On sharp performance bounds for robust sparse signal recoveries

It is well known in compressive sensing that l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> minimization can recover the sparsest solution for a large class of underdetermined systems of linear equations, provided the signal is sufficiently sparse. In this paper, we compute sharp performance bounds for several different notions of robustness in sparse signal recovery via l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> minimization. In particular, we determine necessary and sufficient conditions for the measurement matrix A under which l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> minimization guarantees the robustness of sparse signal recovery in the “weak”, “sectional” and “strong” senses (e.g., robustness for “almost all” approximately sparse signals, or instead for “all” approximately sparse signals). Based on these characterizations, we are able to compute sharp performance bounds on the tradeoff between signal sparsity and signal recovery robustness in these various senses. Our results are based on a high-dimensional geometrical analysis of the null-space of the measurement matrix A. These results generalize the thresholds results for purely sparse signals [1], [3] and also present generalized insights on l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> minimization for recovering purely sparse signals from a null-space perspective.