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.APPROX-RANDOM.2019.47
URN: urn:nbn:de:0030-drops-112628
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/11262/
Emiris, Ioannis Z. ;
Margonis, Vasilis ;
Psarros, Ioannis
Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1
Abstract
Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (l_2) metric, but much less for the Manhattan (l_1) metric. Our primary motivation is the approximate nearest neighbor problem in l_1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both l_2 and l_1 metrics, as well as for doubling subsets of l_2. The case that remained open were doubling subsets of l_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of l_1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest.
BibTeX - Entry
@InProceedings{emiris_et_al:LIPIcs:2019:11262,
author = {Ioannis Z. Emiris and Vasilis Margonis and Ioannis Psarros},
title = {{Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
pages = {47:1--47:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-125-2},
ISSN = {1868-8969},
year = {2019},
volume = {145},
editor = {Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2019/11262},
URN = {urn:nbn:de:0030-drops-112628},
doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.47},
annote = {Keywords: Approximate nearest neighbor, Manhattan metric, randomized embedding}
}
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Keywords: |
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Approximate nearest neighbor, Manhattan metric, randomized embedding |
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Collection: |
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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019) |
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Issue Date: |
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2019 |
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Date of publication: |
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17.09.2019 |
