An Optimal Transport Approach to Some Problems in Frame Theory

Abstract

A probabilistic frame is a probability measure on Euclidean space which has finite second moment and support spanning that space. These objects generalize finite frames for Euclidean space, which are redundant spanning sets. Working in the Wasserstein space of probability measures on Euclidean space with finite second moment, we investigate the properties of these measures, finding geodesics of frames in the Wasserstein space and using machinery from probability theory to define more general concepts of duality, analysis, and synthesis. We then use the Otto calculus to construct gradient flows for the probabilistic p-frame potential and a related potential which we term the (p-)tightness potential, the minimizers of which are the tight probabilistic p-frames. We demonstrate the well-posedness of the minimization problem via the minimizing movement scheme, with a focus on the case p=2. We link this result to earlier approaches to solving the Paulsen Problem for finite frames which involved differential calculus.