Sub-linear Time Compressed Sensing for Support Recovery 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 compressed sensing framework through carefully designed sparse measurement matrices associated with low measurement costs and low-complexity recovery algorithms. The measurement system in our framework captures observations of the signal through well-designed measurement matrices sparsified by capacity-approaching sparse-graph codes, and then recovers 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, by structuring the measurements through sparse-graph codes, we propose two families of measurement matrices, the Fourier family and the binary family respectively, which lead to different measurement and computational costs. In the noiseless setting, our framework recovers the sparse support of any K-sparse signal in time2 O(K) with 2K measurements obtained by the Fourier family, or in time O(K logN) using K log2N + K measurements obtained by the binary family. In the presence of noise, both measurement and computational costs are reduced to O(K logN) in the case of the Fourier family. More importantly, the binary family achieves O(K logN) for both the measurement cost and the computational complexity in the presence of noise, which maintains the same measurement and computational scaling as the noiseless case. Therefore, when the signal sparsity K is sub-linear in the signal dimension N , our framework achieve sublinear time support recovery. Further, our framework also admits a wide class of random matrix family that achieves O(K logN) measurements with near-linear run-time O(N logN). In terms of recovery performance, we show that our framework succeeds with probability one asymptotically under finite signal-to-noise ratios.