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Abstract

Symmetry plays a central role in the sciences, machine learning, and statistics. Generally, a model that accounts for symmetry in the data often acquires various statistical benefits over models that do not. However, incorrectly assuming the presence of symmetry in the data can be detrimental, and so having an inferential tool to verify the existence of a specific symmetry given data is crucial. Those tools currently exist in the form of nonparametric hypothesis tests for group invariance of a probability measure. That is, given a single sample of independent and identically distributed data from some unknown marginal or joint distribution, a test for invariance can determine whether the distribution is invariant with respect to the action of some user-specified compact group of transformations. In contrast, tests for checking equivariant symmetries in conditional distributions are absent from the literature. Equivariance is a form of symmetry in the relationship between two variables: the distribution of one variable conditioned on the transformed value of another is equivalent to a transformed conditional distribution conditional on the untransformed value. In this dissertation, we initiate the study of nonparametric randomization tests for equivariance under the action of a specified locally compact group. The key result underlying our tests is a conditional independence characterization of equivariance. That characterization enables us to adapt existing conditional randomization methodologies for testing conditional independence into a general framework for testing equivariance with finite-sample Type I error control. Moreover, we show that existing tests for marginal invariance can be viewed as conditional randomization tests, unifying those tests and our tests for equivariance under the conditional randomization framework. We propose kernel-based implementations of our randomization tests for equivariance by drawing on ideas from the two-sample kernel hypothesis testing literature. We derive finite-sample power lower bounds for these tests, and we describe approximate versions of these tests that are asymptotically consistent and appropriate for use in practice. We study their properties empirically on synthetic examples and on applications from high-energy particle physics and deep learning. We also provide a high-level documentation for our Julia code implementation of these tests and experiments.