Gradient Flows for Frame Potentials on the Wasserstein Space

TL;DR

This paper integrates frame theory and optimal transport by developing gradient flows in Wasserstein space for a new potential, enabling algorithms to evolve frames toward tight probabilistic frames.

Contribution

It introduces the tightness potential and constructs gradient flows in Wasserstein space to optimize frame properties, bridging two mathematical fields.

Findings

Defined the tightness potential as a modification of the probabilistic frame potential.

Constructed gradient flows in Wasserstein space for the new potential.

Proposed an algorithm for evolving frames toward tight probabilistic frames.

Abstract

In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct gradient flows in the Wasserstein space $P_2(\mathbb{R}^d)$ for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. It is shown that the potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve an existing frame toward a tight probabilistic frame.