Sub-linear Time Support Recovery for Compressed Sensing using Sparse-Graph Codes

We address the problem of robustly recovering the support of high-dimensional sparse signals1 from linear measurements in a low-dimensional subspace. We introduce a new family of sparse measurement matrices associated with low-complexity recovery algorithms. Our measurement system is designed to capture observations of the signal through sparse-graph codes, and to recover the signal by using a simple peeling decoder. As a result, we can simultaneously reduce both the measurement cost and the computational complexity. In this paper, we formally connect general sparse recovery problems in compressed sensing with sparse-graph decoding in packet-communication systems, and analyze our design in terms of the measurement cost, computational complexity and recovery performance. Specifically, in the noiseless setting, our scheme requires 2K measurements asymptotically to recover the sparse support of any K-sparse signal with O(K) arithmetic operations. In the presence of noise, both measurement and computational costs are O(K logN) for recovering any K-sparse signal of dimension N . When the signal sparsity K is sub-linear in the signal dimension N , our design achieves sub-linear time support recovery. Further, the measurement cost for noisy recovery can also be reduced to O(K logN) by increasing the computational complexity to near-linear time O(N logN). In terms of recovery performance, we show that the support of any K-sparse signal can be stably recovered under finite signal-to-noise ratios with probability one asymptotically. ∗This work was supported by grants NSF CCF EAGER 1439725, and NSF CCF 1116404 and MURI CHASE Grant No. 556016. 1In this paper, the signal of interest can be sparse with respect to any known basis.