Random projections of regular simplices
Discrete & Computational Geometry
Decoding by linear programming
IEEE Transactions on Information Theory
Signal Reconstruction From Noisy Random Projections
IEEE Transactions on Information Theory
Breaking the l1 recovery thresholds with reweighted l1 optimization
Allerton'09 Proceedings of the 47th annual Allerton conference on Communication, control, and computing
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It is well known in compressive sensing that l1 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 l1 minimization. In particular, we determine necessary and sufficient conditions for the measurement matrix A under which l1 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 nullspace of the measurement matrix A. These results generalize the thresholds results for purely sparse signals [1], [3] and also present generalized insights on l1 minimization for recovering purely sparse signals from a null-space perspective.