  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.
