Zig-zag and replacement product expander graphs for Compressive Sensing

Compressive Sensing (CS) asserts that one can recover a sparse signal from a limited number of random or deterministic projections exactly if the measurement matrix satisfies the so-called RIP. People try to design deterministic matrices for CS because of the small storage, high efficiency and low complexity compared with the random matrices in practical applications. Recent works explore expander graphs for efficient CS reconstruction, but the existing expander graphs for CS are either difficult to obtain or restricted on the number of the vertices. In this paper, we propose an algorithm named zig-zag and replacement product expander graphs whose main idea is to produce another expander graph or an explicit family of expander graphs using two or more known expander graphs. Based on the proposed algorithm, the expander graphs are easy to obtain and the vertices of the graphs, corresponding to the length of the original signal and the measurement times, aren't restricted too much. Finally, numerical simulations are conducted to verify the better performance of the zig-zag product matrices compared with the random matrices.