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DOI: 10.4230/LIPIcs.ICALP.2018.90
URN: urn:nbn:de:0030-drops-90945
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Nakos, Vasileios ; Shi, Xiaofei ; Woodruff, David P. ; Zhang, Hongyang

Improved Algorithms for Adaptive Compressed Sensing

LIPIcs-ICALP-2018-90.pdf (0.6 MB)


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

  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 =		{},
  URN =		{urn:nbn:de:0030-drops-90945},
  doi =		{10.4230/LIPIcs.ICALP.2018.90},
  annote =	{Keywords: Compressed Sensing, Adaptivity, High-Dimensional Vectors}

Keywords: Compressed Sensing, Adaptivity, High-Dimensional Vectors
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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