Skip to main content
SpringerLink
Log in

Low Distortion Metric Embedding into Constant Dimension

  • Conference paper
Book cover Theory and Applications of Models of Computation (TAMC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6648))

  • 842 Accesses

Abstract

We investigate the possibility of embedding an n-point metric space into a constant dimensional vector space with the maximum norm, such that the embedding is almost isometric, that is, the distortion of distances is kept arbitrarily close to 1. When the source metric is generated by any fixed norm on a finite dimensional vector space, we prove that this embedding is always possible, such that the dimension of the target space remains constant, independent of n. While this possibility has been known in the folklore, we present the first fully detailed proof, which, in addition, is significantly simpler and more transparent, then what was available before. Furthermore, our embedding can be computed in deterministic linear time in n, given oracle access to the norm.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abraham, I., Bartal, Y., Neiman, O.: Advances in Metric Embedding Theory. In: 38th Annual ACM Symp. on Theory of Computing (STOC 2006), Seattle, WA, USA, pp. 271–286 (May 2006)

    Google Scholar 

  2. Abraham, I., Bartal, Y., Neiman, O.: Embedding Metric Spaces in Their Intrinsic Dimension. In: 19th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA 2008), San Francisco, CA, USA, pp. 363–372 (January 2008)

    Google Scholar 

  3. Alon, N.: Problems and Results in Extremal Combinatorics – I. Discrete Math. 273, 31–53 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assouad, P.: Plongements Lipschitziens dans R n. Bull. Soc. Math. 111, 429–448 (1983)

    MathSciNet  MATH  Google Scholar 

  5. Bourgain, J.: On Lipschitz Embedding of Finite Metric Spaces in Hilbert Space. Israel J. Math. 52, 46–52 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brinkman, B., Charikar, M.: On the Impossibility of Dimension Reduction in ℓ1. Journal of the ACM 52, 766–788 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dudley, R.M.: Metric Entropy of Some Classes of Sets with Differentiable Boundaries. Journal of Approximation Theory 10, 227–236 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. Faragó, A., Linder, T., Lugosi, G.: Fast Nearest Neighbor Search in Dissimilarity Spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence 15(9), 957–962 (1993)

    Article  Google Scholar 

  9. Gottlieb, L.-A., Krauthgamer, R.: A Nonlinear Approach to Dimension Reduction, arXiv:0907.5477v2 [cs.CG] (April 2010)

    Google Scholar 

  10. Gupta, A., Krauthgamer, R., Lee, J.R.: Bounded Geometries, Fractals, and Low-Distortion Embeddings. In: 44th Annual IEEE Symp. on Foundations of Computer Science (FOCS 2003), Cambridge, MA, USA, pp. 534–543 (October 2003)

    Google Scholar 

  11. Howard, R.: The John Ellipsoid Theorem, Lecture Note, Dept. of Math., Univ. of South Carolina (November 1997), http://www.math.sc.edu/~howard/Notes/john.pdf

  12. Indyk, P.: Algorithmic Applications of Low-Distortion Geometric Embeddings. In: 42nd Annual IEEE Symp. on Foundations of Computer Science (FOCS 2001), Las Vegas, NV, pp. 10–33 (October 2001)

    Google Scholar 

  13. Indyk, P., Matoušek, J.: Low Distortion Embeddings of Finite Metric Spaces. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Combinatorial Geometry, pp. 177–196. Chapman and Hall/CRC, Boca Raton (2004)

    Google Scholar 

  14. Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz Mappings into a Hilbert Space. Contemp. Math. 26, 189–206 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Linial, N., London, E., Rabinovich, Y.: The Geometry of Graphs and Some of Its Algorithmic Applications. Combinatorica 15, 215–245 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Matoušek, J.: Note on Bi-Lipschitz Embeddings Into Low Dimensional Euclidean Spaces. Comment. Math. Univ. Carolinae 31, 589–600 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Matoušek, J.: On the Distortion Required for Embedding Finite Metric Spaces Into Normed Spaces. Israel J. Math. 93, 333–344 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)

    Book  MATH  Google Scholar 

  19. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  20. Semmes, S.: On the Nonexistence Bilipschitz Parametrizations and Geometric Problems about a  ∞  Weights. Revista Mathemática Iberoamericana 12, 337–410 (1996)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, 75080, USA

    András Faragó

Authors
  1. András Faragó
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Miami, 1365 Memorial Drive, 33146, Coral Gables, FL, USA

    Mitsunori Ogihara

  2. Department of Information and Communication Engineering, University of Electro-Comm, Chofugaoga 1-5-1, Chofu, 182-8585, Tokyo, Japan

    Jun Tarui

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Faragó, A. (2011). Low Distortion Metric Embedding into Constant Dimension. In: Ogihara, M., Tarui, J. (eds) Theory and Applications of Models of Computation. TAMC 2011. Lecture Notes in Computer Science, vol 6648. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20877-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-20877-5_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-20876-8

  • Online ISBN: 978-3-642-20877-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation