Sub-linear Time Compressed Sensing Using Sparse-Graph Codes

We consider the problem of recovering the support of an arbitrary K-sparse N-length vector in the presence of noise, where the sparsity K = O(N δ ) is sub-linear in N for some 0 < δ < 1. A new family of sparse measurement matrices is introduced with a low-complexity recovery algorithm, which achieves a sub-linear measurement cost O(K log 1.3̇ N) and sub-linear computational complexity O(K log 1.3̇ N). Our measurement system is designed to capture observations of the signal through the parity constraints of sparse-graph codes, and to recover the signal by using a simple peeling decoder. We formally connect general sparse recovery problems with sparse-graph decoding, and showcase our design in terms of the measurement cost, computational complexity and recovery performance.