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Abstract:
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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. Existing approaches based on the tangent spaces of the manifold and log maps 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 change the tangent bundle viewpoint and move towards the dual spaces of the tangent spaces, i.e., the cotangent spaces, and build subspaces based on initial covectors. Furthermore, 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 or from affinely independent points.
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