Near-Optimal (Euclidean) Metric Compression

The metric sketching problem is defined as follows. Given a metric on $n$ points, and $ε>0$, we wish to produce a small size data structure (sketch) that, given any pair of point indices, recovers the distance between the points up to a $1+ε$ distortion. In this paper we consider metrics induced by $\ell_2$ and $\ell_1$ norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by $Φ>0$. A well-known dimensionality reduction theorem due to Johnson and Lindenstrauss yields a sketch of size $O(ε^{-2} \log (Φn) n\log n)$, i.e., $O(ε^{-2} \log (Φn) \log n)$ bits per point. We show that this bound is not optimal, and can be substantially improved to $O(ε^{-2}\log(1/ε) \cdot \log n + \log\log Φ)$ bits per point. Furthermore, we show that our bound is tight up to a factor of $\log(1/ε)$. We also consider sketching of general metrics and provide a sketch of size $O(n\log(1/ε)+ \log\log Φ)$ bits per point, which we show is optimal.

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