A Land of Oblique Duality for Frames and Probabilistic Frames

TL;DR

This paper introduces a new framework for oblique dual frames, including probabilistic variants, providing tools for their analysis, optimization, and stability under perturbations in signal processing contexts.

Contribution

It develops the concept of oblique dual probabilistic frames, introduces the oblique dual frame potential, and studies their minimization and stability properties.

Findings

Oblique dual frame potential is minimized at the canonical oblique dual.

Oblique dual probabilistic frames are characterized by tightness and can be optimized.

Perturbations close in Wasserstein topology preserve the oblique dual probabilistic frame structure.

Abstract

Functions or distributions used to sample and to reconstruct signals often occur in different domains, like the Dirac delta and a band-limited bump function in classical sampling. Oblique dual frames generalize this phenomenon. In this paper, we provide new tools to study oblique dual frames and introduce a probabilistic variant of oblique dual frames. We first present the oblique dual frame potential and show that it is minimized precisely when the oblique dual coincides with the canonical oblique dual. We then define oblique dual probabilistic frames and oblique approximately dual probabilistic frames. In particular, we prove that for a given oblique dual probabilistic frame, the associated oblique dual probabilistic frame potential is minimized if and only if the frame is tight and the oblique dual is canonical. Moreover, the tightness assumption can be removed when the minimization is restricted to oblique dual probabilistic frames of pushforward type. Finally, we investigate perturbations of oblique dual probabilistic frames and show that if a probability measure is sufficiently close to an oblique dual probabilistic frame pair in the $2$-Wasserstein topology, then it forms an oblique approximately dual probabilistic frame.