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

Slides:

Advertisements

Similar presentations

The Mathematics of Information Retrieval 11/21/2005 Presented by Jeremy Chapman, Grant Gelven and Ben Lakin.

Advertisements

Eigen Decomposition and Singular Value Decomposition

Introduction to Information Retrieval Outline ❶ Latent semantic indexing ❷ Dimensionality reduction ❸ LSI in information retrieval 1.

Eigen Decomposition and Singular Value Decomposition

Latent Semantic Analysis

Generalised Inverses Modal Analysis and Modal Testing S. Ziaei Rad.

Using Latent Semantic Analysis to Find Different Names for the Same Entity in Free Text Tim Oates, Vinay Bhat, Vishal Shanbhag, Charles Nicholas University.

Dimensionality Reduction PCA -- SVD

Least Squares example There are 3 mountains u,y,z that from one site have been measured as 2474 ft., 3882 ft., and 4834 ft.. But from u, y looks 1422 ft.

Solving Linear Systems (Numerical Recipes, Chap 2)

Comparison of information retrieval techniques: Latent semantic indexing (LSI) and Concept indexing (CI) Jasminka Dobša Faculty of organization and informatics,

Nequalities Takagi Factorization on a GPU using CUDA Gagandeep S. Sachdev, Vishay Vanjani & Mary W. Hall School of Computing, University of Utah What is.

Lecture 19 Singular Value Decomposition

Hinrich Schütze and Christina Lioma

Latent Semantic Indexing via a Semi-discrete Matrix Decomposition.

Unsupervised Learning - PCA The neural approach->PCA; SVD; kernel PCA Hertz chapter 8 Presentation based on Touretzky + various additions.

1 Latent Semantic Indexing Jieping Ye Department of Computer Science & Engineering Arizona State University

Vector Space Information Retrieval Using Concept Projection Presented by Zhiguo Li

Optimization of Sparse Matrix Kernels for Data Mining Eun-Jin Im and Katherine Yelick U.C.Berkeley.

Information Retrieval in Text Part III Reference: Michael W. Berry and Murray Browne. Understanding Search Engines: Mathematical Modeling and Text Retrieval.

7th IEEE Technical Exchange Meeting 2000 Hybrid Wavelet-SVD based Filtering of Noise in Harmonics By Prof. Maamar Bettayeb and Syed Faisal Ali Shah King.

Singular Value Decomposition in Text Mining Ram Akella University of California Berkeley Silicon Valley Center/SC Lecture 4b February 9, 2011.

TFIDF-space  An obvious way to combine TF-IDF: the coordinate of document in axis is given by  General form of consists of three parts: Local weight.

Lecture 21 SVD and Latent Semantic Indexing and Dimensional Reduction

Singular Value Decomposition

1/ 30. Problems for classical IR models Introduction & Background(LSI,SVD,..etc) Example Standard query method Analysis standard query method Seeking.

The Terms that You Have to Know! Basis, Linear independent, Orthogonal Column space, Row space, Rank Linear combination Linear transformation Inner product.

SLIDE 1IS 240 – Spring 2007 Prof. Ray Larson University of California, Berkeley School of Information Tuesday and Thursday 10:30 am - 12:00.

10-603/15-826A: Multimedia Databases and Data Mining SVD - part I (definitions) C. Faloutsos.

3D Geometry for Computer Graphics

Kathryn Linehan Advisor: Dr. Dianne O’Leary

Multimedia Databases LSI and SVD. Text - Detailed outline text problem full text scanning inversion signature files clustering information filtering and.

Singular Value Decomposition and Data Management

Introduction The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations.

CS246 Topic-Based Models. Motivation  Q: For query “car”, will a document with the word “automobile” be returned as a result under the TF-IDF vector.

Latent Semantic Indexing Debapriyo Majumdar Information Retrieval – Spring 2015 Indian Statistical Institute Kolkata.

CS 231A Section 1: Linear Algebra & Probability Review

1 Information Retrieval through Various Approximate Matrix Decompositions Kathryn Linehan Advisor: Dr. Dianne O’Leary.

Presentation to VII International Workshop on Advanced Computing and Analysis Techniques in Physics Research October, 2000.

Indices Tomasz Bartoszewski. Inverted Index Search Construction Compression.

On Scaling Latent Semantic Indexing for Large Peer-to-Peer Systems Chunqiang Tang, Sandhya Dwarkadas, Zhichen Xu University of Rochester; Yahoo! Inc. ACM.

An Introduction to Latent Semantic Analysis. 2 Matrix Decompositions Definition: The factorization of a matrix M into two or more matrices M 1, M 2,…,

SVD: Singular Value Decomposition

CpSc 881: Information Retrieval. 2 Recall: Term-document matrix This matrix is the basis for computing the similarity between documents and queries. Today:

Rozhen 2010, June Singular Value Decomposition of images from scanned photographic plates Vasil Kolev Institute of Computer and Communications Systems.

Ill-posed Computational Algebra with Approximate Data Zhonggang Zeng Northeastern Illinois University, USA Feb. 21, 2006, Radon Institute for Computational.

SINGULAR VALUE DECOMPOSITION (SVD)

Orthogonalization via Deflation By Achiya Dax Hydrological Service Jerusalem, Israel

Accelerating the Singular Value Decomposition of Rectangular Matrices with the CSX600 and the Integrable SVD September 7, 2007 PaCT-2007, Pereslavl-Zalessky.

Scientific Computing Singular Value Decomposition SVD.

A Note on Rectangular Quotients By Achiya Dax Hydrological Service Jerusalem, Israel

Singular Value Decomposition

LATENT SEMANTIC INDEXING BY SINGULAR VALUE DECOMPOSITION

1. Systems of Linear Equations and Matrices (8 Lectures) 1.1 Introduction to Systems of Linear Equations 1.2 Gaussian Elimination 1.3 Matrices and Matrix.

On Computing Multiple Eigenvalues, the Jordan Structure and their Sensitivity Zhonggang Zeng Northeastern Illinois University Sixth International Workshop.

Image Retrieval and Ranking using L.S.I and Cross View Learning Sumit Kumar Vivek Gupta

Singular Value Decomposition and its applications

Introduction The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations.

Introduction The central problems of Linear Algebra are to study the properties of matrices and to investigate the solutions of systems of linear equations.

Elementary Linear Algebra

Matrix Sketching over Sliding Windows

Nonnegative Matrix Factorization via Rank-one Downdate

Nonnegative polynomials and applications to learning

LSI, SVD and Data Management

Parallelism in High-Performance Computing Applications

Singular Value Decomposition

Lecture 21 SVD and Latent Semantic Indexing and Dimensional Reduction

Algebraic Techniques for Analysis of Large Discrete-Valued Datasets

Non-Negative Matrix Factorization

Latent Semantic Analysis

Presentation transcript:

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 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 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

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

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

HighRankRev and LowRankRev Package

Thank you