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Kernelized Discriminant Analysis and Adaptive Methods for Discriminant Analysis

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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

- Classification:
- assign a class label to new unseen data

- 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

- 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

..

..

..

Maximize class separability

in the reduced dimensional space

Maximize class separability

in the reduced dimensional space

Nonlinear Dimension Reduction

- Linear Discriminant Analysis
- Nonlinear Dimension Reduction based on Kernel Methods
- Nonlinear Discriminant Analysis

- Application to Fingerprint Classification

For a given data set {a1,┉,an }

Centroids :

- Within-class scatter matrix
- trace(Sw)

- Between-class scatter matrix
- trace(Sb)

a1┉ an

GTa1┉ GTan

GT

→

trace(GTSbG)

maximize

minimize

trace(GTSwG)

Eigenvalue problem

G

=

Sw-1 Sb

Sw-1Sb X = X

rank(Sb) number of classes - 1

dimension reduction to maximize

the distances among classes.

…

92 x 112

?

10304

GT

…

…

…

- 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

- Undersampled problems:
- high dimensionality & small number of data
- Can’t compute Sw-1Sb

Sb

Sw

Nonlinear Dimension Reductionbased on Kernel Methods

nonlinear mapping

linear dimension

reduction

GT

- 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.

Given a kernel function k(x,y)

linear dimension

reduction

GT

- Gaussian kernel
- Polynomial kernel

{a1,a2,…,an}

{(a1),…,(an)}

Want to apply LDA

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

Sb x= Sw x

{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

Sb

Sw

Minimizetrace(xT Sw x)

xT Sw x = 0

x null(Sw)

Maximizetrace(xT Sb x)

xT Sb x ≠ 0

x range(Sb)

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

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)

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

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

accuracies

methods

- Each color represents different data sets

Data sets

Application of Nonlinear Discriminant Analysis to Fingerprint Classification

Fingerprint Classification

Left Loop Right Loop Whorl

Arch Tented Arch

From NIST Fingerprint database 4

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

Left Loop Right Loop Whorl

Core Point

- 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

512 x 512

105 x 105

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

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 (%)

- 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

- Core point detection

- 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

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- 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
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..

- 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
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- 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