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 $\min \|\bZ\|_1 ~ s.t. ~ \bY=\bG \bZ$, and quantizing/thresholding the resulting solution $\bZ$. 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 $\bX$ can be recovered with $SNR=O(\log n)$ and $m= O(k \log{n/k})$ measurements, where $k$ is the sparsity level and satisfies $0< k \leq \alpha n$, where, $\alpha$ is some non-zero constant. Our proof technique is based on perturbation of the noiseless $\ell_1$ 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 $\bX$ can be recovered from $m= O(k \log{n/k})$ random projections for certain classes of signals.