Abstract
In nonlinear dimensionality reduction, the kernel dimension is the square of the vector number in the data set. In many applications, the number of data vectors is very large. The spectral decomposition of a large dimensioanl kernel encounters difficulties in at least three aspects: large memory usage, high computational complexity, and computational instability. Although the kernels in some nonlinear DR methods are sparse matrices, which enable us to overcome the difficulties in memory usage and computational complexity partially, yet it is not clear if the instability issue can be settled. In this chapter, we study some fast algorithms that avoid the spectral decomposition of large dimensional kernels in DR processing, dramatically reducing memory usage and computational complexity, as well as increasing numerical stability. In Section 15.1, we introduce the concepts of rank revealings. In Section 15.2, we present the randomized low rank approximation algorithms. In Section 15.3, greedy lank-revealing algorithms (GAT) and randomized anisotropic transformation algorithms (RAT), which approximate leading eigenvalues and eigenvectors of DR kernels, are introduced. Numerical experiments are shown in Section 15.4 to illustrate the validity of these algorithms. The justification of RAT algorithms is included in Section 15.5.
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References
Cheng, H., Gimbutas, Z., Martinsson, P.G., Rokhlin, V.: On the compression of low rank matrices. SIAM Journal on Scientific Computing 26(4), 1389–1404 (2005).
Woolfe, F., Liberty, E., Rokhlin, V., Tygert, M.: A randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25(3), 335–366 (2008).
Hansen, P.C.: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia (1998).
Stewart, G.W.: Matrix Algorithms Volume I: Basic Decompositions. SIAM, Philadelphia (1998).
Chan, T.F., Hansen, P.C.: Some applications of the rank revealing QR factorization. SIAM J. Sci. Statist. Comput. 13, 727–741 (1992).
Gu, M., Eisenstat, S.C.: Efficient algorithms for computing a strong rankrevealing QR factorization. SIAM J. Sci. Comput. 17, 848–869 (1996).
Hong, Y.P., Pan, C.T.: Rank-revealing QR factorizations and the singular value decomposition. Mathematics of Computation 58(197), 213–232 (1992).
Berry, M., Pulatova, S., Stewart, G.: Algorithm 844: computing sparse reducedrank approximations to sparse matrices. ACM Trans Math Softw 31(2), 252–269 (2005).
Goreinov, S.A., Tyrtyshnikov, E.E., Zamarashkin, N.L.: A theory of pseudoskeleton approximations. Linear Algebra and Its Applications 261, 1–21 (1997).
Tyrtyshnikov, E.: Matrix bruhat decompositions with a remark on the QR (GR) algorithm. Linear Algebra Appl. 250, 61–68 (1997).
Tyrtyshnikov, E., Zamarashkin, N.: Thin structure of eigenvalue clusters for non-hermitian Toeplitz matrices. Linear Algebra Appl. 292, 297–310 (1999).
Zamarashkin, N., Tyrtyshnikov, E.: Eigenvalue estimates for Hankel matrices. Sbornik: Mathematics 192, 59–72 (2001).
Fierro, R., Bunch, J.: Bounding the subspaces from rank revealing two-sided orthogonal decompositions. SIAM Matrix Anal. Appl. 16, 743–759 (1995).
Fierro, R., Hansen, P.: Low-rank revealing UTV decompositions. Numerical Algorithms 15, 37–55 (1997).
Fierro, R., Hansen, P.C., Hansen, P.S.K.: UTV Tools: Matlab templates for rank-revealing UTV decompositions. Numerical Algorithms 20, 165–194 (1999).
Golub, G.H., van Loan, C.F.: Matrix Computations, third edn. Johns Hopkins Press, Baltimore (1996).
Fierro, R., Hansen, P.: UTV Expansion Pack: Special-purpose rank revaling algorithms. Numerical Algorithms 40, 47–66 (2005).
Hansen, P.C., Yalamov, P.Y.: Computing symmetric rank-revealing decompositions via triangular factorization. SIAM J. Matrix Anal. Appl. 28, 443–458 (2001).
Luk, F.T., S, Q.: A symmetric rank-revealing Toeplitz matrix decomposition. J. VLSI Signal Proc. 14, 19–28 (1996).
Belabbas, M.A., Wolfe, P.J.: Fast low-rank approximation for covariance matrices. In: Proceedings of the 2nd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (2007).
Belabbas, M.A., Wolfe, P.J.: On sparse representations of linear operators and the approximation of matrix products. In: Proceedings of the 42nd Annual Conference on Information Sciences and Systems, pp. 258–263 (2008).
Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral grouping using the Nyström method. IEEE Trans. Patt. Anal. Mach. Intell. pp. 214–225 (2004).
Parker, P., Wolfe, P.J., Tarokh, V.: A signal processing application of randomized low-rank approximations. In: IEEE Worksh. Statist. Signal Process., pp. 345–350 (2005).
Williams, C.K.I., Seeger, M.: Using the Nyström method to speed up kernel machines. In: Neural Information Processing Systems, pp. 682–688 (2000).
Martinsson, P.G., Rokhlin, V., Tygert, M.: A randomized algorithm for the approximation of matrices. Tech. Rep. 1361, Dept. of Computer Science, Yale University (2006).
Martinsson, P.G., Rokhlin, V., Tygert, M.: On interpolation and integration in finite-dimensional spaces of bounded functions. Comm. Appl. Math. Comput. Sci. pp. 133–142 (2006).
Belabbas, M.A., Wolfe, P.J.: Spectral methods in machine learning: New strategies for very large data sets. PANS 106(2), 369–374 (2009).
Chui, C., Wang, J.: Dimensionality reduction of hyper-spectral imagery data for feature classification. In: W. Freeden, Z. Nashed, T. Sonar (eds.) Handbook of Geomathematics. Springer, Berlin (2010).
Chui, C., Wang, J.: Randomized anisotropic transform for nonlinear dimensionality reduction. International Journal on Geomathematics 1(1), 23–50 (2010).
Xiao, L., Sun, J., Boyd, S.P.: A duality view of spectral methods for dimensionality reduction. In: W.W. Cohen, A. Moore (eds.) Machine Learning: Proceedings of the Twenty-Third International Conference, ACM International Conference Proceeding Series, vol. 148, pp. 1041–1048. ICML, Pittsburgh, Pennsylvania, USA (2006).
Goldstine, H.H., von Neumann, J.: Numerical inverting of matrices of high order II. Amer. Math. Soc. Proc. 2, 188–202 (1951).
Chen, Z., Dongarra, J.J.: Condition numbers of Gaussian random matrices. SIAM J. on Matrix Anal. Appl. 27, 603–620 (2005).
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© 2012 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg
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Wang, J. (2012). Fast Algorithms for DR Approximation. In: Geometric Structure of High-Dimensional Data and Dimensionality Reduction. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27497-8_15
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DOI: https://doi.org/10.1007/978-3-642-27497-8_15
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