Near-Optimal (Euclidean) Metric Compression

The metric sketching problem is defined as follows. Given a metric on n points, and > 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 + distortion. In this paper we consider metrics induced by `2 and `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( −2 log(Φn)n log n), i.e., O( −2 log(Φn) log n) bits per point. We show that this bound is not optimal, and can be substantially improved to O( −2 log(1/ ) · log n+ log log Φ) bits per point. Furthermore, we show that our bound is tight up to a factor of log(1/ ). We also consider sketching of general metrics and provide a sketch of size O(n log(1/ ) + log log Φ) bits per point, which we show is optimal. ∗[email protected]. †[email protected]. ar X iv :1 60 9. 06 29 5v 2 [ cs .C G ] 1 8 N ov 2 01 6