Document Open Access Logo

Dimension Reduction Techniques for l_p (1<p<2), with Applications

Authors Yair Bartal, Lee-Ad Gottlieb

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2016.16.pdf
  • Filesize: 0.55 MB
  • 15 pages

Document Identifiers

Subject Classification

Keywords
  • Dimension reduction
  • embeddings

Metrics

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

Abstract

For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms. 

In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.

Cite As Get BibTex

Yair Bartal and Lee-Ad Gottlieb. Dimension Reduction Techniques for l_p (1<p<2), with Applications. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.16

Author Details

Yair Bartal
Lee-Ad Gottlieb

References

  1. I. Abraham, Y. Bartal, and O. Neiman. Embedding metric spaces in their intrinsic dimension. In SODA'08, pages 363-372. SIAM, 2008. Google Scholar
  2. D. Achlioptas. Database-friendly random projections. In PODS'01, pages 274-281, New York, NY, USA, 2001. ACM. Google Scholar
  3. P. K. Agarwal and C. M. Procopiuc. Exact and approximation algorithms for clustering. Algorithmica, 33(2):201-226, 2002. Google Scholar
  4. N. Ailon and B. Chazelle. Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. In STOC'06, pages 557-563. ACM, 2006. Google Scholar
  5. N. Alon. Problems and results in extremal combinatorics, part I. Disc. Math, 273, 2003. Google Scholar
  6. N. Alon, M. Bădoiu, E.D. Demaine, M. Farach-Colton, M. Hajiaghayi, and A. Sidiropoulos. Ordinal embeddings of minimum relaxation: General properties, trees, and ultrametrics. ACM Transactions on Algorithms, 4(4), 2008. Google Scholar
  7. A. Andoni, M. Charikar, O. Neiman, and H.L. Nguyen. Near linear lower bounds for dimension reduction in 𝓁₁. In FOCS'11. IEEE Computer Society, 2011. Google Scholar
  8. Alexandr Andoni. Nearest Neighbor Search: the Old, the New, and the Impossible. PhD thesis, MIT, 2009. Google Scholar
  9. M. Bādoiu, S. Har-Peled, and P. Indyk. Approximate clustering via core-sets. In STOC'02, pages 250-257, New York, NY, USA, 2002. ACM. Google Scholar
  10. Y. Bartal, L. Gottlieb, T. Kopelowitz, M. Lewenstein, and L. Roditty. Fast, precise and dynamic distance queries. In SODA'11, pages 840-853. SIAM, 2011. Google Scholar
  11. Y. Bartal, B. Recht, and L. Schulman. Dimensionality reduction: beyond the Johnson-Lindenstrauss bound. In SODA'11, pages 868-887. SIAM, 2011. Google Scholar
  12. Yair Bartal and Lee-Ad Gottlieb. A linear time approximation scheme for euclidean tsp. In FOCS'13, pages 698-706, Washington, DC, USA, 2013. IEEE Computer Society. Google Scholar
  13. T. Batu, F. Ergun, and C. Sahinalp. Oblivious string embeddings and edit distance approximations. In SODA'06, pages 792-801, 2006. Google Scholar
  14. M. Bădoiu, E. Demaine, M. Hajiaghayi, A. Sidiropoulos, and M. Zadimoghaddam. Ordinal embedding: Approximation algorithms and dimensionality reduction. In APPROX'08, pages 21-34. 2008. Google Scholar
  15. H. Chan, A. Gupta, and K. Talwar. Ultra-low-dimensional embeddings for doubling metrics. J. ACM, 57(4):1-26, 2010. URL: http://dx.doi.org/10.1145/1734213.1734215.
  16. R. Cole and L. Gottlieb. Searching dynamic point sets in spaces with bounded doubling dimension. In 38th annual ACM symposium on Theory of computing, pages 574-583, 2006. Google Scholar
  17. M. M. Deza and M. Laurent. Geometry of Cuts and Metrics. Springer-Verlag, Berlin, 1997. Google Scholar
  18. Tomás Feder and Daniel Greene. Optimal algorithms for approximate clustering. In STOC'88, pages 434-444, New York, NY, USA, 1988. ACM. Google Scholar
  19. L. Gottlieb and R. Krauthgamer. A nonlinear approach to dimension reduction. In SODA'11, pages 888-899. SIAM, 2011. Google Scholar
  20. A. Gupta, R. Krauthgamer, and J. R. Lee. Bounded geometries, fractals, and low-distortion embeddings. In FOCS'03, pages 534-543, 2003. Google Scholar
