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Computer Science > Information Theory

arXiv:2109.07782 (cs)
[Submitted on 16 Sep 2021]

Title:An Open Problem on Sparse Representations in Unions of Bases

Authors:Yi Shen, Chenyun Yu, Yuan Shen, Song Li
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Abstract:We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of sparks are generally combinatorial. For unions of several orthonormal bases, two lower bounds on the spark via the mutual coherence were established in previous work. We constructively prove that both of them are tight. Our main results give positive answers to Gribonval and Nielsen's open problem on sparse representations in unions of orthonormal bases. Constructive proofs rely on a family of mutually unbiased bases which first appears in quantum information theory.
Comments: 17 pages, 5 figures
Subjects: Information Theory (cs.IT)
Cite as: arXiv:2109.07782 [cs.IT]
  (or arXiv:2109.07782v1 [cs.IT] for this version)
  https://doi.org/10.48550/arXiv.2109.07782

Submission history

From: Yi Shen [view email]
[v1] Thu, 16 Sep 2021 08:00:53 UTC (342 KB)
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