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Kernelized Discriminant Analysis and Adaptive Methods for Discriminant Analysis. Haesun Park Georgia Institute of Technology, Atlanta, GA, USA (joint work with C. Park) KAIST, Korea, June 2007. Clustering. Clustering : grouping of data based on similarity measures. Classification.

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Kernelized discriminant analysis and adaptive methods for discriminant analysis

Kernelized Discriminant Analysis and Adaptive Methods for Discriminant Analysis

Haesun Park

Georgia Institute of Technology,

Atlanta, GA, USA

(joint work with C. Park)

KAIST, Korea, June 2007


Clustering Discriminant Analysis


  • Clustering : Discriminant Analysis

  • grouping of data based on similarity measures


Classification Discriminant Analysis

  • Classification:

  • assign a class label to new unseen data


Data mining
Data Mining Discriminant Analysis

  • Mining or discovery of new information - patterns

  • or rules - from large databases

Data Preparation

Data Reduction

  • Dimension reduction

  • Feature Selection

  • -

Preprocessing

Feature Extraction

  • Association Analysis

  • Regression

  • Probabilistic modeling …

Classification

Clustering


Feature extraction
Feature Extraction Discriminant Analysis

  • Optimal feature extraction

  • - Reduce the dimensionality of data space

  • - Minimize effects of redundant features and noise

Curse of dimensionality

number of features

new data

..

..

..

feature extraction

Apply a classifier

to predict a class

label of new data

..

..

..


Linear dimension reduction
Linear dimension reduction Discriminant Analysis

Maximize class separability

in the reduced dimensional space


Linear dimension reduction1
Linear dimension reduction Discriminant Analysis

Maximize class separability

in the reduced dimensional space


What if data is not linear separable
What if data is not linear separable? Discriminant Analysis

Nonlinear Dimension Reduction


Contents
Contents Discriminant Analysis

  • Linear Discriminant Analysis

  • Nonlinear Dimension Reduction based on Kernel Methods

    - Nonlinear Discriminant Analysis

  • Application to Fingerprint Classification


Linear discriminant analysis lda
Linear Discriminant Analysis (LDA) Discriminant Analysis

For a given data set {a1,┉,an }

Centroids :

  • Within-class scatter matrix

  • trace(Sw)


a1┉ an

GTa1┉ GTan

GT

trace(GTSbG)

maximize

minimize

trace(GTSwG)


Eigenvalue problem Discriminant Analysis

G

=

Sw-1 Sb

Sw-1Sb X =  X

rank(Sb) number of classes - 1


Face recognition
Face Recognition Discriminant Analysis

dimension reduction to maximize

the distances among classes.

92 x 112

?

10304

GT


Text classification
Text Classification Discriminant Analysis

  • A bag of words: each document is represented with frequencies of words contained

Education

Recreation

Faculty

Student

Syllabus

Grade

Tuition

….

Movie

Music

Sport

Hollywood

Theater

…..

GT


Generalized LDA Algorithms Discriminant Analysis

  • Undersampled problems:

  • high dimensionality & small number of data

  •  Can’t compute Sw-1Sb

Sb

Sw


Nonlinear dimension reduction based on kernel methods

Nonlinear Dimension Reduction Discriminant Analysisbased on Kernel Methods


Nonlinear dimension reduction
Nonlinear Dimension Reduction Discriminant Analysis

nonlinear mapping

linear dimension

reduction

GT


Kernel method
Kernel Method Discriminant Analysis

  • If a kernel function k(x,y) satisfies Mercer’s condition, then there exists a mapping 

    for which <(x),(y)>= k(x,y) holds

A (A)

< x, y > < (x), (y) > = k(x,y)

  • For a finite data set A=[a1,…,an], Mercer’s condition can be rephrased as the kernel matrix

  • is positive semi-definite.


