Full Spark Frames

Abstract

Finite frame theory has a number of real-world applications. In applications like sparse signal processing, data transmission with robustness to erasures, and reconstruction without phase, there is a pressing need for deterministic constructions of frames with the following property: every size-M subcollection of the M -dimensional frame elements is a spanning set. Such frames are called full spark frames, and this paper provides new constructions using the discrete Fourier transform. Later, we prove that full spark Parseval frames are dense in the entire set of Parseval frames, meaning full spark frames are abundant, even if one imposes an additional tightness constraint. Finally, we prove that testing whether a given matrix is full spark is hard for NP under randomized polynomial-time reductions, indicating that deterministic full spark constructions are particularly significant because they guarantee a property which is otherwise difficult to check.2000 Mathematics Subject Classification. 42C15, 68Q17.

Reconstruction of Signals from Magnitudes of Redundant Representations: The Complex Case

“…For (2) notice that the following four functions defined on the real vector space Sym(H) are convex: X → |y k − X, F k | 2 , X → X 1 , X → −tr{X}, X → a max (X), whereas X → a min (X) is concave. The last two statements are consequences of the Weyl's Inequality, Theorem III.2.1 in [10] with i = j = 1 in (III.5), and i − j = n in (III.6).…”

Compressive Identification of Active OFDM Subcarriers in Presence of Timing Offset

“…H is a full spark matrix [24], therefore its product with the non-singular diagonal matrix Λ is also full spark [22]; i.e.…”

“…On the other hand, if M ≤ N and every M × M subset of A is invertible, then A is called a full-spark matrix [24], i.e. spark(A) = M + 1.…”

Sparse matrices in frame theory

“…This shows that P(n, m) is open in the Zariski topology 1 . Since the set P(n, m) is non-empty by Example 4.6, it is thereby open and dense in the standard topology, see Alexeev et al (2012). Finally, since P(n, m) c is a proper subset and closed in the Zariski topology, it is of measure zero.…”

“…Recall that an n × m matrix is said to be in general position if any sub-collection of n (column) vectors is linearly independent, that is, if any n × n submatrix is invertible. Such matrices are sometimes called full spark frames, such as in Alexeev et al (2012), since their spark is maximal, i.e., n +1.…”