Randomness conductors and constant-degree lossless expanders
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Explicit constructions for compressed sensing of sparse signals
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Almost Euclidean subspaces of ℓN1 via expander codes
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Near-Optimal Sparse Recovery in the L1 Norm
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes
Journal of the ACM (JACM)
A Frame Construction and a Universal Distortion Bound for Sparse Representations
IEEE Transactions on Signal Processing
IEEE Transactions on Information Theory - Part 1
Performance bounds for expander-based compressed sensing in the presence of poisson noise
Asilomar'09 Proceedings of the 43rd Asilomar conference on Signals, systems and computers
Reed muller sensing matrices and the LASSO
SETA'10 Proceedings of the 6th international conference on Sequences and their applications
Randomization of data acquisition and l1-optimization (recognition with compression)
Automation and Remote Control
Strengthening hash families and compressive sensing
Journal of Discrete Algorithms
Compressed sensing construction of spectrum map for routing in cognitive radio networks
Wireless Communications & Mobile Computing
Sketching via hashing: from heavy hitters to compressed sensing to sparse fourier transform
Proceedings of the 32nd symposium on Principles of database systems
Hi-index | 754.84 |
Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any n-dimensional vector that is k-sparse can be fully recovered using O(k log n) measurements and only O(k log n) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond 3/4 and show that, with the same number of measurements, only O(k) recovery iterations are required, which is a significant improvement when n is large. In fact, full recovery can be accomplished by at most 2k very simple iterations. The number of iterations can be reduced arbitrarily close to k, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time O(n log(n/k)). We also show that by tolerating a small penalty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the k significant elements of an almost k-sparse signal and then, using very simple optimization techniques, finds a k-sparse signal which is close to the best k-term approximation of the original signal.