Principal curve algorithms for partitioning high-dimensional data spaces

IEEE Transactions on Neural Networks

Volumn 22, Issue 3, 2011, Pages 367-380

  Zhang, Junping ^a   Wang, Xiaodan ^a   Kruger, Uwe ^b   Wang, Fei Yue ^c,d  

^a FUDAN UNIVERSITY   (China)

^b KHALIFA UNIVERSITY   (United Arab Emirates)

^c INSTITUTE OF AUTOMATION   (China)

^d UNIVERSITY OF ARIZONA   (United States)

Author keywords

Manifold learning; principal component analysis; principal curves; space partitioning; tree based algorithms

Indexed keywords

ARC LENGTH; AVERAGE COVERAGE; COMPACT STRUCTURES; COMPLEX DATA; DATA DISTRIBUTION; HIGH DIMENSIONAL DATA; HIGH-DIMENSIONAL DATA SPACE; INTRINSIC DIMENSIONS; INTRINSIC STRUCTURES; LINEAR METHODS; MANIFOLD LEARNING; NEAREST NEIGHBOR SEARCH; NONLINEAR STRUCTURE; PARTITIONING ALGORITHMS; PARTITIONING STRATEGIES; PRINCIPAL CURVE; PRINCIPAL CURVE ALGORITHMS; RESIDUAL PROPERTIES; SPACE PARTITIONING; TREE ALGORITHMS; TREE-BASED ALGORITHMS;

ITERATIVE METHODS; LEARNING ALGORITHMS; NONLINEAR ANALYSIS; PARTITIONS (BUILDING); PRINCIPAL COMPONENT ANALYSIS; STRUCTURE (COMPOSITION); VECTOR QUANTIZATION;

TREES (MATHEMATICS);

ALGORITHM; ARTICLE; ARTIFICIAL INTELLIGENCE; ARTIFICIAL NEURAL NETWORK; AUTOMATED PATTERN RECOGNITION; BIOLOGICAL MODEL; DECISION TREE; METHODOLOGY; NONLINEAR SYSTEM; PRINCIPAL COMPONENT ANALYSIS; STATISTICAL ANALYSIS;

ALGORITHMS; ARTIFICIAL INTELLIGENCE; DATA INTERPRETATION, STATISTICAL; DECISION TREES; MODELS, NEUROLOGICAL; NEURAL NETWORKS (COMPUTER); NONLINEAR DYNAMICS; PATTERN RECOGNITION, AUTOMATED; PRINCIPAL COMPONENT ANALYSIS;

EID: 79952190421     PISSN: 10459227     EISSN: None     Source Type: Journal    
DOI: 10.1109/TNN.2010.2100408     Document Type: Article

References (39)