  21. S. Har-Peled, P. Indyk, and R. Motwani. Approximate nearest neighbor: Towards removing the curse of dimensionality. Theory of Computing, 8(14):321-350, 2012. Google Scholar
  22. S. Har-Peled and M. Mendel. Fast construction of nets in low-dimensional metrics and their applications. SIAM Journal on Computing, 35(5):1148-1184, 2006. Google Scholar
  23. P. Indyk. Algorithmic applications of low-distortion geometric embeddings. In FOCS'01, pages 10-33, 2001. Google Scholar
  24. P. Indyk. Stable distributions, pseudorandom generators, embeddings, and data stream computation. J. ACM, 53(3):307-323, 2006. Google Scholar
  25. P. Indyk and A. Naor. Nearest-neighbor-preserving embeddings. ACM Trans. Algorithms, 3(3):31, 2007. Google Scholar
  26. W.B. Johnson and J. Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conf. in modern analysis and probability, pages 189-206. AMS, 1984. Google Scholar
  27. W.B. Johnson and A. Naor. The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. In SODA'09, pages 885-891. SIAM, 2009. Google Scholar
  28. W.B. Johnson and G. Schechtman. Embedding l_p^m into l₁ⁿ. Acta Math., 149(1-2):71-85, 1982. Google Scholar
  29. R. Krauthgamer and J.R. Lee. Navigating nets: Simple algorithms for proximity search. In SODA'04, pages 791-801, January 2004. Google Scholar
  30. E. Kushilevitz, R. Ostrovsky, and Y. Rabani. Efficient search for approximate nearest neighbor in high dimensional spaces. STOC'98, pages 614-623. ACM, 1998. Google Scholar
  31. J.R. Lee and A. Naor. Extending Lipschitz functions via random metric partitions. Inventiones Mathematicae, 160:59-95, 2005. Google Scholar
  32. Ping Li. Estimators and tail bounds for dimension reduction in 𝓁_α (0 < α ≤ 2) using stable random projections. In SODA, pages 10-19, 2008. Google Scholar
  33. G. Lugosi. Concentration-of-measure inequalities. Manuscript, available at http://www.econ.upf.edu/~lugosi/anu.ps, 2004.
  34. J. Matoušek. On the distortion required for embedding finite metric spaces into normed spaces. Israel J. Math., 93:333-344, 1996. Google Scholar
  35. M. Mendel and A. Naor. Euclidean quotients of finite metric spaces. Advances in Mathematics, 189(2):451-494, 2004. Google Scholar
  36. V. D. Milman and G. Schechtman. Asymptotic Theory of Finite-dimensional Normed Spaces. Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov. Google Scholar
  37. H. Nguyen. Approximate nearest neighbor search in 𝓁_p. Manuscript, available at http://arxiv.org/pdf/1306.3601v1, 2013.
  38. J.P. Nolan. Stable Distributions - Models for Heavy Tailed Data. Birkhauser, Boston, 2012. In progress, Chapter 1 online at academic2.american.edu/∼jpnolan. Google Scholar
  39. R. Ostrovsky and Y. Rabani. Polynomial time approximation schemes for geometric k-clustering. In FOCS'00, pages 349-358. IEEE Computer Society, 2000. Google Scholar
  40. R. Ostrovsky and Y. Rabani. Polynomial-time approximation schemes for geometric min-sum median clustering. J. ACM, 49(2):139-156, 2002. Google Scholar
  41. R. Panigrahy. Minimum enclosing polytope in high dimensions. CoRR, 2004. Google Scholar
  42. A. Rahimi and B. Recht. Random features for large-scale kernel machines. In NIPS, 2007. Google Scholar
  43. G. Schechtman. Dimension reduction in l_p, 0 < p < 2. Manuscript, available at http://arxiv.org/pdf/1110.2148v1.pdf, 2011.
  44. L. J. Schulman. Clustering for edge-cost minimization (extended abstract). In STOC, pages 547-555. ACM, 2000. Full version available as ECCC report TR99-035. URL: http://dx.doi.org/10.1145/335305.335373.
  45. M. Talagrand. Embedding subspaces of L_1 into l^N_1. Proc. Amer. Math. Soc., 108(2):363-369, 1990. Google Scholar
  46. M. Talagrand. Embedding subspaces of L_p in l^N_p. In Geometric aspects of functional analysis, volume 77 of Oper. Theory Adv. Appl., pages 311-325. Birkhäuser, Basel, 1995. Google Scholar
  47. V.M. Zolotarev. One-dimensional Stable Distributions. AMS, 1986. Google Scholar
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

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