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 Communicated by Qiyu Sun. P.G.C. and A.H. were supported by NSF DMS 0704216. G.K. would like to thank the Department of Statistics at Stanford University and the Mathematics Department at Yale University for their hospitality and support during her visits. She was supported by Deutsche Forschungsgemeinschaft (DFG) Heisenberg-Fellowship KU 1446/8 and by DFG Grants KU 1446/13 and KU 1446/14. R.C. and A.P. were supported in part by NSF under Grant CCF-0916314, by ONR under Grant N00173-06-1-G006, and by AFOSR under Grant FA9550-05-1-0443. The authors would like to thank the American Institute of Mathematics in Palo Alto, CA, for sponsoring the workshop on “Frames for the finite world: Sampling, coding and quantization” in August 2008, which provided an opportunity for the authors to complete a major part of this work. The authors also thank the anonymous referees for their constructive suggestions, for pointing out a mistake in an earlier version of the paper, and for bringing [32] to their attention. R. Calderbank Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA e-mail: [email protected] P. G. Casazza · A. Heinecke Department of Mathematics, University of Missouri, Columbia, MO 65211-4100, USA P. G. Casazza e-mail: [email protected] A. Heinecke e-mail: [email protected] G. Kutyniok (B) Institute of Mathematics, University of Osnabrück, 49069 Osnabrück, Germany e-mail: [email protected] R. Calderbank et al. 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.