Karhunen-Loeve decomposition in the presence of symmetry. I

The Karhunen–Lo`eve (KL) decomposition is widely used to data which very often exhibit some symmetry, afforded by a group action. For a finite group, we derive an algorithm using group representation theory to reduce the cost of determining the KL basis. We demonstrate the technique on a Lorenz-type ODE system. For a compact group such as tori or SO(3, IR) the method also applies, and we extend results to these cases. As a short example, we consider the circle group S1.