Weak Recovery Conditions from Graph Partitioning Bounds and Order Statistics

TL;DR

This paper introduces a probabilistic nullspace property for sparse signal recovery, linking it to graph partitioning problems like k-Dense-Subgraph and MaxCut, with practical bounds and testing on coding matrices.

Contribution

It establishes a weaker nullspace condition for l_1 recovery based on graph partitioning bounds, providing new approximation results and testing methods.

Findings

Bound the optimal values of graph partitioning problems for recovery guarantees.

Efficient relaxations for k-Dense-Subgraph and MaxCut are used.

Performance tested on various coding matrix families.

Abstract

We study a weaker formulation of the nullspace property which guarantees recovery of sparse signals from linear measurements by l_1 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.