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Pursuit Algorithms – Practice

In: Sparse and Redundant Representations

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  • Michael Elad

    (The Technion – Israel Institute of Technology, Computer Science Department)

Abstract

It is now time to consider reliable and effcient methods for solving (P 0), as a straightforward approach seems hopeless.We now discuss methods which, it seems, have no hope of working – but which, under specific conditions, will work. Looking at the problem (P 0), $${\left(P_o\right):\quad \min\limits_X \parallel \mathbf{X}\parallel_0 \,{\rm subject\,\, to} \mathbf \quad \mathbf{b}=\mathbf{A\mathbf{x}}},$$ one observes that the unknown X is composed of two effective parts to be found – the support of the solution, and the non-zero values over this support. Thus, one way to attack the numerical solution of (P 0) is to focus on the support, with the understanding that once found, the non-zero values of X are easily detected by plain Least-Squares. As the support is discrete in nature, algorithms that seek it are discrete as well. This line of reasoning leads to the family of greedy algorithms that will be presented hereafter.

Suggested Citation

  • Michael Elad, 2010. "Pursuit Algorithms – Practice," Springer Books, in: Sparse and Redundant Representations, chapter 0, pages 35-54, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-7011-4_3
    DOI: 10.1007/978-1-4419-7011-4_3
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