Approximately Dual and Pseudo-Dual Probabilistic Frames
Dongwei Chen, Emily J. King, Clayton Shonkwiler
TL;DR
This work analyzes dual structures for probabilistic frames through the lens of optimal transport. It establishes a hierarchy of dual concepts—dual, approximately dual, and pseudo-dual—and ties redundancy to the existence and form of these duals, proving that zero redundancy forces a unique pushforward dual while finite redundancy implies atomic, finitely supported measures. The authors show every probabilistic frame has a discrete finite approximately dual, develop convex constructions and perturbation results to generate approximate duals, and provide pushforward-type and general pseudo-dual characterizations. Overall, the paper blends frame theory with optimal transport to reveal structural insights and practical pathways for robust reconstruction from probabilistic frames.
Abstract
This paper studies properties of dual probabilistic frames -- in particular in relation to redundancy -- and introduces both approximately dual probabilistic frames and pseudo-dual probabilistic frames. We show that the canonical dual probabilistic frame is the only dual frame of pushforward type of a probabilistic frame with zero redundancy. Furthermore, we show that probabilistic frames with finite redundancy are atomic and finite. Approximately dual probabilistic frames generalize duality, with pseudo-duality being a further generalization. We introduce these concepts and prove certain structural results. In particular, every probabilistic frame has a discrete finite frame as an approximate dual.