Renegar's Condition Number, Sharpness and Compressed Sensing Performance

  • TITLE: Renegar's Condition Number, Sharpness and Compressed Sensing Performance.

  • AUTHORS: Vincent Roulet, Nicolas Boumal, Alexandre d'Aspremont

  • ABSTRACT: We show that several quantities controlling compressed sensing performance also directly control algorithmic complexity. We describe linearly convergent restart schemes solving a broad range of compressed sensing problems using first-order methods. The key term controlling convergence measures the sharpness of the optimum and can be interpreted as a condition number, computed as the ratio between the true signal sparsity and the maximum signal size that can be recovered by the observation matrix. In a similar vein, Renegar's condition number is a data-driven computational complexity measure for convex programs, generalizing classical condition numbers for linear systems. We provide evidence that for a broad class of compressed sensing problems, the worst case value of this algorithmic complexity measure taken over all signals matches the restricted eigenvalue of the observation matrix, which controls compressed sensing performance. This condition number also measures the robustness of the recovered solution with respect to a misspecification of the observation matrix A, a point rarely addressed by classical recovery results. Overall, this means that, in compressed sensing problems, a single parameter directly controls computational complexity and recovery performance.

  • STATUS: Submitted.

  • ArXiv PREPRINT: ArXiv 1506.03295

  • PAPER: Renegar's Condition Number and Compressed Sensing Performance in pdf.