Optimally Sparse Fusion Frames: Existence and Construction

Fusion frame theory is an emerging mathematical theory that provides a natural framework for performing hierarchical data processing. 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 then provide an algorithmic construction to compute fusion frames with desired fusion frame operators, including tight fusion frames. Surprisingly, we can even prove that our algorithm constructs optimally sparse fusion frames. Keywords— Computational complexity, frame decompositions, frame operator, frames, redundancy, sparse approximations, sparse matrices, tight frames.