Local procrustes for manifold embedding: A measure of embedding quality and embedding algorithms

Machine Learning

Volumn 77, Issue 1, 2009, Pages 1-25

  Goldberg, Yair ^a   Ritov, Ya'Acov ^a  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

Author keywords

Dimension reducing; Local PCA; Manifold learning; Procrustes analysis; Simulated annealing

Indexed keywords

DIMENSION REDUCING; DIMENSION-REDUCTION; EMBEDDING ALGORITHMS; LOCAL PCA; MANIFOLD LEARNING; NATURAL TOOLS; PROCRUSTES; PROCRUSTES ANALYSIS; QUANTITATIVE COMPARISON; TENENBAUM;

ALGORITHMS;

SIMULATED ANNEALING;

EID: 63649089799     PISSN: 08856125     EISSN: 15730565     Source Type: Journal    
DOI: 10.1007/s10994-009-5107-9     Document Type: Article

References (30)

1. M. Belkin P. Niyogi 2003 Laplacian eigenmaps for dimensionality reduction and data representation Neural Computation 15 6 1373 1396
2. Bernstein, M., de Silva, V., Langford, J. C., & Tenenbaum, J. B. (2000). Graph approximations to geodesics on embedded manifolds (Technical Report). Stanford University, Stanford. Available at http://isomap.stanford.edu.
3. Chen, L. (2006) Local multidimensional scaling for nonlinear dimension reduction, graph layout and proximity analysis. PhD thesis, University of Pennsylvania.
4. B. Cipra 1987 An introduction to the Ising model American Mathematical Monthly 94 10 937 959
5. Corman, T., Leiserson, C., & Rivest, R. (1990). Introduction to algorithms. Cambridge: MIT Press.
6. Dollar, P., Rabaud, V., & Belongie, S. J. (2007). Non-isometric manifold learning: analysis and an algorithm. In Z. Ghahramani (Ed.), Proceedings of the 24th annual international conference on machine learning (ICML) (pp. 241-248). Omnipress.
7. D. Donoho C. Grimes 2004 Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data Proceedings of the National Academy of Sciences of the United States of America 100 10 5591 5596
8. Y. Goldberg A. Zakai D. Kushnir Y. Ritov 2008 Manifold learning: The price of normalization The Journal of Machine Learning Research 9 1909 1939
9. Golub, G. H., & Loan, C. F. V. (1983). Matrix computations. Baltimore: Johns Hopkins University Press.
10. Hamm, J., Lee, D., & Saul, L. K. (2005). Semisupervised alignment of manifolds. In: R. G. Cowell & Z. Ghahramani (Eds.), Proceedings of the tenth international workshop on artificial intelligence and statistics (pp. 120-127).
11. J. J. Hull 1994 A database for handwritten text recognition research IEEE Transactions on Pattern Analysis and Machine Intelligence 16 5 550 554
12. S. Kirkpatrick C. D. Gelatt M. P. Vecchi 1983 Optimization by simulated annealing Science 220 4598 671 680
13. Mardia, K., Kent, J., & Bibby, J. (1979). Multivariate analysis. New York: Academic.
14. N. Munksgaard 1980 Solving sparse symmetric sets of linear equations by preconditioned conjugate gradients ACM Transactions on Mathematical Software 6 2 206 219
15. Roweis, S. (retrieved Nov. 2006). Frey face on Sam Roweis' page. http://www.cs.toronto.edu/roweis/data.html.
16. S. T. Roweis L. K. Saul 2000 Nonlinear dimensionality reduction by locally linear embedding Science 290 5500 2323 2326
17. Roweis, S. T., Saul, L. K., & Hinton, G. E. (2001). Global coordination of local linear models. In Advances in neural information processing systems, 14 (pp. 889-896). Cambridge: MIT Press.
18. Sha, F., & Saul, L. K. (2005). Analysis and extension of spectral methods for nonlinear dimensionality reduction. In Machine learning, proceedings of the twenty-second international conference (ICML) (pp. 784-791).
19. Shaw, B., & Jebara, T. (2007). Minimum volume embedding. In Proceedings of the international conference on artificial intelligence and statistics (Vol. 11).
20. P. Siarry G. Berthiau F. Durdin J. Haussy 1997 Enhanced simulated annealing for globally minimizing functions of many-continuous variables ACM Transactions on Mathematical Software 23 2 209 228
21. R. Sibson 1978 Studies in robustness of multidimensional-scaling: Procrustes statistics Journal of the Royal Statistical Society 40 2 234 238
22. R. Sibson 1979 Studies in the robustness of multidimensional-scaling: Perturbational analysis of classical scaling Journal of the Royal Statistical Society 41 2 217 229
23. de Silva, V., & Tenenbaum, J. B. (2003). Global versus local methods in nonlinear dimensionality reduction. In Advances in neural information processing systems (p. 15). Cambridge: MIT Press.
24. G. W. Stewart 1980 The efficient generation of random orthogonal matrices with an application to condition estimators SIAM Journal on Numerical Analysis 17 3 403 409
25. Teh, Y. W., & Roweis, S. (2003). Automatic alignment of local representations. In S. Becker, S. Thrun, & K. Obermayer (Eds.), Advances in neural information processing systems (p. 15). Cambridge: MIT Press.
26. J. B. Tenenbaum V. de Silva J. C. Langford 2000 A global geometric framework for nonlinear dimensionality reduction Science 290 5500 2319 2323
27. J. Venna S. Kaski 2006 Local multidimensional scaling Neural Networks 19 6 889 899
28. Verbeek, J., Vlassis, N., & Kröse, B. (2002) Coordinating principal component analyzers. In Proceedings of international conference on artificial neural networks.
29. K. Weinberger L. Saul 2006 Unsupervised learning of image manifolds by semidefinite programming International Journal of Computer Vision 70 1 77 90
30. Z. Zhang H. Zha 2004 Principal manifolds and nonlinear dimensionality reduction via tangent space alignment SIAM Journal of Scientific Computing 26 1 313 338