Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1

Authors Ioannis Z. Emiris, Vasilis Margonis, Ioannis Psarros

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Author Details

Ioannis Z. Emiris
  • Department of Informatics & Telecommunications, National & Kapodistrian University of Athens, Greece
  • ATHENA Research & Innovation Center, Greece
Vasilis Margonis
  • Department of Informatics & Telecommunications, National & Kapodistrian University of Athens, Greece
Ioannis Psarros
  • Institute of Computer Science, University of Bonn, Germany


IZE is member of team AROMATH, joint between INRIA Sophia-Antipolis and NKUA. IP thanks Robert Krauthgamer for useful discussions on the topic.

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Ioannis Z. Emiris, Vasilis Margonis, and Ioannis Psarros. Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of l_1. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 47:1-47:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Nearest neighbor algorithms
  • Mathematics of computing → Dimensionality reduction
  • Approximate nearest neighbor
  • Manhattan metric
  • randomized embedding


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