Duality and Geodesics for Probabilistic Frames

TL;DR

This paper extends the concept of finite frames to the Wasserstein space of probability measures, introducing duality, analysis, and geodesic path theories for probabilistic frames with applications to optimal transport.

Contribution

It develops a new theory of probabilistic frames in Wasserstein space, including duality, analysis, synthesis, and geodesic path properties, expanding the mathematical framework of frame theory.

Findings

Formulated transport duals for probabilistic frames.

Identified conditions for geodesic paths to be probabilistic frames.

Linked discrete and continuous cases to matrix ranks and transport plan continuity.

Abstract

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their properties. In particular, we formulate a theory of transport duals for probabilistic frames and prove certain properties of this class. We also investigate paths of probabilistic frames, identifying conditions under which geodesic paths between two such measures are themselves probabilistic frames. In the discrete case, this is related to ranks of convex combinations of matrices, while, in the continuous case, this is related to the continuity of the optimal transport plan.