Mean breakdown points for compressed sensing by uniformly distributed matrices

概要

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It is graphically observed that curves of mean breakdown points obtained by $\ell_1$ optimization for compressed sensing defined by underdetermined systems $y=Aw$ with uniformly distributed random matrices $A\in{\mathbb R}^{d\times m}$ and sparse $w$ almost coincide with the curves obtained by normally distributed random matrices, both with sparse vectors $w^+$ with nonnegative components and $w^\pm$ with components of either sign. Three-dimensional figures illustrate asymptotic phase transition cliffs. These and the standard deviation of the mean breakdown points can be used to define a level of sparseness of $w$ below which a unique solution is expected to a high probability.