Low Dimensional Embeddings of Doubling Metrics

We study several embeddings of doubling metrics into low dimensional normed spaces, in particular into ℓ₂ and ℓ_∞. 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 ℓ₂ with the best possible distortion, which has implications for possible approaches to the above problem. Since arbitrary doubling metrics require high distortion for embedding into ℓ₂ and even into ℓ₁, we turn to the ℓ_∞ space that enables us to obtain arbitrarily small distortion. We show embeddings of doubling metrics and their ”snowflakes” into low dimensional ℓ_∞ space that simplify and extend previous results.