Probabilistic frames and Wasserstein distances

TL;DR

This paper explores the use of Wasserstein distances to analyze probabilistic frames, establishing topological properties and characterizations of transport duals, thereby advancing the mathematical understanding of probabilistic frame theory.

Contribution

It generalizes recent probabilistic frame results using Wasserstein distances and characterizes the structure of transport duals, including those not arising as push-forwards.

Findings

Sets of probabilistic frames with fixed frame operator are homeomorphic via optimal linear push-forward.

The set of probabilistic frames with a fixed operator is path-connected.

Characterization of transport duals that are push-forwards and those that are not.

Abstract

We use Wasserstein distances to characterize and study probabilistic frames. Adapting results from Olkin and Pukelsheim, from Gelbrich and from Cuesta-Albertos, Matran-Bea and Tuero-Diaz to frame operators, we show that the sets of probabilistic frames with given frame operator are homeomorphic by an optimal linear push-forward. Using the Wasserstein distances, we generalize several recent results in probabilistic frame theory and show path connectedness of the set of probabilistic frames with a fixed frame operator. We also describe transport duals that do not arise as push-forwards and characterize those that are push-forwards.