Near-Optimal Sparse Recovery in the L1 Norm

• Published in: 2008 49th Annual IEEE Symposium on Foundations of Computer Science pp. 199 - 207
• Main Authors: Indyk, P., Ruzic, M.
• Format: Conference Proceeding
• Language: English; Japanese
• Published: IEEE 01.10.2008
• Subjects: Analog computers; Approximation error; Compressed sensing; Computer errors; Computer science; Data acquisition; Encoding; expanders; Hardware; l1 norm; Linearity; sparse recovery; streaming algorithms; Vectors
• ISBN: 0769534368; 9780769534367
• ISSN: 0272-5428
• DOI: 10.1109/FOCS.2008.82

We consider the approximate sparse recovery problem, where the goal is to (approximately) recover a high-dimensional vector xisinRopf n from its lower-dimensional sketch AxisinRopf m . Specifically, we focus on the sparse recovery problem in the L 1 norm: for a parameter k, given the sketch Ax, compute an approximation xcirc of x such that the L 1 approximation error parx-xcircpar 1 is close to min x' parx-x'par 1 , where x' ranges over all vectors with at most k terms. The sparse recovery problem has been subject to extensive research over the last few years. Many solutions to this problem have been discovered, achieving different trade-offs between various attributes, such as the sketch length, encoding and recovery times. In this paper we provide a sparse recovery scheme which achieves close to optimal performance on virtually all attributes (see Figure 1). In particular, this is the first recovery scheme that guarantees O(k log(n/k)) sketch length, and near-linear O(n log (n/k)) recovery time simultaneously. It also features low encoding and update times, and is noise-resilient.