Testing for Geometric Invariance and Equivariance

TL;DR

This paper introduces a framework for testing geometric invariance and equivariance in models, enabling validation of symmetry assumptions in non-parametric regression before model fitting.

Contribution

It provides a model-independent, computationally efficient method for testing $G$-equivariance for any semi-group $G$, addressing a key challenge in symmetry-based modeling.

Findings

Tests are independent of the model used.

Tests are computationally quick and easy to implement.

Framework applies to any semi-group $G$.

Abstract

Invariant and equivariant models incorporate the symmetry of an object to be estimated (here non-parametric regression functions $f : \mathcal{X} \rightarrow \mathbb{R}$). These models perform better (with respect to $L^2$ loss) and are increasingly being used in practice, but encounter problems when the symmetry is falsely assumed. In this paper we present a framework for testing for $G$-equivariance for any semi-group $G$. This will give confidence to the use of such models when the symmetry is not known a priori. These tests are independent of the model and are computationally quick, so can be easily used before model fitting to test their validity.