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 and 1 norms whose spread (the ratio of the diameter to the closest pair distance) is bounded by 0. A well-known dimensionality reduction theorem due to Johnson and Lindenstrauss yields a sketch of size O(2 log(n)n log n), i.e., O(2 log(n) 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 ) 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 ) bits per point, which we show is optimal.