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Uniqueness and Uncertainty

In: Sparse and Redundant Representations

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  • Michael Elad

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

Abstract

We return to the basic problem (P 0), which is at the core of our discussion, $${\left(P_o\right):\quad \min\limits_X \parallel \mathbf{X}\parallel_0 \,{\rm subject\,\, to} \mathbf \quad \mathbf{b}=\mathbf{A\mathbf{x}}}.$$ While we shall refer hereafter to this problem as our main goal, we stress that we are quite aware of its two major shortcomings in leading to any practical tool. 1. The equality requirement b = A X is too strict, as there are small chances for any vector b to be represented by a few columns from A. A better requirement would be one that allows for small deviation. 2. The sparsity measure is too sensitive to very small entries in X, and a better measure would adopt a more forgiving approach towards such small entries.

Suggested Citation

  • Michael Elad, 2010. "Uniqueness and Uncertainty," Springer Books, in: Sparse and Redundant Representations, chapter 0, pages 17-33, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4419-7011-4_2
    DOI: 10.1007/978-1-4419-7011-4_2
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