Abstract

Abstract. We can recover approximately a sparse signal with limited noise, i.e, a vector of length d with at least d − m zeros or near-zeros, using little more than m log(d) nonadaptive linear measurements rather than the d measurements needed to recover an arbitrary signal of length d. Several research communities are interested in techniques for measuring and recovering such signals and a variety of approaches have been proposed. We focus on two important properties of such algorithms. • Uniformity. A single measurement matrix should work simultaneously for all signals. • Computational Efficiency. The time to recover such an m-sparse signal should be close to the obvious lower bound, m log(d/m). To date, algorithms for signal recovery that provide a uniform measurement matrix with approximately the optimal number of measurements, such as first proposed by Donoho and his collaborators, and, separately, by Candès and Tao, are based on linear programming and require time poly(d) instead of m polylog(d). On the other hand, fast decoding algorithms to date from the Theoretical Computer Science and Database communities fail with probability at least 1 / poly(d), whereas we need failure probability no more than around 1/d m to achieve a uniform failure guarantee. This paper develops a new method for recovering m-sparse signals that is simultaneously uniform

Citations