Nonlinear dimension reduction by kernel methods
Nonlinear Dimension Reduction by Kernel Methods Discriminant Analysis

Given a kernel function k(x,y)

linear dimension

reduction

GT


Positive definite kernel functions
Positive Definite Kernel Functions Discriminant Analysis

  • Gaussian kernel

  • Polynomial kernel


Nonlinear discriminant analysis using kernel methods
Nonlinear Discriminant Analysis using Kernel Methods Discriminant Analysis

{a1,a2,…,an}

{(a1),…,(an)}

Want to apply LDA

<(x),(y)>= k(x,y)

Sb x= Sw x


Nonlinear discriminant analysis using kernel methods1
Nonlinear Discriminant Analysis using Kernel Methods Discriminant Analysis

{a1,a2,…,an}

{(a1),…,(an)}

k(a1,a1) k(a1,an)

… ,…, …

k(an,a1) k(an,an)

Sbu= Swu

Sb x= Sw x

Apply Generalized LDA

Algorithms


Generalized LDA Algorithms Discriminant Analysis

Sb

Sw

Minimizetrace(xT Sw x)

xT Sw x = 0

x null(Sw)

Maximizetrace(xT Sb x)

xT Sb x ≠ 0

x range(Sb)


Generalized lda algorithms
Generalized LDA algorithms Discriminant Analysis

RLDA

  • Add a positive diagonal matrix I

    to Swso that Sw+I is nonsingular

  • Apply the generalized singular value

  • decomposition (GSVD) to {Hw , Hb}

  • in Sb = Hb HbT and Sw=Hw HwT

LDA/GSVD

To-N(Sw)

  • Projection to null space of Sw

  • Maximize between-class scatter

  • in the projected space


Generalized lda algorithms1
Generalized LDA Discriminant AnalysisAlgorithms

To-R(Sb)

  • Transformation to range space of Sb

  • Diagonalize within-class scatter matrix

    in the transformed space

  • Reduce data dimension by PCA

  • Maximize between-class scatter

  • in range(Sw) and null(Sw)

To-NR(Sw)


Data sets
Data sets Discriminant Analysis

From Machine Learning Repository Database

Data dim no. of data no. of classes

Musk 166 6599 2

Isolet 617 7797 26

Car 6 1728 4

Mfeature 649 2000 10

Bcancer 9 699 2

Bscale 4 625 3


Experimental settings
Experimental Settings Discriminant Analysis

Original data

Split

Training data

Test data

kernel function k and a linear transf. GT

Dimension reducing

Predict class labels of test data using training data


Prediction Discriminant Analysis

accuracies

methods

  • Each color represents different data sets


Linear and nonlinear discriminant analysis
Linear and Nonlinear Discriminant Analysis Discriminant Analysis

Data sets


Face recognition1
Face Recognition Discriminant Analysis



Fingerprint Classification Fingerprint Classification

Left Loop Right Loop Whorl

Arch Tented Arch

From NIST Fingerprint database 4


Previous works in fingerprint classification
Previous Works in Fingerprint Classification Fingerprint Classification

Apply Classifiers:

Neural Networks

Support Vector

Machines

Probabilistic NN

Feature representation

Minutiae

Gabor filtering

Directional partitioning

Our Approach

Construct core directional images by DFT

Dimension Reduction by Nonlinear Discriminant Analysis


Construction of core directional images
Construction of Core Directional Images Fingerprint Classification

Left Loop Right Loop Whorl


Construction of core directional images1
Construction of Core Directional Images Fingerprint Classification

Core Point


Discrete fourier transform dft
Discrete Fourier transform (DFT) Fingerprint Classification


Discrete fourier transform dft1
Discrete Fourier transform (DFT) Fingerprint Classification


Construction of directional images
Construction of Directional Images Fingerprint Classification

  • Computation of local dominant directions by DFT and directional filtering

  • Core point detection

  • Reconstruction of core directional images

  • Fast computation of DFT by FFT

  • Reliable for low quality images



Construction of directional images1
Construction of Directional Images directional filtering

512 x 512

105 x 105


Nonlinear discriminant analysis
Nonlinear discriminant Analysis directional filtering

