Consistency of invariance-based randomization tests

Invariance-based randomization tests -- such as permutation tests -- are an important and widely used class of statistical methods. They allow drawing inferences with few assumptions on the data distribution. Most work focuses on their type I error control properties, while their consistency properties are much less understood. We develop a general framework and a set of results on the consistency of invariance-based randomization tests in signal-plus-noise models. Our framework is grounded in the deep mathematical area of representation theory. We allow the transforms to be general compact topological groups, such as rotation groups. Moreover, we allow actions by general linear group representations. We apply our framework to a number of fundamental and highly important problems in statistics, including sparse vector detection, testing for low-rank matrices in noise, sparse detection in linear regression, symmetric submatrix detection, and two-sample testing. Perhaps surprisingly, we find that randomization tests can adapt to problem structure and detect signals at the same rate as tests with full knowledge of the noise distribution.