License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.90
URN: urn:nbn:de:0030-drops-90945
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9094/
Nakos, Vasileios ;
Shi, Xiaofei ;
Woodruff, David P. ;
Zhang, Hongyang
Improved Algorithms for Adaptive Compressed Sensing
Abstract
In the problem of adaptive compressed sensing, one wants to estimate an approximately k-sparse vector x in R^n from m linear measurements A_1 x, A_2 x,..., A_m x, where A_i can be chosen based on the outcomes A_1 x,..., A_{i-1} x of previous measurements. The goal is to output a vector x^ for which |x-x^|_p <=C * min_{k-sparse x'} |x-x'|_q, with probability at least 2/3, where C > 0 is an approximation factor. Indyk, Price and Woodruff (FOCS'11) gave an algorithm for p=q=2 for C = 1+epsilon with O((k/epsilon) loglog (n/k)) measurements and O(log^*(k) loglog (n)) rounds of adaptivity. We first improve their bounds, obtaining a scheme with O(k * loglog (n/k) + (k/epsilon) * loglog(1/epsilon)) measurements and O(log^*(k) loglog (n)) rounds, as well as a scheme with O((k/epsilon) * loglog (n log (n/k))) measurements and an optimal O(loglog (n)) rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for (p,p) for every 0 < p < 2. We show that the improvement from O(k log(n/k)) measurements to O(k log log (n/k)) measurements in the adaptive setting can persist with a better epsilon-dependence for other values of p and q. For example, when (p,q) = (1,1), we obtain O(k/sqrt{epsilon} * log log n log^3 (1/epsilon)) measurements. We obtain nearly matching lower bounds, showing our algorithms are close to optimal. Along the way, we also obtain the first nearly-optimal bounds for (p,p) schemes for every 0 < p < 2 even in the non-adaptive setting.
BibTeX - Entry
@InProceedings{nakos_et_al:LIPIcs:2018:9094,
author = {Vasileios Nakos and Xiaofei Shi and David P. Woodruff and Hongyang Zhang},
title = {{Improved Algorithms for Adaptive Compressed Sensing}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {90:1--90:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-076-7},
ISSN = {1868-8969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9094},
URN = {urn:nbn:de:0030-drops-90945},
doi = {10.4230/LIPIcs.ICALP.2018.90},
annote = {Keywords: Compressed Sensing, Adaptivity, High-Dimensional Vectors}
}
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Keywords: |
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Compressed Sensing, Adaptivity, High-Dimensional Vectors |
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Collection: |
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45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) |
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Issue Date: |
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2018 |
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Date of publication: |
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04.07.2018 |
