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Mathematical Problems in Engineering
Volume 2016, Article ID 5815429, 9 pages
http://dx.doi.org/10.1155/2016/5815429
Research Article

Parallelization of Eigenvalue-Based Dimensional Reductions via Homotopy Continuation

Institute of Electronics, Chinese Academy of Sciences, North Fourth Ring Road West 98, Beijing 100190, China

Received 12 December 2015; Accepted 22 February 2016

Academic Editor: Jinyun Yuan

Copyright © 2016 Size Bi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. De Silva and J. B. Tenenbaum, Global Versus Local Methods in Nonlinear Dimensionality Reduction, vol. 15 of Advances in Neural Information Processing Systems, 2002.
  2. J. C. Platt, “Fastmap, metricmap, and landmark mds are all nystrom algorithms,” in Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics, 2005.
  3. C. Orsenigo, “An improved set covering problem for isomap supervised landmark selection,” Pattern Recognition Letters, vol. 49, pp. 131–137, 2014. View at Publisher · View at Google Scholar · View at Scopus
  4. I. Borg and P. J. Groenen, Modern Multidimensional Scaling, Theory and Applications, Springer, Mannheim, Germany, 2nd edition, 2005.
  5. U. Brandes and C. Pich, “Eigensolver methods for progressive multidimensional scaling of large data,” in Graph Drawing: 14th International Symposium, GD 2006, Karlsruhe, Germany, September 18–20, 2006. Revised Papers, vol. 4372 of Lecture Notes in Computer Science, pp. 42–53, Springer, Berlin, Germany, 2007. View at Publisher · View at Google Scholar
  6. M. Williams, “Steerable, progressive multidimensional scaling,” in Proceedings of the IEEE Symposium on Information Visualization (INFOVIS '04), pp. 57–64, Austin, Tex, USA, 2004. View at Publisher · View at Google Scholar
  7. L. van der Maaten, “Accelerating t-SNE using tree-based algorithms,” Journal of Machine Learning Research, vol. 15, pp. 3221–3245, 2014. View at Google Scholar · View at MathSciNet
  8. A. Civril, M. Magdon-Ismail, and E. Bocek-Rivele, “SDE: graph drawing using spectral distance embedding,” in Graph Drawing: 13th International Symposium, GD 2005, Limerick, Ireland, September 12–14, 2005. Revised Papers, vol. 3843 of Lecture Notes in Computer Science, pp. 512–513, Springer, Berlin, Germany, 2006. View at Publisher · View at Google Scholar
  9. K.-I. Lin and C. Faloutsos, “FastMap—a fast algorithm for indexing, data-mining and visualization of traditional and multimedia datasets,” in Proceedings of the ACM SIGMOD International Conference on Management of Data, pp. 163–174, San Jose, Calif, USA, May 1995.
  10. D. Harel and Y. Koren, “Graph drawing by high-dimensional embedding,” in Proceedings of the Symposium on Graph Drawing, pp. 207–219, Irvine, Calif, USA, August 2002.
  11. A. Morrison and M. Chalmers, “Improving hybrid MDS with pivot-based searching,” in Proceedings of the Ninth Annual IEEE Conference on Information Visualization (INFOVIS '03), pp. 85–90, 2003.
  12. J. G. Silva, J. S. Marques, and J. M. Lemos, “Selecting landmark points for sparse manifold learning,” in Neural Information Processing Systems, 2006. View at Google Scholar
  13. C. Campos and J. E. Roman, “Strategies for spectrum slicing based on restarted Lanczos methods,” Numerical Algorithms, vol. 60, no. 2, pp. 279–295, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. J. Beisheim, “Introducing the PCG lanczos eigensolver,” ANSYS Advantage, vol. 1, no. 1, pp. 42–43, 2007. View at Google Scholar
  15. S. H. Lui, H. B. Keller, and T. W. C. Kwok, “Homotopy method for the large, sparse, real nonsymmetric eigenvalue problem,” SIAM Journal on Matrix Analysis and Applications, vol. 18, no. 2, pp. 312–333, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. A. Chesnokov, M. V. Barel, N. Mastronardi, and R. Vandebril, “Homotopy algorithm for the symmetric diagonal-plus-semiseparable eigenvalue problem,” in Proceedings of the 16th International Conference on Mathematical Modelling and Analysis (MMMA '11), pp. 1–28, Sigulda, Latvia, May 2011.
  17. M. T. Chu, “A simple application of the homotopy method to symmetric eigenvalue problems,” Linear Algebra and Its Applications, vol. 59, pp. 85–90, 1984. View at Google Scholar
  18. M. T. Chu, “A note on the homotopy method for linear algebraic eigenvalue problems,” Linear Algebra and its Applications, vol. 105, pp. 225–236, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. T. Y. Li and N. H. Rhee, “Homotopy algorithm for symmetric eigenvalue problems,” Numerische Mathematik, vol. 55, no. 3, pp. 265–280, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. P. Brockman, T. Carson, Y. Cheng et al., “Homotopy method for the eigenvalues of symmetric tridiagonal matrices,” Journal of Computational and Applied Mathematics, vol. 237, no. 1, pp. 644–653, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. S. T. Roweis and L. K. Saul, “Nonlinear dimensionality reduction by locally linear embedding,” Science, vol. 290, no. 5500, pp. 2323–2326, 2000. View at Publisher · View at Google Scholar · View at Scopus
  22. M. Belkin and P. Niyogi, “Laplacian eigenmaps and spectral techniques for embedding and clustering,” in Proceedings of the Advances in Neural Information Processing Systems 14 (NIPS '02), MIT Press, December 2002.
  23. 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, 2000. View at Publisher · View at Google Scholar · View at Scopus
  24. W. Ford, “Implementing the QR decomposition,” in Numerical Linear Algebra with Applications Using MATLAB, chapter 17, pp. 351–378, Academic Press, 2015. View at Google Scholar
  25. P. Bogacki and L. F. Shampine, “A 3(2) pair of runge—kutta formulas,” Applied Mathematics Letters, vol. 2, no. 4, pp. 321–325, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. C. B. Moler, Numerical Computing with MATLAB, SIAM Press, Philadelphia, Pa, USA, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  27. J. A. Lee and M. Verleysen, “Scale-independent quality criteria for dimensionality reduction,” Pattern Recognition Letters, vol. 31, no. 14, pp. 2248–2257, 2010. View at Publisher · View at Google Scholar · View at Scopus