Estimating Maximal Symmetries of Regression Functions via Subgroup Lattices

We present a method for estimating the maximal symmetry of a regression function. Knowledge of such a symmetry can be used to significantly improve modelling by removing the modes of variation resulting from the symmetries. Symmetry estimation is carried out using hypothesis testing for invariance strategically over the subgroup lattice of a search group G acting on the feature space. We show that the estimation of the unique maximal invariant subgroup of G can be achieved by testing on only a finite portion of the subgroup lattice when G_max is a compact subgroup of G, even for infinite search groups and lattices (such as for the 3D rotation group SO(3)). We then show that the estimation is consistent when G is finite. We demonstrate the performance of this estimator in low dimensional simulations, on a synthetic image classification on MNIST data, and apply the methods to an application using satellite measurements of the earth's magnetic field.