Advertisement

Approximate Nearest Neighbor Search for \(\ell _p\)-Spaces \((2<p<\infty )\) via Embeddings

  • Yair Bartal
  • Lee-Ad Gottlieb
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

Abstract

While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and \(\ell _1\), few non-trivial algorithms are known for \(\ell _p\) when \(2<p<\infty \). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the best known approximation factor guarantees in this setting.

Notes

Acknowledgements

We thank Sariel Har-Peled, Piotr Indyk, Robi Krauthgamer, Assaf Naor and Gideon Schechtman for helpful conversations.

References

  1. 1.
    Ailon, N., Chazelle, B.: Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In: STOC 2006, pp. 557–563 (2006)Google Scholar
  2. 2.
    Andoni, A., Croitoru, D., Patrascu, M.: Hardness of nearest neighbor under L-infinity. In: Foundations of Computer Science, pp. 424–433 (2008)Google Scholar
  3. 3.
    Andoni, A.: Nearest neighbor search: the old, the new, and the impossible. Ph.D. thesis, MIT (2009)Google Scholar
  4. 4.
    Arya, S., Malamatos, T.: Linear-size approximate Voronoi diagrams. In: SODA 2002, pp. 147–155 (2002)Google Scholar
  5. 5.
    Ball, K.: Isometric embedding in \(l_p\)-spaces. Eur. J. Comb. 11(4), 305–311 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Batu, T., Ergun, F., Sahinalp, C.: Oblivious string embeddings and edit distance approximations. In: SODA 2006, pp. 792–801 (2006)Google Scholar
  7. 7.
    Bellman, R.E.: Adaptive Control Processes: A Guided Tour. Princeton University Press, Princeton (1961)CrossRefzbMATHGoogle Scholar
  8. 8.
    Bern, M.: Approximate closest-point queries in high dimensions. Inf. Process. Lett. 45(2), 95–99 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Beygelzimer, A., Kakade, S., Langford, J.: Cover trees for nearest neighbor. In: ICML 2006, pp. 97–104 (2006)Google Scholar
  10. 10.
    Binyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Colloquium Publications (American Mathematical Society), Providence (2000)Google Scholar
  11. 11.
    Chan, T.M.: Approximate nearest neighbor queries revisited. Discret. Comput. Geom. 20(3), 359–373 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Clarkson, K.L.: A randomized algorithm for closest-point queries. SIAM J. Comput. 17(4), 830–847 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Cole, R., Gottlieb, L.: Searching dynamic point sets in spaces with bounded doubling dimension. In: STOC 2006, pp. 574–583 (2006)Google Scholar
  14. 14.
    de Amorim, R.C., Mirkin, B.: Minkowski metric feature weighting and anomalous cluster initializing in k-means clustering. Pattern Recogn. 45(3), 1061–1075 (2012)CrossRefGoogle Scholar
  15. 15.
    Fichet, B.: \(l_p\)-spaces in data analysis. In: Bock, H.H. (ed.) Classification and Related Metods of Data Analysis, pp. 439–444. North-Holland, Amsterdam (1988)Google Scholar
  16. 16.
    Finlayson, G.D., Rey, P.A.T., Trezzi, E.: General \(l_p\) constrained approach for colour constancy. In: ICCV Workshops 2011, pp. 790–797 (2011)Google Scholar
  17. 17.
    Finlayson, G.D., Trezzi, E.: Shades of gray and colour constancy. In: CIC 2004, pp. 37–41 (2004)Google Scholar
  18. 18.
    Har-Peled, S., Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. Theory Comput. 8(1), 321–350 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Har-Peled, S., Mendel, M.: Fast construction of nets in low-dimensional metrics and their applications. SIAM J. Comput. 35(5), 1148–1184 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Har-Peled, S., Kumar, N.: Approximating minimization diagrams and generalized proximity search. SIAM J. Comput. 44(4), 944–974 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Indyk, P.: On approximate nearest neighbors in non-Euclidean spaces. In: FOCS, pp. 148–155 (1998)Google Scholar
  22. 22.
    Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: STOC 1998, pp. 604–613 (1998)Google Scholar
  23. 23.
    Indyk, P., Naor, A.: Nearest-neighbor-preserving embeddings. ACM Trans. Algorithms 3(3), 31 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Johnson, W.B., Lindenstrauss, J.: Extensions of Lipschitz mappings into a Hilbert space. In: Conference in Modern Analysis and Probability, New Haven, Connecticut, 1982, pp. 189–206. American Mathematical Society, Providence (1984)Google Scholar
  25. 25.
    Johnson, W.B., Schechtman, G.: Embedding \(l_p^m\) into \(l_1^n\). Acta Math. 149(1–2), 71–85 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Krauthgamer, R., Lee, J.R.: Navigating nets: simple algorithms for proximity search. In: SODA 2004, pp. 791–801 (2004)Google Scholar
  27. 27.
    Kushilevitz, E., Ostrovsky, R., Rabani, Y.: Efficient search for approximate nearest neighbor in high dimensional spaces. In: STOC 1998, pp. 614–623 (1998)Google Scholar
  28. 28.
    Naor, A.: Comparison of metric spectral gaps. Anal. Geome. Metr. Spaces 2(1), 1–52 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Naor, A., Rabani, Y.: On approximate nearest neighbor search in \(\ell _p\), \(p >2\) (2006, manuscript)Google Scholar
  30. 30.
    Neylon, T.: A locality-sensitive hash for real vectors. In: SODA 2010, pp. 1179–1189 (2010)Google Scholar
  31. 31.
    Ostrovsky, R., Rabani, Y.: Polynomial time approximation schemes for geometric k-clustering. In: FOCS 2000, pp. 349–358 (2000)Google Scholar
  32. 32.
    Ostrovsky, R., Rabani, Y.: Polynomial-time approximation schemes for geometric min-sum median clustering. J. ACM 49(2), 139–156 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Yu, D., Yu, X., Wu, A.: Making the nearest neighbor meaningful for time series classification. In: CISP 2011, pp. 2481–2485 (2011)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Hebrew UniversityJerusalemIsrael
  2. 2.Ariel UniversityArielIsrael

Personalised recommendations