An Efficient Approach Toward the Asymptotic Analysis of Node-Based Recovery Algorithms in Compressed Sensing

In this paper, we propose a general framework for the asymptotic analysis of node-based verification-based algorithms. In our analysis we tend the signal length n to infinity. We also let the number of non-zero elements of the signal k scale linearly with n. Using the proposed framework, we study the asymptotic behavior of the recovery algorithms over random sparse matrices (graphs) in the context of compressive sensing. Our analysis shows that there exists a success threshold on the density ratio k/n, before which the recovery algorithms are successful, and beyond which they fail. This threshold is a function of both the graph and the recovery algorithm. We also demonstrate that there is a good agreement between the asymptotic behavior of recovery algorithms and finite length simulations for moderately large values of n.