Down the rabbit hole: Robust proximity search and density estimation in sublinear space

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

Volumn , Issue , 2012, Pages 430-439

  Har Peled, Sariel ^a   Kumar, Nirman ^a  

^a University of Illinois at Urbana Champaign   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DENSITY ESTIMATION; DISTANCE-BASED; GEOMETRIC DATA; INPUT SIZE; K CLOSEST POINTS; K NEAREST NEIGHBOR QUERIES; LINEAR SPACES; LOGARITHMIC TIME; POINT SET; PROXIMITY QUERIES; QUERY POINTS; SQUARED DISTANCES; SUBLINEAR; SUM OF DISTANCES;

DATA STRUCTURES;

COMPUTER SCIENCE;

EID: 84871940254     PISSN: 02725428     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/FOCS.2012.31     Document Type: Conference Paper

References (45)

1. G. Shakhnarovich, T. Darrell, and P. Indyk, Nearest-Neighbor Methods in Learning and Vision: Theory and Practice (Neural Information Processing). The MIT Press, 2006.
2. B. Chazelle, "Technical perspective: finding a good neighbor, near and fast," Commun. ACM, vol. 51, no. 1, p. 115, 2008.
3. A. Andoni and P. Indyk, "Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions," Commun. ACM, vol. 51, no. 1, pp. 117-122, 2008.
4. K. L. Clarkson, "Nearest-neighbor searching and metric space dimensions," in Nearest-Neighbor Methods for Learning and Vision: Theory and Practice, G. Shakhnarovich, T. Darrell, and P. Indyk, Eds. The MIT Press, 2006, pp. 15-59.
5. E. Fix and J. Hodges, "Discriminatory analysis. nonparametric discrimination: Consistency properties," Technical Report 4, Project Number 21-49-004, USAF School of Aviation Medicine, Randolph Field, TX, 1949.
6. T. Cover and P. Hart, "Nearest neighbor pattern classification, " IEEE Transactions on Information Theory, vol. 13, pp. 21-27, 1967.
7. T. Kohonen, Self-Organizing Maps, ser. Springer Series in Information Sciences. Springer, Berlin, 2001, vol. 30.
8. G. Salton, A. Wong, and C. S. Yang, "A vector space model for automatic indexing," Commun. ACM, vol. 18, pp. 613-620, 1975.
9. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Kluwer Academic Publishers, 1991.
10. L. Devroye and T. Wagner, "Handbook of statistics," in Nearest neighbor methods in discrimination, P. R. Krishnaiah and L. N. Kanal, Eds. North-Holland, 1982, vol. 2.
11. R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd ed. New York: Wiley-Interscience, 2001.
12. K. L. Clarkson, "A randomized algorithm for closest-point queries," SIAM J. Comput., vol. 17, pp. 830-847, 1988.
13. P. K. Agarwal and J. Matoušek, "Ray shooting and parametric search," SIAM J. Comput., vol. 22, pp. 540-570, 1993.
14. S. Meiser, "Point location in arrangements of hyperplanes," Inform. Comput., vol. 106, pp. 286-303, 1993.
15. M. Minsky and S. Papert, Perceptrons. The MIT Press, 1969.
16. S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Y. Wu, "An optimal algorithm for approximate nearest neighbor searching in fixed dimensions," J. Assoc. Comput. Mach., vol. 45, no. 6, pp. 891-923, 1998. [Online]. Available: http://www.cs.umd.edu/ mount/Papers/dist.pdf
17. P. Indyk and R. Motwani, "Approximate nearest neighbors: Towards removing the curse of dimensionality," in Proc. 30th Annu. ACM Sympos. Theory Comput., 1998, pp. 604-613. [Online]. Available: http://theory.lcs.mit. edu/ indyk/nndraft.ps
18. E. Kushilevitz, R. Ostrovsky, and Y. Rabani, "Efficient search for approximate nearest neighbor in high dimensional spaces," SIAM J. Comput., vol. 2, no. 30, pp. 457-474, 2000. [Online]. Available: http://epubs.siam.org/ sambin/dbq/article/34717
19. S. Har-Peled, P. Indyk, and R. Motwani, "Approximate nearest neighbors: Towards removing the curse of dimensionality," Theory Comput., p. to appear, 2012.
20. S. Har-Peled, Geometric Approximation Algorithms. Amer. Math. Soc., 2011.
