One condition for solution uniqueness and robustness of both l1-synthesis and l1-analysis minimizations

The $\ell_1$-synthesis model and the $\ell_1$-analysis model recover structured signals from their undersampled measurements. The solution of former is a sparse sum of dictionary atoms, and that of the latter makes sparse correlations with dictionary atoms. This paper addresses the question: when can we trust these models to recover specific signals? We answer the question with a condition that is both necessary and sufficient to guarantee the recovery to be unique and exact and, in presence of measurement noise, to be robust. The condition is one--for--all in the sense that it applies to both of the $\ell_1$-synthesis and $\ell_1$-analysis models, to both of their constrained and unconstrained formulations, and to both the exact recovery and robust recovery cases. Furthermore, a convex infinity--norm program is introduced for numerically verifying the condition. A comprehensive comparison with related existing conditions are included.