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Bounded geometries, fractals, and low-distortion embeddings
Open accessAbstract
The doubling constant of a metric space (X, d) is the smallest value /spl lambda/ such that every ball in X can be covered by /spl lambda/ balls of half the radius. The doubling dimension of X is then defined as dim (X) = log/sub 2//spl lambda/. 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 according to T. J. Laakso (2002). Finally, we discuss some applications and point out a central open question regarding dimensionality reduction in L/sub 2/.
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Cites
A Global Geometric Framework for Nonlinear Dimensionality Reduction
Tenenbaum, J. B., Silva, V. D., & Langford, J. C. (2000) · Science · doi:10.1126/science.290.5500.2319 · Cited by 13,585
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Self-citationRandom projection trees and low dimensional manifolds
Dasgupta, S., & Freund, Y. (2008) · Proceedings of the fortieth annual ACM symposium on Theory of computing · doi:10.1145/1374376.1374452 · Cited by 375
Graph cluster randomization: network exposure to multiple universes
Ugander, J., Karrer, B., Backstrom, L., & Kleinberg, J. (2013) · Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining · doi:10.1145/2487575.2487695 · Cited by 183
Metadata
- Authors
- Gupta, A., Krauthgamer, R., & Lee, J. R.
- Venue
- 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- Publisher
- IEEE Computer. Soc
- Published
- 2004-03-01
- Pages
- 534-543
- ISSN
- 0272-5428 (ISSN-L)
- OpenAlex
- W2127248062
- Cited by
- 405
- References
- 1
- FWCI
- 54.25
- Type
- conference paper
- Language
- en
- Open access
- green
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- Gupta, A., Krauthgamer, R., & Lee, J. R. (2004). Bounded geometries, fractals, and low-distortion embeddings. 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings. https://doi.org/10.1109/sfcs.2003.1238226
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