Hypothesis Tests for Distributional Group Symmetry with Applications to Particle Physics
Keywords: invariance, equivariance, symmetry, exact test, kernel methods
TL;DR: We develop a general formulation of hypothesis tests for distributional group symmetry and evaluate kernel-based instantiations of these tests on problems in particle physics.
Abstract: Symmetry plays a central role in the sciences, machine learning, and statistics. When data are known to obey a symmetry, various methods that exploit symmetry have been developed. However, statistical tests for the presence of group invariance focus on a handful of specialized situations, and tests for equivariance are largely non-existent. This work formulates non-parametric hypothesis tests, based on a single independent and identically distributed sample, for distributional symmetry under a specified group. We provide a general formulation of tests for symmetry within two broad settings. Generalizing existing theory for group-based randomization tests, the first setting tests for the invariance of a marginal or joint distribution under the action of a compact group. The second setting tests for the invariance or equivariance of a conditional distribution under the action of a locally compact group. We show that the test for conditional symmetry can be formulated as a test for conditional independence. We implement our tests using kernel methods and apply them to testing for symmetry in problems from high-energy particle physics.
Submission Track: Original Research
Submission Number: 66
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