## Section: New Results

### Near-Neighbor Preserving Dimension Reduction for Doubling Subsets of ${L}_{1}$

Participants : Ioannis Emiris, Ioannis Psarros.

In [21], we study randomized dimensionality reduction which 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.

This is joint work with Vassilis Margonis (NKUA), and is based on his MSc thesis.