Probabilistic Dual Frames and Minimization of Dual Frame Potentials

TL;DR

This paper explores probabilistic dual frames and their potentials, demonstrating that tightness and canonical duality minimize the dual frame potential, with implications for optimal transport and frame theory.

Contribution

It introduces conditions under which probabilistic dual frames minimize their potentials, extending classical frame theory with probabilistic and optimal transport perspectives.

Findings

Dual frame potential minimized by tight probabilistic frames

Canonical dual is optimal when minimizing dual frame potential

Tightness condition can be relaxed under certain dual frame types

Abstract

This paper studies probabilistic dual frames and the associated dual frame potentials from the perspective of optimal mass transport. The main contribution of this work shows that given a probabilistic frame, its associated dual frame potential is minimized if and only if the probabilistic frame is tight and the probabilistic dual frame is the canonical dual. In particular, the tightness condition can be dropped if the probabilistic dual frame potential is minimized only among probabilistic dual frames of pushforward type.