Expander Recovery Performance of Bipartite Graphs With Girth Greater Than 4

Abstract

Expander recovery is an iterative algorithm designed to recover sparse signals measured with binary matrices with linear complexity. In the paper, we study the expander recovery performance of the bipartite graph with girth greater than 4, which can be associated with a binary matrix with column correlations equal to either 0 or 1. For such a graph, expander recovery is proved to achieve the same performance as the traditional basis pursuit recovery, as the signal is dissociated. Compared to random graphs widely used for expander recovery, the graph we study tends to present better empirical performance. Furthermore, its special structure enables reducing the iteration number of expander recovery from $O(n\log k)$ times to exactly $k$ times in serial recovery, and from $\mathcal {O}(\log k)$ times to exactly one time in parallel recovery.