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The $\ell_1$-synthesis model and the $\ell_1$-analysis model recoverstructured signals from their undersampled measurements. The solution of formeris a sparse sum of dictionary atoms, and that of the latter makes sparsecorrelations with dictionary atoms. This paper addresses the question: when canwe trust these models to recover specific signals? We answer the question witha condition that is both necessary and sufficient to guarantee the recovery tobe unique and exact and, in presence of measurement noise, to be robust. Thecondition 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 constrainedand unconstrained formulations, and to both the exact recovery and robustrecovery cases. Furthermore, a convex infinity--norm program is introduced fornumerically verifying the condition. A comprehensive comparison with relatedexisting conditions are included.

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