105 x 105

Maximizing class separability

in the reduced dimensional space

Right loop

Whorl

Left loop

GT

Tented arch

Arch

4-dim. space

11025-dim. space


Comparison of experimental results
Comparison of Experimental Results directional filtering

NIST Database 4

Rejection rate (%) 0 1.8 8.5 20.0

Nonlinear LDA/GSVD90.791.392.8 95.3

PCASYS +  89.7 90.5 92.895.6

Jain et.al. [1999,TPAMI] - 90.0 91.2 93.5

Yao et al. [2003,PR] - 90.0 92.2 95.6

prediction accuracies (%)


Summary
Summary directional filtering

  • Nonlinear Feature Extraction based on Kernel Methods

    - Nonlinear Discriminant Analysis

    - Kernel Orthogonal Centroid Method (KOC)

  • A comparison of Generalized Linear and Nonlinear Discriminant Analysis Algorithms

  • Application to Fingerprint Classification


  • Dimension reduction - feature transformation : directional filtering

    linear combination of original features

  • Feature selection :

    select a part of original features

    gene expression microarray data anaysis

    -- gene selection

  • Visualization of high dimensional data

  • Visual data mining


  • θi,j:dominant direction on the neighborhood centered at (i, j)

  • Measure consistency of local dominant directions

    | ΣΣi,j=-1,0,1[cos(2θi,j), sin(2θi,j)] |

    :distance from the starting point to finishing point

  • the lowest value -> Core point


References
References directional filtering

  • L.Chen et al., A new LDA-based face recognition system which can solve the small sample size problem, Pattern Recognition, 33:1713-1726, 2000

  • P.Howland et al., Structure preserving dimension reduction for clustered text data based on the generalized singular value decomposition, SIMAX, 25(1):165-179, 2003

  • H.Yu and J.Yang, A direct LDA algorithm for high-dimensional data-with application to face recognition, Pattern Recognition, 34:2067-2070, 2001

  • J.Yang and J.-Y.Yang, Why can LDA be performed in PCA transformed space?, Pattern Recognition, 36:563-566, 2003

  • H. Park et al., Lower dimensional representation of text data based on centroids and least squares, BIT Numerical Mathematics, 43(2):1-22, 2003

  • S. Mika et al., Fisher discriminant analysis with kernels, Neural networks for signal processing IX, J.Larsen and S.Douglas, pp.41-48, IEEE, 1999

  • B. Scholkopf et al., Nonlinear component analysis as a kernel eigenvalue problem, Neural computation, 10:1299-1319, 1998

  • G. Baudat and F. Anouar, Generalized discriminant analysis using a kernel approach, Neural computation, 12:2385-2404, 2000

  • V. Roth and V. Steinhage, Nonlinear discriminant analysis using a kernel functions, Advances in neural information processing functions, 12:568-574, 2000

..


  • S.A. Billings and K.L. Lee, Nonlinear fisher discriminant analysis using a minimum squared error cost function and the orthogonal least squares algorithm, Neural networks, 15(2):263-270, 2002

  • C.H. Park and H. Park, Nonlinear discriminant analysis based on generalized singular value decomposition, SIMAX, 27-1, pp. 98-102, 2005

  • A.K.Jain et al., A multichannel approach to fingerprint classification, IEEE transactions on Pattern Analysis and Machine Intelligence, 21(4):348-359,1999

  • Y.Yao et al., Combining flat and structural representations for fingerprint classifiaction with recursive neural networks and support vector machines, Pattern recognition, 36(2):397-406,2003

  • C.H.Park and H.Park, Nonlinear feature extraction based on cetroids and kernel functions, Pattern recognition, 37(4):801-810

  • C.H.Park and H.Park, A Comparison of Generalized LDA algorithms for undersampled problems, Pattern Recognition, to appear

  • C.H.Park and H.Park, Fingerprint classification using fast fourier transform and nonlinear discriminant analysis, Pattern recognition, 2006


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