Optimally Sparse Frames

Abstract

Frames have established themselves as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. However, when the signal dimension is large, the computation of the frame measurements of a signal typically requires a large number of additions and multiplications, and this makes a frame decomposition intractable in applications with limited computing budget. To address this problem, in this paper, we focus on frames in finite-dimensional Hilbert spaces and introduce sparsity for such frames as a new paradigm. In our terminology, a sparse frame is a frame whose elements have a sparse representation in an orthonormal basis, thereby enabling low-complexity frame decompositions. To introduce a precise meaning of optimality, we take the sum of the numbers of vectors needed of this orthonormal basis when expanding each frame vector as sparsity measure. We then analyze the recently introduced algorithm Spectral Tetris for construction of unit norm tight frames and prove that the tight frames generated by this algorithm are in fact optimally sparse with respect to the standard unit vector basis. Finally, we show that even the generalization of Spectral Tetris for the construction of unit norm frames associated with a given frame operator produces optimally sparse frames. *

An Automatic Sparse Pruning Endmember Extraction Algorithm with a Combined Minimum Volume and Deviation Constraint

“…A sparse estimate of p in [55,56] is defined as one in which irrelevant or redundant components are exactly zero, which is estimated by:…”

Sparse matrices in frame theory

“…Frames as sparse matrices were first analyzed in 2011 by Casazza et al (2011b), whereas sparse duals were first considered and discussed independently by Krahmer et al (2013) and Li et al (2013). For both situations, results on the optimal sparsity were derived, existence results were proven, and algorithms provided.…”

“…Proposition 3.4 (Casazza et al (2011b)) Let m > n > 0 and Φ = (ϕ i ) m i=1 ∈ F (n, m). Then the frame matrix Φ has block decomposition of order at most gcd(n, m).…”

“…For this, we remind the reader that the tight spectral tetris frame as constructed by Algorithm 1 is denoted by STTF(n, m). Theorem 3.6 (Casazza et al (2011b)) Let m ≥ 2n > 0. Let (e j ) n j=1 be an orthonormal basis for R n .…”

“…While it is shown in Casazza et al (2011b) that given a diagonal frame operator a variant of Algorithm 1 for non-tight frames always constructs the optimally sparse frame associated to that operator, there may exist a sparser frame associated with a non-diagonal frame operator having the same spectrum. Hence in such cases spectral tetris does not always find the sparsest frame of a given spectrum.…”