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In this paper we present a linear programming solution for sign patternrecovery of a sparse signal from noisy random projections of the signal. Weconsider two types of noise models, input noise, where noise enters before therandom projection; and output noise, where noise enters after the randomprojection. Sign pattern recovery involves the estimation of sign pattern of asparse signal. Our idea is to pretend that no noise exists and solve thenoiseless $\ell_1$ problem, namely, $\min \|\beta\|_1 ~ s.t. ~ y=G \beta$ andquantizing the resulting solution. We show that the quantized solutionperfectly 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 $SNR=\Omega(\log n)$ andmeasurements scaling as $m= \Omega(k \log{n/k})$ for all sparsity levels $k$satisfying $0< k \leq \alpha n$, where $\alpha$ is a sufficiently smallpositive constant. Surprisingly, this bound matches the optimal\emph{Max-Likelihood} performance bounds in terms of $SNR$, required number ofmeasurements, and admissible sparsity level in an order-wise sense. In contrastto our results, previous results based on LASSO and Max-Correlation techniqueseither assume significantly larger $SNR$, sublinear sparsity levels orrestrictive assumptions on signal sets. Our proof technique is based on noisyperturbation of the noiseless $\ell_1$ problem, in that, we estimate themaximum admissible noise level before sign pattern recovery fails.

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