Symplectic Geometry and Connectivity of Spaces of Frames

TL;DR

This paper establishes a novel connection between symplectic geometry and frame theory, providing new proofs and insights into the connectivity of spaces of frames with prescribed properties.

Contribution

It introduces a symplectic approach to prove the connectedness of complex frames with arbitrary norms and frame operators, generalizing previous results.

Findings

Connectedness of spaces of complex frames with prescribed norms

Symplectic proof of the generalized frame homotopy conjecture

New questions in frame theory approachable via symplectic methods

Abstract

Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper we focus on the frame homotopy conjecture posed in 2002 and recently proved by Cahill, Mixon, and Strawn, which says that the space of finite unit norm tight frames is connected. We give a simple symplectic proof of a double generalization of the frame homotopy conjecture, showing that spaces of complex frames with arbitrary prescribed norms and frame operators are connected. To spark further investigation, we also suggest a number of fundamental questions in frame theory which seem amenable to a symplectic approach.