Design of continuous-time flows on intertwined orbit spaces

Consider a space M endowed with two or more Lie group actions. Under a certain condition on the orbits of the Lie group actions, we show how to construct a flow on M that projects to prescribed flows on the orbit spaces of the group actions. Hence, in order to design a flow that converges to the intersection of given orbits, it suffices to design flows on the various orbit spaces that display convergence to the desired orbits, and then to lift these flows to M using the proposed procedure. We illustrate the technique by creating a flow for principal component analysis. The flow projects to a flow on the Grassmann manifold that achieves principal subspace analysis and to a flow on the "shape" manifold that converges to the set of orthonormal matrices.