Near neighbor preserving dimension reduction for doubling subsets of ℓ_1
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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 and, 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. In this paper, we propose a dimension reduction, near neighbor-preserving embedding for doubling subsets of ℓ_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.
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