1. A. W. Moore and T. Hall, "Efficient memory-based learning for robot control," Comput. Lab., Univ. Cambridge, Cambridge, U.K., Tech. Rep. UCAM-CL-TR-209, Nov. 1990.
2. J. McNames, "A fast nearest-neighbor algorithm based on a principal axis search tree," IEEE Trans. Pattern Anal. Mach. Intell., vol. 23, no. 9, pp. 964-976, Sep. 2001. (Pubitemid 32981616)
3. Y. Freund, S. Dasgupta, M. Kabra, and N. Verma, "Learning the structure of manifolds using random projections," in Proc. Adv. Neural Inf. Process. Syst. 20, Vancouver, BC, Canada, 2007, pp. 1-8.
4. S. Dasgupta and Y. Freund, "Random projection trees and low dimensional manifolds," in Proc. 40th Annu. ACM Symp. Theory Comput., Victoria, BC, Canada, May 2008, pp. 1-10.
5. T. Hastie, "Principal curves and surfaces," Dept. Stat., Stanford Univ., Stanford, CA, Tech. Rep. 11, Nov. 1984.
6. T. Hastie and W. Stuetzle, "Principal curves," J. Amer. Stat. Assoc., vol. 84, no. 406, pp. 502-516, Jun. 1989.
7. P. Dollár, V. Rabaud, and S. Belongie, "Non-isometric manifold learning: Analysis and an algorithm," in Proc. 24th Int. Conf. Mach. Learn., Corvallis, OR, 2007, pp. 1-8.
8. R. Urtasun, D. J. Fleet, A. Geiger, J. Popovíc, T. J. Darrell, and N. D. Lawrence, "Topologically-constrained latent variable models," in Proc. 25th Int. Conf. Mach. Learn., Helsinki, Finland, 2008, pp. 1-8.
9. B. Shaw and T. Jebara, "Structure preserving embedding," in Proc. 26th Int. Conf. Mach. Learn., Montreal, QC, Canada, 2009, pp. 1-8.
10. W.-S. Zheng, J. Lai, and P. C. Yuen, "Penalized preimage learning in kernel principal component analysis," IEEE Trans. Neural Netw., vol. 21, no. 4, pp. 551-570, Apr. 2010.
11. D. Zeng, Z. Zhou, and S. Xie, "Coarse-to-fine boundary location with a SOM-like method," IEEE Trans. Neural Netw., vol. 21, no. 3, pp. 481-493, Mar. 2010.
12. K. Tasdemir, "Graph based representations of density distribution and distances for self-organizing maps," IEEE Trans. Neural Netw., vol. 21, no. 3, pp. 520-526, Mar. 2010.
13. R. Tibshirani, "Principal curves revisited," Stat. Comput., vol. 2, no. 4, pp. 183-190, 1992.
14. J. D. Banfield and A. E. Raftery, "Ice floe identification in satellite images using mathematical morphology and clustering about principal curves," J. Amer. Stat. Assoc., vol. 87, no. 417, pp. 7-16, Mar. 1992.
15. T. Duchamp and W. Stuetzle, "Extremal properties of principal curves in the plane," Ann. Stat., vol. 24, no. 4, pp. 1511-1520, 1996.
16. P. Delicado, "Another look at principal curves and surfaces," J. Multivariate Anal., vol. 77, no. 1, pp. 84-116, Apr. 2001. (Pubitemid 33633904)
17. J. Einbeck, G. Tutz, and L. Evers, "Local principal curves," Stat. Comput., vol. 15, no. 4, pp. 301-313, Oct. 2005. (Pubitemid 41442618)
18. B. Kégl, "Principal curves: Learning, design and applications," Ph.D. thesis, Dept. Comput. Sci., Concordia Univ., Montreal, QC, Canada, 2000.
19. B. Kégl, A. Krzyzak, T. Linder, and K. Zeger, "Learning and design of principal curves," IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 3, pp. 281-297, Mar. 2000.
20. S. Sandilya and S. R. Kulkarni, "Principal curves with bounded turn," IEEE Trans. Inf. Theory, vol. 48, no. 10, pp. 2789-2793, Oct. 2002. (Pubitemid 35170338)
21. A. J. Smola, R. C. Williamson, S. Mika, and B. Schölkopf, "Regularized principal manifolds," J. Mach. Learn. Res., vol. 1, pp. 179-209, Jun. 2001. (Pubitemid 33687202)
22. K.-Y. Chang and J. Ghosh, "A unified model for probabilistic principal surfaces," IEEE Trans. Pattern Anal. Mach. Intell., vol. 23, no. 1, pp. 22-41, Jan. 2001.
23. A. N. Gorban, N. R. Sumner, and A. Y. Zinovyev, "Beyond the concept of manifolds: Principal trees, metro maps, and elastic cubic complexes," in Principal Manifolds for Data Visualization and Dimension Reduction (Lecture Notes in Computational Science and Engineering), vol. 58. Berlin, Germany: Springer-Verlag, 2007, pp. 223-241.
24. D. J. H. Wilson, G. W. Irwin, and G. Lightbody, "RBF principal manifolds for process monitoring," IEEE Trans. Neural Netw., vol. 10, no. 6, pp. 1424-1434, Nov. 1999. (Pubitemid 30536584)
25. S. Gerber, T. Tasdizen, and R. Whitaker, "Dimensionality reduction and principal surfaces via kernel map manifolds," in Proc. 12th Int. Conf. Comput. Vis., Kyoto, Japan, Sep.-Oct. 2009, pp. 529-536.
26. N. Verma, S. Kpotufe, and S. Dasgupta, "Which spatial partition trees are adaptive to intrinsic dimension?" in Proc. 25th Conf. Uncertainty Artif. Intell., Corvallis, OR, 2009, pp. 565-574.
27. T. Liu, A. Moore, W. Gray, and K. Yang, "An investigation of practical approximate nearest neighbor algorithms," in Advances in Neural Information Processing Systems. Cambridge, MA: MIT Press, 2005, pp. 825-832.
28. K. Q. Weinberger and L. K. Saul, "Fast solvers and efficient implementations for distance metric learning," in Proc. 25th Int. Conf. Mach. Learn., 2008, pp. 1-8.
29. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, 2nd ed. New York: Springer-Verlag, 2008.
30. S. Gerber, T. Tasdizen, and R. Whitaker, "Robust non-linear dimensionality reduction using successive 1-D Laplacian eigenmaps," in Proc. 24th Int. Conf. Mach. Learn., Corvallis, OR, 2007, pp. 1-8.
31. A. Asuncion and D. J. Newman. (2007). UCI Machine Learning Repository. School Inf. Comput. Sci., Univ. California, Los Angles [Online]. Available: http://www.ics.uci.edu/∼mlearn/MLRepository.html
32. J. B. Tenenbaum, V. de Silva, and J. C. Langford, "A global geometric framework for nonlinear dimensionality reduction," Science, vol. 290, no. 5500, pp. 2319-2323, Dec. 2000. (Pubitemid 32041577)
33. S. T. Roweis and L. K. Saul, "Nonlinear dimensionality reduction by locally linear embedding, Science, vol. 290, no. 5500, pp. 2323-2326, Dec. 2000. (Pubitemid 32041578)
34. A. Ittner and M. Schlosser, "Non-linear decision trees-NDT," in Proc. Int. Conf. Mach. Learn., 1996, pp. 252-257.
35. J. Liu and J. Ye, "Efficient Euclidean projections in linear time," in Proc. 26th Annu. Int. Conf. Mach. Learn., Montreal, QC, Canada, 2009, pp. 1-8.
36. Y. Aksu, D. J. Miller, G. Kesidis, and Q. X. Yang, "Margin- maximizing feature elimination methods for linear and nonlinear kernel-based discriminant functions," IEEE Trans. Neural Netw., vol. 21, no. 5, pp. 701-717, May 2010.
37. J. Liu, S. Chen, Z.-H. Zhou, and X. Tan, "Generalized low-rank approximations of matrices revisited," IEEE Trans. Neural Netw., vol. 21, no. 4, pp. 621-632, Apr. 2010.
38. C. M. Bishop and M. E. Tipping, "A hierarchical latent variable model for data visualization," IEEE Trans. Pattern Anal. Mach. Intell., vol. 20, no. 3, pp. 281-293, Mar. 1998. (Pubitemid 128741316)
39. B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. Cambridge, U.K.: Cambridge Univ. Press, 1992.