Abstract
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.
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Neiman, O. (2014). Low Dimensional Embeddings of Doubling Metrics. In: Kaklamanis, C., Pruhs, K. (eds) Approximation and Online Algorithms. WAOA 2013. Lecture Notes in Computer Science, vol 8447. Springer, Cham. https://doi.org/10.1007/978-3-319-08001-7_2
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DOI: https://doi.org/10.1007/978-3-319-08001-7_2
Publisher Name: Springer, Cham
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