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Sparse Solutions of Underdetermined Linear Systems

In: Handbook of Geomathematics

Author

Listed:
  • Inna Kozlov

    (Holon Institute of Technology, Department of Computer Science)

  • Alexander Petukhov

    (University of Georgia, Department of Mathematics)

Abstract

Our main goal here is to discuss perspectives of applications based on solving underdetermined systems of linear equations (SLE). This discussion will include interconnection of underdetermined SLE with global problems of information theory and with data measuring and representation. The preference will be given to the description of the hypothetic destination point of the undertaken efforts, the current status of the problem, and possible methods to overcome difficulties on the way to that destination point. We do not pretend on a full survey of the current state of the theoretical researches which are very extensive now. We are going to discuss only some fundamental theoretical results justifying main applied ideas.In the end of the chapter, we give numerical results related to the popular algorithms like ℓ 1 minimization, stagewise orthogonal greedy algorithm, reweighted ℓ 1 algorithm, and the new ℓ 1 greedy algorithm.

Suggested Citation

  • Inna Kozlov & Alexander Petukhov, 2015. "Sparse Solutions of Underdetermined Linear Systems," Springer Books, in: Willi Freeden & M. Zuhair Nashed & Thomas Sonar (ed.), Handbook of Geomathematics, edition 2, pages 2773-2798, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-54551-1_42
    DOI: 10.1007/978-3-642-54551-1_42
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