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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, Chicago joint work with Tien-Yien Li and Zhonggang Zeng

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

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

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Mirsky Theorem:

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The numerical rank w.r.t. threshold :

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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

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

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Updating problem :

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Downdating problem :

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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

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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.

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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. =+

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Stop when Power iteration on Random

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0 approxi-range

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The implicit singular value deflation:

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USV+E decomp. LQ

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

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Numerical experiments and comparisons Matlab 7.0, on Dell PC Pentium D 3.2MHz CPU, 1GB RAM 2n n 400x200800x4001600x8003200x1600 timeerrortimeerrortimeerrortimeerror SVD0.313e-92.194e-916.63e-9144.7e-9 lurv0.663e-91.524e-95.973e-932.57e-9 lulv0.564e-91.526e-96.035e-931.95e-9 larank0.053e-90.114e-90.393e-91.814e-9

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

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row downdating deflate R =+

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

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

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

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= +

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

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

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j

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Rank 20 approximation imageRank 480 image

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7.477.920.735.38 : 1510.8% 2.014.945.140.388.07 : 1341.3% k SVD 2.87 lulv 3.02 lurvlarank 0.17 Running time (seconds) 15.2 : 1 Compression ratio 18 Approxi-rank 2.1% Threshold

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

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Thank you

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