Verifiable and computable ℓ_∞ performance evaluation of ℓ₁ sparse signal recovery

In this paper, we develop verifiable and computable performance analysis of the ℓ_∞ norms of the recovery errors for ℓ₁ minimization algorithms. We define a family of goodness measures for arbitrary sensing matrices as a set of optimization problems, and design algorithms with a theoretical global convergence guarantee to compute these goodness measures. The proposed algorithms solve a series of second-order cone programs, or linear programs. As a by-product, we implement an efficient algorithm to verify a sufficient condition for exact ℓ₁ recovery in the noise-free case. This implementation performs orders-of-magnitude faster than the state-of-the-art techniques. We derive performance bounds on the ℓ_∞ norms of the recovery errors in terms of these goodness measures. We establish connections between other performance criteria (e.g., the ℓ₂ norm, ℓ₁ norm, and support recovery) and the ℓ_∞ norm in a tight manner. We also analytically demonstrate that the developed goodness measures are non-degenerate for a large class of random sensing matrices, as long as the number of measurements is relatively large. Numerical experiments show that, compared with the restricted isometry based performance bounds, our error bounds apply to a wider range of problems and are tighter, when the sparsity levels of the signals are relatively low.