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: assign a class label to new unseen data

Classification

Data Mining

Data Preparation

Preprocessing

Classification Clustering •Association Analysis• Regression• Probabilistic modeling …

Dimension reduction-Feature Selection

-

Data Reduction

• Mining or discovery of new information - patterns or rules - from large databases

Feature Extraction

Feature Extraction

• Optimal feature extraction - Reduce the dimensionality of data space - Minimize effects of redundant features and noise

Apply a classifier to predict a class label of new data

feature extraction

.. .. ..

......

number of features

new data

Curse of dimensionality

Linear dimension reduction

Maximize class separability

in the reduced dimensional space

Linear dimension reduction

Maximize class separability

in the reduced dimensional space

What if data is not linear separable?

Nonlinear Dimension Reduction

Contents

• Linear Discriminant Analysis

• Nonlinear Dimension Reduction based on Kernel Methods

- Nonlinear Discriminant Analysis

• Application to Fingerprint Classification

n

iia

nc

1

1

For a given data set {a1,┉,an }

• Within-class scatter matrix

• trace(Sw)

r

ii

iclassa

ca1

2||||

r

i

Ti

iclassaiw cacaS

1

))((

Centroids :

iclassai

i an

c 1

Linear Discriminant Analysis (LDA)

• Between-class scatter matrix

• trace(Sb)2

1

|||| ccr

ii

Tiii

r

ib ccccnS ))((

1

GT→

maximize minimize trace(GTSwG)

trace(GTSbG)

a1┉ an GTa1┉ GTan

Eigenvalue problem xxSS bw 1

Sw-1 Sb

G

=

Sw-1Sb X = X

))()(( tracemax)( 1 GSGGSGGJ bT

wT

G

rank(Sb) number of classes - 1

Face Recognition

…

…

92 x 112

10304

…

GT

…

?

dimension reduction to maximize the distances among classes.

Text Classification

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

Education

FacultyStudentSyllabusGradeTuition….

Recreation

MovieMusicSportHollywoodTheater…..

GT

SbSw

Generalized LDA Algorithms

xxSS bw 1

xSxS wb

• Undersampled problems:

high dimensionality & small number of data

Can’t compute Sw-1Sb

Nonlinear Dimension Reductionbased on Kernel Methods

Nonlinear Dimension Reduction

GT

nonlinear mapping linear dimension reduction ),2,(),( 2

2212121 xxxxxx

Kernel Method

• 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.

njiji aakK ,1)],([

Nonlinear Dimension Reduction by Kernel Methods

GT

),()(),( yxkyx

Given a kernel function k(x,y)

linear dimension reduction

Positive Definite Kernel Functions

• Gaussian kernel

• Polynomial kernel

)/exp(),(2 yxyxk

),,0(),(),( 2121 Rdyxyxk d

Nonlinear Discriminant Analysis using Kernel Methods

{a1,a2,…,an}

Sb x= Sw x

{(a1),…,(an)}

Want to apply LDA

<(x),(y)>= k(x,y)

Nonlinear Discriminant Analysis using Kernel Methods

{a1,a2,…,an}

Sb x= Sw x

{(a1),…,(an)}

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

Sb u= Sw u

Apply Generalized LDA

Algorithms

SbSw

Generalized LDA Algorithms

xSxS wb

Minimize trace(xT Sw x)

xT Sw x = 0

x null(Sw)

Maximize trace(xT Sb x)

xT Sb x ≠ 0

x range(Sb)

Generalized LDA algorithms

• Add a positive diagonal matrix I

to Sw so that Sw+I is nonsingularRLDA

LDA/GSVD • Apply the generalized singular value

decomposition (GSVD) to {Hw , Hb}

in Sb = Hb HbT and Sw=Hw Hw

T

To-N(Sw) • Projection to null space of Sw

• Maximize between-class scatter in the projected space

Generalized LDA Algorithms

To-R(Sb)• Transformation to range space of Sb

• Diagonalize within-class scatter matrix in the transformed space

To-NR(Sw)• Reduce data dimension by PCA• Maximize between-class scatter

in range(Sw) and null(Sw)

Data sets

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

From Machine Learning Repository Database

Experimental Settings

Split

kernel function k and a linear transf. GT

Dimension reducing

Predict class labels of test data using training data

Original data

Training data Test data

• Each color represents different data sets

methods

Prediction accuracies

Linear and Nonlinear Discriminant Analysis

Data sets

Face Recognition

Application of Nonlinear Discriminant Analysis to Fingerprint Classification

Left Loop Right Loop Whorl

Arch Tented Arch

Fingerprint Classification

From NIST Fingerprint database 4

Previous Works in Fingerprint Classification

Feature representation

Minutiae

Gabor filtering

Directional partitioning

Apply Classifiers:

Neural Networks

Support Vector

Machines

Probabilistic NN

Our Approach Construct core directional images by DFT Dimension Reduction by Nonlinear Discriminant Analysis

Construction of Core Directional Images

Left Loop Right Loop Whorl

Construction of Core Directional Images

Core Point

Discrete Fourier transform (DFT)

Discrete Fourier transform (DFT)

Construction of Directional Images

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

Computation of local dominant directions by DFT and directional filtering

Construction of Directional Images

105 x 105

512 x 512

Nonlinear discriminant Analysis

…

…

105 x 105

11025-dim. space

GT

Left loop

WhorlRight loop

Tented archArch

Maximizing class separability in the reduced dimensional space

4-dim. space

Comparison of Experimental Results

NIST Database 4

Rejection rate (%) 0 1.8 8.5 20.0

Nonlinear LDA/GSVD 90.7 91.3 92.8 95.3PCASYS + 89.7 90.5 92.8 95.6

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

• 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 :

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

Core point detection

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..

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