A Unified Framework for Principal Subspace Analysis from the Hamiltonian Viewpoint

Abstract

Extended Abstract In this work, we develop a general and unified framework for principal component analysis (PCA) applicable to Riemannian, sub-Riemannian and symplectic manifold-valued data and functional data. Almost all existing statistical methods for manifold data rely on the tangent bundle of the manifold, with the purpose of transforming the nonlinear manifold to the linear tangent spaces. However, such methods become invalid when the tangent vectors are constrained to lie in a subspace of the tangent space since the exponential map will no longer be a local diffeomorphism. This scenario, known as the sub-Riemannian geometry, has attracted considerable attention in recent years. We propose to shift the tangent space viewpoint and move towards the dual spaces of the tangent spaces, i.e., the cotangent spaces, and build subspaces based on initial covectors. More generally, motivated by the Arnold-Liouville theorem we propose the anchor-compatible identification for subspaces with first integrals (ACISFI), which constructs a properly nested sequence of subspaces as the fibres of a carefully chosen set of functionally independent functions defined on the cotangent bundle, i.e., the first integrals of the Hamiltonian system, generalising the ideas of obtaining subspaces from linearly independent tangent vectors [1] or from affinely independent points [2]. There are several advantages of the ACISFI over the existing PCA on manifolds. First, the subspaces can be learnt from sample points in a completely data-driven way. We do not impose a particular form for the Hamiltonian, e.g., the Hamiltonian which induces the geodesic flow, but it can be chosen as any smooth function on the manifold, and there is not any a priori restriction on the form of first integrals as well. Second, the submanifolds can be