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2003
IEEE

Bounded Geometries, Fractals, and Low-Distortion Embeddings

8 years 9 months ago
Bounded Geometries, Fractals, and Low-Distortion Embeddings
The doubling constant of a metric space (X, d) is the smallest value λ such that every ball in X can be covered by λ balls of half the radius. The doubling dimension of X is then defined as dim(X) = log2 λ. A metric (or sequence of metrics) is called doubling precisely when its doubling dimension is bounded. This is a robust class of metric spaces which contains many families of metrics that occur in applied settings. We give tight bounds for embedding doubling metrics into (low-dimensional) normed spaces. We consider both general doubling metrics, as well as more restricted families such as those arising from trees, from graphs excluding a fixed minor, and from snowflaked metrics. Our techniques include decomposition theorems for doubling metrics, and an analysis of a fractal in the plane due to Laakso [20]. Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L2.
Anupam Gupta, Robert Krauthgamer, James R. Lee
Added 04 Jul 2010
Updated 04 Jul 2010
Type Conference
Year 2003
Where FOCS
Authors Anupam Gupta, Robert Krauthgamer, James R. Lee
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