Union of subspaces, manifold-modeled signals, and differential geometry interpretation

Abstract

Notions of a union of subspaces and manifold-modeled signals play key roles in modern signal processing. In this paper, we explore the differential geometry structures of these signal spaces. To this end, a sparse generalized Prony signal model is used as a toy model. We show that the family of K-dimensional generalized Prony (GP) coefficient spaces associated with pairwise-distinct spectral configurations forms a smooth vector bundle. This bundle is isomorphic to the tangent bundle of the Prony manifold, which is an embedded submanifold of a K-sphere in RK+1. For physically parameterized GP models, the corresponding family is naturally identified with the pullback of this tangent bundle to the physical parameter domain. Consequently, the associated bundle total spaces are diffeomorphic. As an example, these manifold structures are applied to shallow water soundfields, and the parameter inversion problem in shallow water acoustics is reformulated as a manifold optimization which shows local geodesic convexity.