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A Rank-Revealing Method for Low Rank Matrices with Updating, Downdating, and Applications Tsung-Lin Lee (Michigan State University) 2007 AMS Session Meeting,

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Presentation on theme: "A Rank-Revealing Method for Low Rank Matrices with Updating, Downdating, and Applications Tsung-Lin Lee (Michigan State University) 2007 AMS Session Meeting,"— Presentation transcript:

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

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 The numerical rank w.r.t. threshold :  

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

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

8  Updating problem :

9  Downdating problem :

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

11 2005, T.Y. Li and Zhonggang Zeng => rank-revealing algorithm for high rank matrices 1.The approxi-rank. 2.The approxi-kernel. 3.The method is more efficient and robust. 4.Algorithms for updating and downdating problems are straightforward, stable and efficient.

12

13 Tsung-Lin Lee, T.Y. Li and Zhonggang Zeng => rank-revealing algorithm for low rank matrices 1.The approxi-rank. 2.The approxi-range. 3.The approxi-rowspace. 4.The projections of left and right kernel. 5.USV+E decomposition. 6.The method is robust and more efficient. 7.Algorithms for updating and downdating problems are straightforward, stable and efficient. =+

14 Stop when Power iteration on Random

15 0 approxi-range

16 The implicit singular value deflation:

17 USV+E decomp. LQ

18 perturbation =+ USV+E decomposition approxi-range approxi-rowspace

19 Numerical experiments and comparisons Matlab 7.0, on Dell PC Pentium D 3.2MHz CPU, 1GB RAM  2n n 400x200800x x x1600 timeerrortimeerrortimeerrortimeerror SVD0.313e e e e-9 lurv0.663e e e e-9 lulv0.564e e e e-9 larank0.053e e e e-9

20 AU = + Row updating AU = + 1 AU = + 1

21 row downdating deflate R =+

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

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

24

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

26 = +

27 Image processing Saving storage of photographs FBI Fingerprint Image Database Face Image Database …

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

29 j

30 Rank 20 approximation imageRank 480 image

31

32 : % : % k SVD 2.87 lulv 3.02 lurvlarank 0.17 Running time (seconds) 15.2 : 1 Compression ratio 18 Approxi-rank 2.1% Threshold

33 HighRankRev and LowRankRev Package

34 Thank you


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