Thresholded Basis Pursuit : Support Recovery for Sparse and Approximately Sparse Signals . ∗

In this paper we present a linear programming solution for support recovery. Support recovery involves the estimation of sign pattern of a sparse signal from a set of randomly projected noisy measurements. Our solution of the problem amounts to solving min ‖Z‖1 s.t. Y = GZ, and quantizing/thresholding the resulting solution Z. We show that this scheme is guaranteed to perfectly reconstruct a discrete signal or control the element-wise reconstruction error for a continuous signal for specific values of sparsity. We show that the sign pattern of X can be recovered with SNR = O(log n) and m = O(k logn/k) measurements, where k is the sparsity level and satisfies 0 < k ≤ αn, where, α is some non-zero constant. Our proof technique is based on perturbation of the noiseless l1 problem. Consequently, the maximum achievable sparsity level in the noisy problem is comparable to that of the noiseless problem. Our result offers a sharp characterization in that neither the SNR nor the sparsity ratio can be significantly improved. In contrast previous results based on LASSO and MAX-Correlation techniques either assume significantly larger SNR or sub-linear sparsity. Our results has implications for approximately sparse problems. We show that the k largest coefficients of a non-sparse signal X can be recovered from m = O(k logn/k) random projections for certain classes of signals.