Sparse fusion frames: existence and construction

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. A fusion frame can be regarded as a frame-like collection of subspaces in a Hilbert space, and thereby generalizes the concept of a frame for signal representation. However, when the signal and/or subspace dimensions are large, the decomposition of the signal into its fusion frame measurements through subspace projections typically requires a large number of additions and multiplications, and this makes the decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we introduce the notion of a sparse fusion frame, that is, a fusion frame whose subspaces are generated by orthonormal basis vectors that are sparse in a `uniform basis' over all subspaces, thereby enabling low-complexity fusion frame decompositions. We study the existence and construction of sparse fusion frames, but our focus is on developing simple algorithmic constructions that can easily be adopted in practice to produce sparse fusion frames with desired (given) operators. By a desired (or given) operator we simply mean one that has a desired (or given) set of eigenvalues for the fusion frame operator. We start by presenting a complete characterization of Parseval fusion frames in terms of the existence of special isometries defined on an encompassing Hilbert space. We then introduce two general methodologies to generate new fusion frames from existing ones, namely the Spatial Complement Method and the Naimark Complement Method, and analyze the relationship between the parameters of the original and the new fusion frame. We proceed by establishing existence conditions for 2-sparse fusion frames for any given fusion frame operator, for which the eigenvalues are greater than or equal to two. We then provide an easily implementable algorithm for computing such 2-sparse fusion frames.