious scales and returns the maximum of the resulting tensor. Scale correctedmultiscale scan statistics apply different standardizations at each scale, andthe limiting distribution under the null hypothesis---that the data is onlynoise---is known for smooth patterns. We consider the problem of simultaneouslylearning and detecting the anomalous pattern from a dictionary of smoothpatterns and a database of many tensors. To this end, we show that themultiscale scan statistic is a subexponential random variable, and prove achaining lemma for standardized suprema, which may be of independent interest.Then by averaging the statistics over the database of tensors we can learn thepattern and obtain Bernstein-type error bounds. We will also provide aconstruction of an $\epsilon$-net of the location and scale parameters,providing a computationally tractable approximation with similar error bounds.

Learning Patterns for Detection with Multiscale Scan Statistics