# Non-parametric Hypothesis Tests for Distributional Group Symmetry

arXiv: https://arxiv.org/abs/2307.15834 · HTML: https://mlsanity.com/paper/2307.15834/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 73 references that are in the map.
- First posted on arXiv 28 Jul 2023.

## Builds on (most cited first, 18 of 18)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [On the Benefits of Invariance in Neural Networks](https://mlsanity.com/paper/2005.00178/) (2020)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Machine Learning in the Search for New Fundamental Physics](https://mlsanity.com/paper/2112.03769/) (2021)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Unsupervised Learning of Group Invariant and Equivariant Representations](https://mlsanity.com/paper/2202.07559/) (2022)
- [Detecting Symmetries with Neural Networks](https://mlsanity.com/paper/2003.13679/) (2020)
- [Provably Strict Generalisation Benefit for Invariance in Kernel Methods](https://mlsanity.com/paper/2106.02346/) (2021)
- [Nystr"om Kernel Mean Embeddings](https://mlsanity.com/paper/2201.13055/) (2022)
- [Local Group Invariant Representations via Orbit Embeddings](https://mlsanity.com/paper/1612.01988/) (2016)
- [Consistency of invariance-based randomization tests](https://mlsanity.com/paper/2104.12260/) (2021)
- [Testing for Geometric Invariance and Equivariance](https://mlsanity.com/paper/2205.15280/) (2022)
- [Estimating Maximal Symmetries of Regression Functions via Subgroup Lattices](https://mlsanity.com/paper/2303.13616/) (2023)

## Built on by (most cited first, 3 of 3)

- [EdgeBench: Unveiling Scaling Laws of Learning from Real-World Environments](https://mlsanity.com/paper/2607.05155/) (2026)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Global optimality under amenable symmetry constraints](https://mlsanity.com/paper/2402.07613/) (2024)

## Abstract (arXiv)

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.