Testing for Geometric Invariance and Equivariance

Invariant and equivariant models incorporate the symmetry of an object to be estimated (here non-parametric regression functions $f : $\mathcal{X}$ \rightarrow $\mathbb{R}$$)\mathrm{.}Thesemodelsperformbetter(withrespectto$L^2$ 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.