Rademacher-Sketch: A Dimensionality-Reducing Embedding for Sum-Product Norms, with an Application to Earth-Mover Distance

  • Elad Verbin
  • Qin Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

Consider a sum-product normed space, i.e. a space of the form \(Y=\ell_1^n \otimes X\), where X is another normed space. Each element in Y consists of a length-n vector of elements in X, and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constant-distortion embedding from the normed space \(\ell_1^n \otimes X\) into a lower-dimensional normed space \(\ell_1^{n'} \otimes X\), where n′ ≪ n is some value that depends on the properties of the normed space X (namely, on its Rademacher dimension). In particular, composing this embedding with another well-known embedding of Indyk [18], we get an O(1/ε)-distortion embedding from the earth-mover metric EMD Δ on the grid [Δ]2 to \(\ell_1^{\Delta^{O(\epsilon )}} \otimes {\sf{EEMD}}_{\Delta^{\epsilon }}\) (where EEMD is a norm that generalizes earth-mover distance). This embedding is stronger (and simpler) than the sketching algorithm of Andoni et al [4], which maps EMD Δ with O(1/ε) approximation into sketches of size Δ O(ε).

Keywords

Normed Space Hash Function Earth Mover Sample Level Pseudorandom Generator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. In: SoCG, pp. 39–50 (1995)Google Scholar
  2. 2.
    Agarwal, P.K., Varadarajan, K.R.: A near-linear constant-factor approximation for euclidean bipartite matching? In: Symposium on Computational Geometry, pp. 247–252 (2004)Google Scholar
  3. 3.
    Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58, 137–147 (1999)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Andoni, A., Ba, K.D., Indyk, P., Woodruff, D.: Efficient sketches for earth-mover distance, with applications. In: FOCS (2009)Google Scholar
  5. 5.
    Andoni, A., Charikar, M.S., Neiman, O., Nguyen, H.L.: Near linear lower bound for dimension reduction in ℓ1. In: IEEE Symposium on Foundations of Computer Science (2011)Google Scholar
  6. 6.
    Andoni, A., Indyk, P., Krauthgamer, R.: Overcoming the ℓ1 non-embeddability barrier: algorithms for product metrics. In: SODA, pp. 865–874 (2009)Google Scholar
  7. 7.
    Andoni, A., Krauthgamer, R., Onak, K.: Streaming algorithms via precision sampling. In: FOCS, pp. 363–372 (2011)Google Scholar
  8. 8.
    Andoni, A., Nguyen, H.L.: Near-optimal sublinear time algorithms for Ulam distance. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 76–86 (2010)Google Scholar
  9. 9.
    Andoni, A., Onak, K.: Approximating edit distance in near-linear time. In: STOC, pp. 199–204 (2009)Google Scholar
  10. 10.
    Brinkman, B., Charikar, M.: On the impossibility of dimension reduction in ℓ1. J. ACM 52, 766–788 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Charikar, M.S.: Similarity estimation techniques from rounding algorithms. In: STOC, pp. 380–388 (2002)Google Scholar
  12. 12.
    Chefd’hotel, C., Bousquet, G.: Intensity-based image registration using earth mover’s distance. In: SPIE (2007)Google Scholar
  13. 13.
    Deng, Y., Du, W.: The Kantorovich metric in computer science: A brief survey. Electr. Notes Theor. Comput. Sci. 253(3), 73–82 (2009)CrossRefGoogle Scholar
  14. 14.
    Grauman, K., Darrell, T.: Fast contour matching using approximate earth movers distance. In: CVPR, pp. 220–227 (2004)Google Scholar
  15. 15.
    Holmes, A.S., Rose, C.J., Taylor, C.J.: Transforming pixel signatures into an improved metric space. Image Vision Comput 20(9-10), 701–707 (2002)CrossRefGoogle Scholar
  16. 16.
    Indyk, P.: Algorithmic aspects of geometric embeddings. In: IEEE Symposium on Foundations of Computer Science (2001)Google Scholar
  17. 17.
    Indyk, P.: Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM 53, 307–323 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Indyk, P.: A near linear time constant factor approximation for euclidean bichromatic matching (cost). In: SODA, pp. 39–42 (2007)Google Scholar
  19. 19.
    Indyk, P., Matousek, J.: Low-distortion embeddings of finite metric spaces. In: Handbook of Discrete and Computational Geometry, pp. 177–196. CRC Press (2004)Google Scholar
  20. 20.
    Indyk, P., Thaper, N.: Fast color image retrieval via embeddings. In: Workshop on Statistical and Computational Theories of Vision, at ICCV (2003)Google Scholar
  21. 21.
    Johnson, W., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in modern analysis and probability (New Haven, Conn., 1982). Contemporary Mathematics, vol. 26, pp. 189–206. American Mathematical Society (1984)Google Scholar
  22. 22.
    Kantorovich, L.V.: On the translocation of masses. Dokl. Akad. Nauk SSSR 37(7-8), 227–229 (1942)Google Scholar
  23. 23.
    Lawler, E.: Combinatorial optimization - networks and matroids. Holt, Rinehart and Winston, New York (1976)Google Scholar
  24. 24.
    Levina, E., Bickel, P.J.: The earth mover’s distance is the mallows distance: Some insights from statistics. In: ICCV, pp. 251–256 (2001)Google Scholar
  25. 25.
    Linial, N.: Finite metric spaces - combinatorics, geometry and algorithms. In: Proceedings of the International Congress of Mathematicians III, pp. 573–586 (2002)Google Scholar
  26. 26.
    Maurey, B.: Type, cotype and k-convexity. In: Handbook of the Geometry of Banach Spaces, vol. 2, pp. 1299–1332. North-Holland (2003)Google Scholar
  27. 27.
    McGregor, A.: Open problems in data streams, property testing, and related topics (2011), http://www.cs.umass.edu/~mcgregor/papers/11-openproblems.pdf
  28. 28.
    Naor, A., Schechtman, G.: Planar earthmover is not in ℓ1. SIAM J. Comput. 37(3), 804–826 (2007)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Nisan, N.: Pseudorandom generators for space-bounded computations. In: Proceedings of the Twenty-Second Annual ACM Symposium on Theory of Computing, STOC 1990, pp. 204–212 (1990)Google Scholar
  30. 30.
    Puzicha, J., Buhmann, J.M., Rubner, Y., Tomasi, C.: Empirical evaluation of dissimilarity measures for color and texture. In: ICCV, pp. 1165–1173 (1999)Google Scholar
  31. 31.
    Rubner, Y., Tomasi, C., Guibas, L.J.: The earth movers distance as a metric for image retrieval. International Journal of Computer Vision 40 (2000)CrossRefGoogle Scholar
  32. 32.
    Vaidya, P.M.: Geometry helps in matching. SIAM J. Comput. 18, 1201–1225 (1989)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Varadarajan, K.R., Agarwal, P.K.: Approximation algorithms for bipartite and non-bipartite matching in the plane. In: SODA, pp. 805–814 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Elad Verbin
    • 1
  • Qin Zhang
    • 2
  1. 1.MADALGO and CTICAarhus UniversityDenmark
  2. 2.MADALGOAarhus UniversityDenmark

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