Algorithmic linear dimension reduction in the l_1 norm for sparse vectors

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dc.creatorGilbert, A. C.
dc.creatorStrauss, M. J.
dc.creatorTropp, J. A.
dc.creatorVershynin, R.
dc.date2006-08-19
dc.date.accessioned2026-07-07T07:20:02Z
dc.date.available2026-07-07T07:20:02Z
dc.descriptionThis paper develops a new method for recovering m-sparse signals that is simultaneously uniform and quick. We present a reconstruction algorithm whose run time, O(m log^2(m) log^2(d)), is sublinear in the length d of the signal. The reconstruction error is within a logarithmic factor (in m) of the optimal m-term approximation error in l_1. In particular, the algorithm recovers m-sparse signals perfectly and noisy signals are recovered with polylogarithmic distortion. Our algorithm makes O(m log^2 (d)) measurements, which is within a logarithmic factor of optimal. We also present a small-space implementation of the algorithm. These sketching techniques and the corresponding reconstruction algorithms provide an algorithmic dimension reduction in the l_1 norm. In particular, vectors of support m in dimension d can be linearly embedded into O(m log^2 d) dimensions with polylogarithmic distortion. We can reconstruct a vector from its low-dimensional sketch in time O(m log^2(m) log^2(d)). Furthermore, this reconstruction is stable and robust under small perturbations.
dc.identifierhttps://arxiv.org/abs/cs/0608079
dc.identifierhttp://arxiv.org/abs/cs/0608079
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/114842
dc.subjectData Structures and Algorithms
dc.titleAlgorithmic linear dimension reduction in the l_1 norm for sparse vectors
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