Thresholded Basis Pursuit: An LP Algorithm for Achieving Optimal Support Recovery for Sparse and Approximately Sparse Signals

In this paper we present a linear programming solution for sign pattern recovery of a sparse signal from noisy random projections of the signal. We consider two types of noise models, input noise, where noise enters before the random projection; and output noise, where noise enters after the random projection. Sign pattern recovery involves the estimation of sign pattern of a sparse signal. Our idea is to pretend that no noise exists and solve the noiseless l1 problem, namely, min ‖β‖1 s.t. y = Gβ and quantizing the resulting solution. We show that the quantized solution perfectly reconstructs the sign pattern of a sufficiently sparse signal. Specifically, we show that the sign pattern of an arbitrary k-sparse, n-dimensional signal x can be recovered with SNR = Ω(logn) and measurements scaling as m = Ω(k logn/k) for all sparsity levels k satisfying 0 < k ≤ αn, where α is a sufficiently small positive constant. Surprisingly, this bound matches the optimal Max-Likelihood performance bounds in terms of SNR, required number of measurements, and admissible sparsity level in an order-wise sense. In contrast to our results, previous results based on LASSO and Max-Correlation techniques either assume significantly larger SNR, sublinear sparsity levels or restrictive assumptions on signal sets. Our proof technique is based on noisy perturbation of the noiseless l1 problem, in that, we estimate the maximum admissible noise level before sign pattern recovery fails.