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.

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Acknowledgements

The authors thank Profs. Peter G. Casazza and Matthew Fickus for discussions on full spark frames, Prof. Dan Edidin and Will Sawin for discussions on algebraic geometry, and the anonymous referees for very helpful comments and suggestions. B.A. was supported by the NSF Graduate Research Fellowship under Grant No. DGE-0646086, J.C. was supported by NSF Grant No. DMS-1008183, DTRA/NSF Grant No. DMS-1042701 and AFOSR Grant No. FA9550-11-1-0245, and D.G.M. was supported by the A.B. Krongard Fellowship. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

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Correspondence to Dustin G. Mixon.

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Communicated by Roman Vershynin.

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Alexeev, B., Cahill, J. & Mixon, D.G. Full Spark Frames. J Fourier Anal Appl 18, 1167–1194 (2012). https://doi.org/10.1007/s00041-012-9235-4

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Keywords

  • Frames
  • Spark
  • Sparsity
  • Erasures

Mathematics Subject Classification (2000)

  • 42C15
  • 68Q17