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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

### Application of Nonlinear Discriminant Analysis to Fingerprint Classification

Haesun Park

Georgia Institute of Technology,

Atlanta, GA, USA

(joint work with C. Park)

KAIST, Korea, June 2007

Clustering :

- grouping of data based on similarity measures

Classification

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

Data Mining

- 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

- 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

..

..

..

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

Linear Discriminant Analysis (LDA)

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)

Text Classification

- 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

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.

Nonlinear Dimension Reduction by Kernel Methods

Given a kernel function k(x,y)

linear dimension

reduction

GT

Positive Definite Kernel Functions

- Gaussian kernel
- Polynomial kernel

Nonlinear Discriminant Analysis using Kernel Methods

{a1,a2,…,an}

{(a1),…,(an)}

Want to apply LDA

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

Sb x= Sw x

Nonlinear Discriminant Analysis using Kernel Methods

{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)

Generalized LDA algorithms

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 LDAAlgorithms

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

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

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

Fingerprint Classification

Left Loop Right Loop Whorl

Arch Tented Arch

From NIST Fingerprint database 4

Previous Works in 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

Left Loop Right Loop Whorl

Construction of Core Directional Images

Core Point

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

Nonlinear discriminant Analysis

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

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

- 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

References

- 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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