Near-Optimal (Euclidean) Metric Compression

The metric sketching problem is defined as follows. Given a metric on n points, and epsilon > 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 + epsilon distortion. In this paper we consider metrics induced by l(2) and l(1) norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by Phi > 0. A well-known dimensionality reduction theorem due to Johnson and Lindenstrauss yields a sketch of size O(epsilon(-2) log(Phi n)n)n log n), i.e., O(epsilon(-2) log(Phi)n) log n) bits per point. We show that this bound is not optimal, and can be substantially improved to O(epsilon(-2) log(1/epsilon). log n + log log Phi) bits per point. Furthermore, we show that our bound is tight up to a factor of log(1/epsilon). We also consider sketching of general metrics and provide a sketch of size O(n log(1/epsilon) + log log Phi) bits per point, which we show is optimal.