Document Open Access Logo

Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces

Authors Arturs Backurs, Anastasios Sidiropoulos

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2016.1.pdf
  • Filesize: 496 kB
  • 15 pages

Document Identifiers

Subject Classification

Keywords
  • metric embeddings
  • Hausdorff metric
  • distortion
  • dimension

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We show that the Hausdorff metric over constant-size pointsets in constant-dimensional Euclidean space admits an embedding into constant-dimensional l_{infinity} space with constant distortion. More specifically for any s,d>=1, we obtain an embedding of the Hausdorff metric over pointsets of size s in d-dimensional Euclidean space, into  l_{\infinity}^{s^{O(s+d)}} with distortion s^{O(s+d)}. We remark that any metric space M admits an isometric embedding into l_{infinity} with dimension proportional to the size of M. In contrast, we obtain an embedding of a space of infinite size into constant-dimensional l_{infinity}.

We further improve the distortion and dimension trade-offs by considering probabilistic embeddings of the snowflake version of the Hausdorff metric. For the case of pointsets of size s in the real line of bounded resolution, we obtain a probabilistic embedding into l_1^{O(s*log(s()} with distortion O(s).

Cite As Get BibTex

Arturs Backurs and Anastasios Sidiropoulos. Constant-Distortion Embeddings of Hausdorff Metrics into Constant-Dimensional l_p Spaces. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 1:1-1:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2016.1

Author Details

Arturs Backurs
Anastasios Sidiropoulos

References

  1. Artūrs Bačkurs and Piotr Indyk. Better embeddings for planar earth-mover distance over sparse sets. In Proceedings of the Thirtieth Annual Symposium on Computational Geometry, SOCG'14, pages 280:280-280:289, New York, NY, USA, 2014. ACM. URL: http://dx.doi.org/10.1145/2582112.2582120.
  2. Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15(6):1373-1396, 2003. Google Scholar
  3. Radu Berinde, Anna C. Gilbert, Piotr Indyk, H. Karloff, and Martin J. Strauss. Combining geometry and combinatorics: A unified approach to sparse signal recovery. In Communication, Control, and Computing, 2008 46th Annual Allerton Conference on, pages 798-805. IEEE, 2008. Google Scholar
  4. J. Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52(1-2):46-52, 1985. URL: http://dx.doi.org/10.1007/BF02776078.
  5. Manuel Costa, Miguel Castro, Antony Rowstron, and Peter Key. Pic: Practical internet coordinates for distance estimation. In Distributed Computing Systems, 2004. Proceedings. 24th International Conference on, pages 178-187. IEEE, 2004. Google Scholar
  6. Aryeh Dvoretzky. Some results on convex bodies and Banach spaces. Hebrew University, 1960. Google Scholar
  7. Martin Farach-Colton and Piotr Indyk. Approximate nearest neighbor algorithms for hausdorff metrics via embeddings. In Foundations of Computer Science, 1999. 40th Annual Symposium on, pages 171-179. IEEE, 1999. Google Scholar
  8. Sariel Har-Peled and Manor Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006. Google Scholar
  9. P. Indyk and N. Thaper. Fast color image retrieval via embeddings. Workshop on Statistical and Computational Theories of Vision (at ICCV), 2003. Google Scholar
  10. Piotr Indyk. Algorithmic applications of low-distortion geometric embeddings. In focs, page 10. IEEE, 2001. Google Scholar
  11. Piotr Indyk. High-dimensional Computational Geometry. PhD thesis, Stanford University, 2001. Google Scholar
  12. Piotr Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. Journal of the ACM (JACM), 53(3):307-323, 2006. Google Scholar
  13. Piotr Indyk, Avner Magen, Anastasios Sidiropoulos, and Anastasios Zouzias. On-line embeddings. In Proc. of APPROX, 2010. Google Scholar
  14. William B Johnson and Joram Lindenstrauss. Extensions of lipschitz mappings into a hilbert space. Contemporary mathematics, 26(189-206):1, 1984. Google Scholar
  15. Nathan Linial, Eran London, and Yuri Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995. Google Scholar
  16. Ofer Neiman. Low dimensional embeddings of doubling metrics. In Christos Kaklamanis and Kirk Pruhs, editors, Approximation and Online Algorithms, volume 8447 of Lecture Notes in Computer Science, pages 12-23. Springer International Publishing, 2014. Google Scholar
  17. Santosh S Vempala. The random projection method, volume 65. American Mathematical Soc., 2005. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact