Weak Recovery Conditions from Graph Partitioning Bounds and Order Statistics

We study a weaker formulation of the nullspace property which guarantees recovery of sparse signals from linear measurements by 𝓁₁ minimization. We require this condition to hold only with high probability, given a distribution on the nullspace of the coding matrix A. Under some assumptions on the distribution of the reconstruction error, we show that testing these weak conditions means bounding the optimal value of two classical graph partitioning problems: the k-Dense-Subgraph and MaxCut problems. Both problems admit efficient, relatively tight relaxations, and we use a randomization argument to produce new approximation bounds for k-Dense-Subgraph. We test the performance of our results on several families of coding matrices.