Karhunen-Loève decomposition in the presence of symmetry

Abstract

The Karhunen-Loève decomposition is widely used with data which very often exhibits 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.R) the method also applies, and we extend the results to these cases. We then investigate the group SO(3) in more detail and follow the algorithm with a specific example of data lying in the space of vector fields that are tangent to the sphere.