Uniqueness and Uncertainty

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.