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 (\ell_2) metric and, much less ,  for the Manhattan (\ell_1) metric. Our primary motivation is the approximate nearest neighbor problem in \ell_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 \ell_2 and \ell_1 metrics, as well as for doubling subsets of \ell_2.  In this paper, we propose a dimension reduction , near neighbor-preserving embedding for doubling subsets of \ell_1. Our approach is to represent the point set with a carefully chosen covering set, and then apply a random projection to that covering set . We study two cases 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  .
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 (\ell_2) metric , but much less for the Manhattan (\ell_1) metric. Our primary motivation is the approximate nearest neighbor problem in \ell_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 \ell_2 and \ell_1 metrics, as well as for doubling subsets of \ell_2. The case that remained open were doubling subsets of \ell_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of \ell_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 .
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The case that remained open were doubling subsets of \ell_1.
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. We employ Cauchy variables: certain concentration bounds derived should be of independent interest