Efficient and Robust Compressed Sensing Using Optimized Expander Graphs

Abstract

Abstract-Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(k log n) measurements and only O(k log n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3 4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(n log( n k ))). We also show that by tolerating a small penalty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal.Index Terms-Compressive sensing, expander graphs, sparse recovery, sparse sensing matrices, unique neighborhood property.

A User's Guide to Compressed Sensing for Communications Systems

“…Since the increase in the number of paths, which corresponds to the number of measurements, results in the increase in the number of probe packets injected into the network, it is desirable to minimize the number of paths in order not to give unnecessary load to the network. As for the number of required measurements for the reconstruction of k-sparse vectors with random binary measurements matrices, it has been known that O(k log n k ) measurements are required [153], [154], while it has been shown that O(k log n) measurements are needed if the binary matrix is accompanied by a graph constraint [151], [155]. Moreover, a deterministic guarantee and a designed method of the routing matrix for the reconstruction of any 1-sparse signal are provided in [149], [150], taking advantage of the knowledge on compressed sensing using expander graphs [154], [156].…”

Robust support recovery using sparse compressive sensing matrices

“…The performance of convex optimizations for sparse estimation in this setting were examined in [34]. Greedy and other ad-hoc procedures for signal estimation in such settings were examined in [35]- [38].…”

A game theoretic approach to expander-based compressive sensing

“…There are two different recovery approaches for e-CS in the literature: Combinatorial approach [4], [5]: In this approach, messagepassing algorithms are proposed for approximating x * . The recovery algorithms rely on the combinatorial properties of these graphs, and have lower computational complexity (e.g., O(N log N k )).…”

“…Messagepassing algorithms [4], [5] exploit the combinatorial structure of the expander graphs. While these algorithms are efficient and rather easy to implement, their approximation guarantees are meaningful only in extremely high-dimensions, as they feature large constants that are not suitable for practical applications.…”