A Multiscale Scan Statistic for Adaptive Submatrix Localization

Abstract

We consider the problem of localizing a submatrix with larger-than-usual entry values inside a data matrix, without the prior knowledge of the submatrix size. We establish an optimization framework based on a multiscale scan statistic, and develop algorithms in order to approach the optimizer. We also show that our estimator only requires a signal strength of the same order as the minimax estimator with oracle knowledge of the submatrix size, to exactly recover the anomaly with high probability. We perform some simulations that show that our estimator has superior performance compared to other estimators which do not require prior submatrix knowledge, while being comparatively faster to compute.