# Consistency of invariance-based randomization tests

arXiv: https://arxiv.org/abs/2104.12260 · HTML: https://mlsanity.com/paper/2104.12260/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2021.

## Built on by (most cited first, 4 of 4)

- [SymmPI: Predictive Inference for Data with Group Symmetries](https://mlsanity.com/paper/2312.16160/) (2023)
- [Pursuit of a Discriminative Representation for Multiple Subspaces via Sequential Games](https://mlsanity.com/paper/2206.09120/) (2022)
- [Non-parametric Hypothesis Tests for Distributional Group Symmetry](https://mlsanity.com/paper/2307.15834/) (2023)
- [Global optimality under amenable symmetry constraints](https://mlsanity.com/paper/2402.07613/) (2024)

## Abstract (arXiv)

Invariance-based randomization tests -- such as permutation tests, rotation tests, or sign changes -- are an important and widely used class of statistical methods. They allow drawing inferences under weak 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, acting by general linear group representations. We study test statistics with a generalized sub-additivity property. 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, and two-sample testing. Comparing with minimax lower bounds, we find perhaps surprisingly that in some cases, randomization tests detect signals at the minimax optimal rate.