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Latin American Symposium on Theoretical Informatics

LATIN 2018: LATIN 2018: Theoretical Informatics pp 120–133Cite as

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

Part of the Lecture Notes in Computer Science book series (LNTCS,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.

Y. Bartal is supported in part by an Israel Science Foundation grant #1817/17.

L.-A. Gottlieb is supported in part by an Israel Science Foundation grant #755/15.

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Notes

  1. 1.

    This is a space equipped with a Minkowski norm, which defines the distance between two d-dimensional vectors x, y as \(\Vert x-y\Vert _p = (\sum _{i=1}^d |x_i-y_i|^p)^{1/p}\).

  2. 2.

    If \(d = O(\log n)\) then AVDs may be used, and if \(d = n^{\varOmega (1)}\) then comparing the query point q to each point in V in a brute-force manner can be done in \(O(dn) = d^{O(1)}\) time. (We recall also that there exists an oblivious mapping for all \(\ell _p\) that embeds \(\ell _p^m\) into \(\ell _p^d\) for \(d = {n \atopwithdelims ()2}\) dimensions [5, 15].) We also assume that \(d=2^{o({{\mathrm{ddim}}})}\), as otherwise a constant-factor approximation can be computed in polynomial time [13, 19].

References

  1. Ailon, N., Chazelle, B.: Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In: STOC 2006, pp. 557–563 (2006)

    Google Scholar 

  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. Andoni, A.: Nearest neighbor search: the old, the new, and the impossible. Ph.D. thesis, MIT (2009)

    Google Scholar 

  4. Arya, S., Malamatos, T.: Linear-size approximate Voronoi diagrams. In: SODA 2002, pp. 147–155 (2002)

    Google Scholar 

  5. Ball, K.: Isometric embedding in \(l_p\)-spaces. Eur. J. Comb. 11(4), 305–311 (1990)

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Batu, T., Ergun, F., Sahinalp, C.: Oblivious string embeddings and edit distance approximations. In: SODA 2006, pp. 792–801 (2006)

    Google Scholar 

  7. Bellman, R.E.: Adaptive Control Processes: A Guided Tour. Princeton University Press, Princeton (1961)

    CrossRef  MATH  Google Scholar 

  8. Bern, M.: Approximate closest-point queries in high dimensions. Inf. Process. Lett. 45(2), 95–99 (1993)

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. Beygelzimer, A., Kakade, S., Langford, J.: Cover trees for nearest neighbor. In: ICML 2006, pp. 97–104 (2006)

    Google Scholar 

  10. Binyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Colloquium Publications (American Mathematical Society), Providence (2000)

    Google Scholar 

  11. Chan, T.M.: Approximate nearest neighbor queries revisited. Discret. Comput. Geom. 20(3), 359–373 (1998)

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Clarkson, K.L.: A randomized algorithm for closest-point queries. SIAM J. Comput. 17(4), 830–847 (1988)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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. 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)

    CrossRef  Google Scholar 

  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. 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. Finlayson, G.D., Trezzi, E.: Shades of gray and colour constancy. In: CIC 2004, pp. 37–41 (2004)

    Google Scholar 

  18. Har-Peled, S., Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. Theory Comput. 8(1), 321–350 (2012)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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)

    MathSciNet  CrossRef  MATH  Google Scholar 

  20. Har-Peled, S., Kumar, N.: Approximating minimization diagrams and generalized proximity search. SIAM J. Comput. 44(4), 944–974 (2015)

    MathSciNet  CrossRef  MATH  Google Scholar 

  21. Indyk, P.: On approximate nearest neighbors in non-Euclidean spaces. In: FOCS, pp. 148–155 (1998)

    Google Scholar 

  22. Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: STOC 1998, pp. 604–613 (1998)

    Google Scholar 

  23. Indyk, P., Naor, A.: Nearest-neighbor-preserving embeddings. ACM Trans. Algorithms 3(3), 31 (2007)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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. Johnson, W.B., Schechtman, G.: Embedding \(l_p^m\) into \(l_1^n\). Acta Math. 149(1–2), 71–85 (1982)

    MathSciNet  CrossRef  MATH  Google Scholar 

  26. Krauthgamer, R., Lee, J.R.: Navigating nets: simple algorithms for proximity search. In: SODA 2004, pp. 791–801 (2004)

    Google Scholar 

  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. Naor, A.: Comparison of metric spectral gaps. Anal. Geome. Metr. Spaces 2(1), 1–52 (2014)

    MathSciNet  MATH  Google Scholar 

  29. Naor, A., Rabani, Y.: On approximate nearest neighbor search in \(\ell _p\), \(p >2\) (2006, manuscript)

    Google Scholar 

  30. Neylon, T.: A locality-sensitive hash for real vectors. In: SODA 2010, pp. 1179–1189 (2010)

    Google Scholar 

  31. Ostrovsky, R., Rabani, Y.: Polynomial time approximation schemes for geometric k-clustering. In: FOCS 2000, pp. 349–358 (2000)

    Google Scholar 

  32. Ostrovsky, R., Rabani, Y.: Polynomial-time approximation schemes for geometric min-sum median clustering. J. ACM 49(2), 139–156 (2002)

    MathSciNet  CrossRef  MATH  Google Scholar 

  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 

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Acknowledgements

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

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Authors and Affiliations

  1. Hebrew University, Jerusalem, Israel

    Yair Bartal

  2. Ariel University, Ariel, Israel

    Lee-Ad Gottlieb

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  1. Yair Bartal
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  2. Lee-Ad Gottlieb
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Correspondence to Lee-Ad Gottlieb .

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Editors and Affiliations

  1. Stony Brook University, Stony Brook, New York, USA

    Prof. Michael A. Bender

  2. Rutgers University, New Brunswick, New Jersey, USA

    Martín Farach-Colton

  3. Pace University, New York, New York, USA

    Miguel A. Mosteiro

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Bartal, Y., Gottlieb, LA. (2018). Approximate Nearest Neighbor Search for \(\ell _p\)-Spaces \((2<p<\infty )\) via Embeddings. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_10

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  • DOI: https://doi.org/10.1007/978-3-319-77404-6_10

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