Symmetry plays a central role in the sciences, machine learning, and statistics. For situations in which data are known to obey a symmetry, a multitude of methods that exploit symmetry have been developed. Statistical tests for the presence or absence of general group symmetry, however, 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 that apply to two broad settings. The first setting tests for the invariance of a marginal or joint distribution under the action of a compact group. Here, an asymptotically unbiased test only requires a computable metric on the space of probability distributions and the ability to sample uniformly random group elements. Building on this, we propose an easy-to-implement conditional Monte Carlo test and prove that it achieves exact $p$-values with finitely many observations and Monte Carlo samples. 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 invariance or equivariance can be formulated as particular tests of conditional independence. We implement these tests from both settings using kernel methods and study them empirically on synthetic data. Finally, we apply them to testing for symmetry in geomagnetic satellite data and in two problems from high-energy particle physics.

翻译：对称性在科学、机器学习及统计学中扮演着核心角色。针对已知数据服从某种对称性的情形，已有大量利用对称性的方法被开发出来。然而，对于一般群对称性存在与否的统计检验，目前在很大程度上仍是空白。本文基于独立同分布的单样本，提出了针对指定群作用下分布对称性的非参数假设检验。我们给出了适用于两类广泛场景的对称性检验通用框架。第一个场景检验边际分布或联合分布在紧致群作用下的不变性。在此情况下，渐近无偏检验仅需概率分布空间上的可计算度量，以及均匀随机采样群元素的能力。基于此，我们提出一种易于实现的条件蒙特卡洛检验，并证明其在有限观测值和蒙特卡洛样本下能实现精确$p$值。第二个场景检验条件分布在局部紧致群作用下的不变性或等变性。我们证明条件不变性或等变性的检验可转化为特定的条件独立性检验。我们使用核方法实现了这两类场景的检验，并在合成数据上进行了实证研究。最后，我们将这些方法应用于地磁卫星数据及高能粒子物理中的两个问题，以检验其对称性。