Nearly-optimal bounds for sparse recovery in generic norms, with applications to k-median sketching
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Nearly-optimal bounds.pdf
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Author(s)
Woodruff, David P.
Backurs, Arturs
Indyk, Piotr
Razenshteyn, Ilya
Date Issued
January 2016
Journal
SODA '16 Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms
Publisher
Association for Computing Machinery
Citation
Backurs, Arturs et al. "Nearly-optimal bounds for sparse recovery in generic norms, with applications to k-median sketching." SODA '16 Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms, 20-12 January, 2016, Philadelphia, Pennsylvania, Association of Computing Machinery, 2016, pp. 318-337.
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Original manuscript
Abstract
We initiate the study of trade-offs between sparsity and the number of measurements in sparse recovery schemes for generic norms. Specifically for a norm ||·||, sparsity parameter k, approximation factor K > 0, and probability of failure P > 0, we ask: what is the minimal value of m so that there is a distribution over m × n matrices A with the property that for any x, given Ax, we can recover a k-sparse approximation to x in the given norm with probability at least 1 -- P? We give a partial answer to this problem, by showing that for norms that admit efficient linear sketches, the optimal number of measurements m is closely related to the doubling dimension of the metric induced by the norm ||·|| on the set of all k-sparse vectors. By applying our result to specific norms, we cast known measurement bounds in our general framework (for the [subscript ℓ]p norms, p ∈ [1, 2]) as well as provide new, measurement-efficient schemes (for the Earth-Mover Distance norm). The latter result directly implies more succinct linear sketches for the well-studied planar k-median clustering problem. Finally our lower bound for the doubling dimension of the EMD norm enables us to resolve the open question of [Frahling-Sohler, STOC'05] about the space complexity of clustering problems in the dynamic streaming model.
MIT Department
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
http://dl.acm.org/citation.cfm?id=2884459
