# 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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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

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

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

Mirsky Theorem:

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

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

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

 Updating problem :

 Downdating problem :

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

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.

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. =+

Stop when Power iteration on Random

0 approxi-range

The implicit singular value deflation:

USV+E decomp. LQ

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

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

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

row downdating deflate R =+

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

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

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

= +

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

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

j

Rank 20 approximation imageRank 480 image

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

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

Thank you

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