21. -, "A replacement for Voronoi diagrams of near linear size," in Proc. 42nd Annu. IEEE Sympos. Found. Comput. Sci., 2001, pp. 94-103. [Online]. Available: http://cs.uiuc.edu/ sariel/papers/01/avoronoi/
22. S. Arya and T. Malamatos, "Linear-size approximate Voronoi diagrams," in Proc. 13th ACM-SIAM Sympos. Discrete Algorithms, 2002, pp. 147-155.
23. P. B. Callahan and S. R. Kosaraju, "A decomposition of multidimensional point sets with applications to k-nearestneighbors and n-body potential fields," J. Assoc. Comput. Mach., vol. 42, pp. 67-90, 1995.
24. S. Arya, T. Malamatos, and D. M. Mount, "Space-time tradeoffs for approximate nearest neighbor searching," J. Assoc. Comput. Mach., vol. 57, no. 1, pp. 1-54, 2009.
25. B. Silverman, Density estimation for statistics and data analysis, ser. Monographs on statistics and applied probability. Chapman and Hall, 1986.
26. J. Ruppert, "A Delaunay refinement algorithm for quality 2-dimensional mesh generation," J. Algorithms, vol. 18, no. 3, pp. 548-585, 1995.
27. J. B. Tenenbaum, "Mapping a manifold of perceptual observations," Adv. Neur. Inf. Proc. Sys. 10, pp. 682-688, 1998.
28. M. Bernstein, V. D. Silva, J. C. Langford, and J. B. Tenenbaum, "Graph approximations to geodesics on embedded manifolds," 2000.
29. T. Martinetz and K. Schulten, "Topology representing networks," Neural Netw., vol. 7, no. 3, pp. 507-522, Mar. 1994.
30. K. Q. Weinberger and L. K. Saul, "Unsupervised learning of image manifolds by semidefinite programming," Int. J. Comput. Vision, vol. 70, no. 1, pp. 77-90, Oct. 2006.
31. L. J. Guibas, Q. Mérigot, and D. Morozov, "Witnessed k-distance," in Proc. 27th Annu. ACM Sympos. Comput. Geom., 2011, pp. 57-64.
32. F. Aurenhammer, "Voronoi diagrams: A survey of a fundamental geometric data structure," ACM Comput. Surv., vol. 23, pp. 345-405, 1991.
33. K. L. Clarkson and P. W. Shor, "Applications of random sampling in computational geometry, II," Discrete Comput. Geom., vol. 4, pp. 387-421, 1989. [Online]. Available: http://cm.bell-labs.com/who/clarkson/rs2m.html
34. J. Matoušek, Lectures on Discrete Geometry. Springer, 2002. [Online]. Available: http://kam.mff.cuni.cz/matousek/dg.html
35. -, "Efficient partition trees," Discrete Comput. Geom., vol. 8, pp. 315-334, 1992.
36. T. M. Chan, "Optimal partition trees," in Proc. 26th Annu. ACM Sympos. Comput. Geom., 2010, pp. 1-10.
37. P. K. Agarwal and J. Erickson, "Geometric range searching and its relatives," in Advances in Discrete and Computational Geometry, B. Chazelle, J. E. Goodman, and R. Pollack, Eds. Amer. Math. Soc., 1998.
38. S. Har-Peled, "Constructing approximate shortest path maps in three dimensions," SIAM J. Comput., vol. 28, no. 4, pp. 1182-1197, 1999.
39. P. Carmi, S. Dolev, S. Har-Peled, M. J. Katz, and M. Segal, "Geographic quorum systems approximations," Algorithmica, vol. 41, no. 4, pp. 233-244, 2005.
40. S. Arya, T. Malamatos, and D. M. Mount, "Space-time tradeoffs for approximate spherical range counting," in Proc. 16th ACM-SIAM Sympos. Discrete Algorithms, 2005, pp. 535-544.
41. B. Aronov and S. Har-Peled, "On approximating the depth and related problems," SIAM J. Comput., vol. 38, no. 3, pp. 899-921, 2008.
42. M. de Berg, H. Haverkort, S. Thite, and L. Toma, "Starquadtrees and guard-quadtrees: I/o-efficient indexes for fat triangulations and low-density planar subdivisions," Comput. Geom. Theory Appl., vol. 43, pp. 493-513, jul 2010.
43. S. Har-Peled and N. Kumar, "Down the rabbit hole: Robust proximity search and density estimation in sublinear space," CoRR, vol. abs/1111.2942, 2011.
44. S. Har-Peled, "Coresets for discrete integration and clustering," in Proc. 26th Conf. Found. Soft. Tech. Theoret. Comput. Sci., 2006, pp. 33-44.
45. S. Har-Peled and N. Kumar, "Approximate nearest neighbor search for low dimensional queries," in Proc. 22nd ACMSIAM Sympos. Discrete Algorithms, 2011, pp. 854-867.