Probabilistic Dual Frames and Minimization of Dual Frame Potentials
Dongwei Chen
TL;DR
The paper develops a probabilistic frame framework grounded in optimal transport to study dual frames and their potentials. It proves a general PDFP lower bound that is saturated only by tight probabilistic frames with canonical duals, and shows that restricting duals to pushforward-type maps yields a stronger, unitary-invariant bound with a clear equality criterion. It also provides mechanisms to convert Bessel measures into tight frames and characterizes the structure of pushforward duals, while connecting these ideas to Wasserstein distances to inform future transport-based minimization approaches. The results unify finite-frame intuition with probabilistic measures and deliver precise conditions for energy-minimizing dual configurations. Practical impact lies in guiding the design of robust, energy-efficient probabilistic frames for signal processing and related OT-based analyses.
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.