Testing for Geometric Invariance and Equivariance
This addresses the issue for practitioners using invariant and equivariant models in non-parametric regression, providing a method to validate symmetry assumptions before model fitting.
The paper tackles the problem of falsely assumed symmetry in invariant and equivariant models by presenting a framework for testing G-equivariance for any semi-group G, enabling confidence in using such models when symmetry is not known a priori.
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.