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

TL;DR

This paper introduces a linear programming method called Thresholded Basis Pursuit for recovering the sign pattern of sparse signals from noisy measurements, achieving optimal bounds comparable to maximum likelihood methods.

Contribution

It presents a novel LP-based algorithm that attains optimal support recovery bounds for noisy sparse signals, matching theoretical performance limits.

Findings

Sign pattern of sparse signals can be recovered with high SNR and measurements scaling as k log(n/k).

The method matches optimal maximum likelihood bounds in terms of SNR and measurement complexity.

Previous methods require higher SNR or restrictive assumptions, unlike our approach.

Abstract

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 $\ell_1$ problem, namely, $\min \|\beta\|_1 ~ s.t. ~ y=G \beta$ 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 $SNR=\Omega(\log n)$ and measurements scaling as $m= \Omega(k \log{n/k})$ for all sparsity levels $k$ satisfying $0< k \leq \alpha n$, where $\alpha$ is a sufficiently small positive constant. Surprisingly, this bound matches the optimal \emph{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 $\ell_1$ problem, in that, we estimate the maximum admissible noise level before sign pattern recovery fails.