Abstract
In the problem of adaptive compressed sensing, one wants to estimate an approximately ksparse 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_{i1} x of previous measurements. The goal is to output a vector x^ for which xx^_p <=C * min_{ksparse x'} xx'_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 epsilondependence 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 nearlyoptimal bounds for (p,p) schemes for every 0 < p < 2 even in the nonadaptive 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 = {90:190:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9094},
URN = {urn:nbn:de:0030drops90945},
doi = {10.4230/LIPIcs.ICALP.2018.90},
annote = {Keywords: Compressed Sensing, Adaptivity, HighDimensional Vectors}
}
Keywords: 

Compressed Sensing, Adaptivity, HighDimensional Vectors 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 