Fusion Frame Homotopy and Tightening Fusion Frames by Gradient Descent

TL;DR

This paper extends the understanding of fusion frames by proving their tightness and path connectivity, and demonstrates that gradient descent can be used to construct tight fusion frames, using advanced geometric techniques.

Contribution

It generalizes results on tight frames to fusion frames, proving path connectivity and the effectiveness of gradient descent for their construction.

Findings

Tight fusion frames with prescribed dimensions are path connected.

Gradient descent can find tight fusion frames when they exist.

The fusion frame potential has no spurious local minima, ensuring convergence.

Abstract

Finite frames, or spanning sets for finite-dimensional Hilbert spaces, are a ubiquitous tool in signal processing. There has been much recent work on understanding the global structure of collections of finite frames with prescribed properties, such as spaces of unit norm tight frames. We extend some of these results to the more general setting of fusion frames -- a fusion frame is a collection of subspaces of a finite-dimensional Hilbert space with the property that any vector can be recovered from its list of projections. The notion of tightness extends to fusion frames, and we consider the following basic question: is the collection of tight fusion frames with prescribed subspace dimensions path connected? We answer (a generalization of) this question in the affirmative, extending the analogous result for unit norm tight frames proved by Cahill, Mixon and Strawn. We also extend a result of Benedetto and Fickus, who defined a natural functional on the space of unit norm frames (the frame potential), showed that its global minimizers are tight, and showed that it has no spurious local minimizers, meaning that gradient descent can be used to construct unit-norm tight frames. We prove the analogous result for the fusion frame potential of Casazza and Fickus, implying that, when tight fusion frames exist for a given choice of dimensions, they can be constructed via gradient descent. Our proofs use techniques from symplectic geometry and Mumford's geometric invariant theory.