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 : X rightarrow ℝ). These models perform better (with respect to L2 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.