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A Rank-Revealing Method for Low Rank Matrices with Updating, Downdating, and Applications

A Rank-Revealing Method for Low Rank Matrices with Updating, Downdating, and Applications. Tsung-Lin Lee (Michigan State University). joint work with Tien-Yien Li and Zhonggang Zeng. 2007 AMS Session Meeting, Chicago. Rank determination problems appear in 1. Image Processing

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A Rank-Revealing Method for Low Rank Matrices with Updating, Downdating, and Applications

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  1. A Rank-Revealing Method for Low Rank Matrices with Updating, Downdating, and Applications Tsung-Lin Lee (Michigan State University) joint work with Tien-Yien Li and Zhonggang Zeng 2007 AMS Session Meeting, Chicago

  2. Rank determination problems appear in 1. Image Processing 2. Information Retrieval 3. Matrix Approximation 4. Least Squares Problems 5. Numerical Polynomial Algebra ……

  3. The rank gap: Numerical rank: the rank decision threshold : the approxi-rank w.r.t. the threshold : (numerical rank)

  4. Mirsky Theorem:

  5. q q The numerical rank w.r.t. threshold :

  6. SVD Algorithm (Golub-Reinsch) => efficient when the matrix size is moderate. In some applications, the matrix is large. -The rank is close to full. (high rank) -The rank is close to zero. (low rank) The goal: An efficient and stable algorithm

  7. 1989 Tony Chan =>Rank Revealing QR algorithm for high rank matrices The updating and downdating problems => It can’t solve them efficiently.

  8. Updating problem: A

  9. Downdating problem: A

  10. F.D. Fierro, P.C. Hansen and P.S. K. Hansen (1999) UTV tools: Matlab templates for rank-revealing UTV decomposition => re-compute the UTV decomposition 1992 G.W. Stewart => rank revealing UTV decomposition. (URV/ULV) 1. Updating problems are applicable. 2. Downdating problems are difficult.

  11. 2005, T.Y. Li and Zhonggang Zeng => rank-revealing algorithm for high rank matrices • The approxi-rank. • The approxi-kernel. • The method is more efficient and robust. • Algorithms for updating and downdating problems • are straightforward, stable and efficient.

  12. = + Tsung-Lin Lee, T.Y. Li and Zhonggang Zeng => rank-revealing algorithm for low rank matrices • The approxi-rank. • The approxi-range. • The approxi-rowspace. • The projections of left and right kernel. • USV+E decomposition. • The method is robust and more efficient. • Algorithms for updating and downdating problems • are straightforward, stable and efficient.

  13. Power iteration on Random Stop when

  14. approxi-range 0

  15. The implicit singular value deflation:

  16. LQ USV+E decomp.

  17. USV+E decomposition approxi-rowspace perturbation = + approxi-range

  18. Numerical experiments and comparisons Matlab 7.0, on Dell PC Pentium D 3.2MHz CPU, 1GB RAM n A 2n

  19. A A A U U U = = = + + + 1 1 Row updating

  20. deflate R = + row downdating

  21. USV+E decomposition A Dominant (signal) Perturbation (noise) = +

  22. Information retrieval Latent Semantic Indexing method (LSI) Library database Webpage search engine (Google) …

  23. rank, revealing, updating, downdating, application 12x8 term by document matrix

  24. + =

  25. Image processing Saving storage of photographs FBI Fingerprint Image Database Face Image Database …

  26. A 480x640 monochrome (baseball picture) Grey levels: 0 => 1 black white

  27. j

  28. Rank 480 image Rank 20 approximation image

  29. Threshold Approxi-rank Compression ratio Running time (seconds) k larank lurv lulv SVD 2.1% 18 15.2 : 1 0.17 3.02 2.87 1.3% 34 8.07 : 1 0.38 5.14 4.94 2.01 0.8% 51 5.38 : 1 0.73 7.92 7.47

  30. HighRankRev and LowRankRev Package http://www.msu.edu/~leetsung/Software.htm

  31. Thank you

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