Low Dimensional Embeddings of Doubling Metrics

摘要

We study several embeddings of doubling metrics into low dimensional normed spaces, in particular into a"" (2) and a"" (a) . Doubling metrics are a robust class of metric spaces that have low intrinsic dimension, and often occur in applications. Understanding the dimension required for a concise representation of such metrics is a fundamental open problem in the area of metric embedding. Here we show that the n-vertex Laakso graph can be embedded into constant dimensional a"" (2) with the best possible distortion, which has implications for possible approaches to the above problem.