A Land of Oblique Duality for Frames and Probabilistic Frames

TL;DR

This work extends frame theory by developing oblique dual frames that realize sampling and reconstruction in distinct subspaces, and it then elevates the framework to probabilistic frames with optimal transport tools. It proves that the oblique dual frame potential is minimized precisely by the canonical oblique dual, and derives analogous minimization results for oblique probabilistic frames, including sharp lower bounds tied to tightness and pushforward duals. A substantial contribution is the introduction of oblique probabilistic frame potentials and their saturation conditions, revealing when cross-subspace reconstruction achieves optimal energy. Finally, the paper establishes perturbation and stability results in the 2-Wasserstein metric, showing that near-dual measures remain near-optimal under small transport-distance perturbations, thereby connecting classical frame theory with modern optimal transport theory.

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.

Figures (1)

• Figure 1: Illustration of oblique projections and consistent reconstruction. Here $W$ and $V$ are closed subspaces of a Hilbert space $\mathcal{H}$ with $\mathcal{H} = W \oplus V^\perp$. $\{\mathbf{w}_i\}_{i \in I} \subset W$ and $\{\mathbf{v}_i\}_{i \in I} \subset V$ are not only Bessel sequences for $\mathcal{H}$, but $\{\mathbf{w}_i\}_{i \in I}$ and $\{\mathbf{v}_i\}_{i \in I}$ are frames for $W$ and $V$, respectively. Furthermore, ${\boldsymbol{\pi}}_{WV^\perp}$ is the oblique projection of $\mathcal{H}$ onto $W$ along $V^\perp$. Consistent reconstruction requires that ${\boldsymbol{\pi}}_{W V^\perp}\mathbf{f} = \hat{\mathbf{f}}$ for all $\mathbf{f} \in \mathcal{H}$, where $\hat{\mathbf{f}} = \sum_{i\in I} \langle \mathbf{f}, \mathbf{v}_i \rangle \mathbf{w}_i$.

Theorems & Definitions (55)

• Example 1.1: li1998theory and li2004pseudo
• Theorem 2.1: tang2000oblique
• Definition 2.2
• Lemma 2.3: heineken2018oblique
• proof
• Definition 2.4: christensen2004oblique
• Definition 2.5
• Theorem 2.6: eldar2003sampling and christensen2004oblique
• Theorem 2.7: christensen2004oblique
• Example 2.8