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 ClassificationClustering 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

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

For a given data set {a 1, ┉, a n } Within-class scatter matrix trace(S w ) Centroids : Linear Discriminant Analysis (LDA)

Between-class scatter matrix trace(S b ) GTGT → maximize minimize trace(G T S w G) trace(G T S b G) a1┉ ana1┉ an GTa1┉ GTanGTa1┉ GTan

Eigenvalue problem S w -1 S b G = S w -1 S b X =  X rank(S b )  number of classes - 1

Face Recognition … … 92 x … GTGT … ? dimension reduction to maximize the distances among classes.

Text Classification A bag of words: each document is represented with frequencies of words contained Education Faculty Student Syllabus Grade Tuition …. Recreation Movie Music Sport Hollywood Theater ….. GTGT

SbSb SwSw Generalized LDA Algorithms Undersampled problems: high dimensionality & small number of data  Can’t compute S w -1 S b

Nonlinear Dimension Reduction based on Kernel Methods

Nonlinear Dimension Reduction GTGT nonlinear mapping linear dimension reduction

Kernel Method If a kernel function k(x,y) satisfies Mercer’s condition, then there exists a mapping  for which = k(x,y) holds A  (A) = k(x,y) For a finite data set A=[a 1, …,a n ], Mercer’s condition can be rephrased as the kernel matrix is positive semi-definite. 

Nonlinear Dimension Reduction by Kernel Methods GTGT Given a kernel function k(x,y) linear dimension reduction

Positive Definite Kernel Functions Gaussian kernel Polynomial kernel

Nonlinear Discriminant Analysis using Kernel Methods {a 1,a 2,…,a n } S b x= S w x {  (a 1 ),…,  (a n )} Want to apply LDA = k(x,y) 

Nonlinear Discriminant Analysis using Kernel Methods {a 1,a 2,…,a n } S b x= S w x {  (a 1 ),…,  (a n )} k(a 1,a 1 ) k(a 1,a n ) …,…, … k(a n,a 1 ) k(a n,a n ) S b u= S w u Apply Generalized LDA Algorithms 

SbSb SwSw Generalized LDA Algorithms Minimize trace(x T S w x) x T S w x = 0 x  null(S w ) Maximize trace(x T S b x) x T S b x ≠ 0 x  range(S b )

Generalized LDA algorithms Add a positive diagonal matrix  I to S w so that S w +  I is nonsingular RLDA LDA/GSVD Apply the generalized singular value decomposition (GSVD) to {H w, H b } in S b = H b H b T and S w =H w H w T To-N(S w ) Projection to null space of S w Maximize between-class scatter in the projected space

Generalized LDA Algorithms To-R(S b ) Transformation to range space of S b Diagonalize within-class scatter matrix in the transformed space To-NR(S w ) Reduce data dimension by PCA Maximize between-class scatter in range(S w ) and null(S w )

Data sets Data dim no. of data no. of classes Musk Isolet Car Mfeature Bcancer Bscale From Machine Learning Repository Database

Experimental Settings Split kernel function k and a linear transf. G T Dimension reducing Predict class labels of test data using training data Original data Training dataTest 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)

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 x 512

Nonlinear discriminant Analysis … … 105 x dim. space GTGT Left loop WhorlRight loop Tented arch Arch Maximizing class separability in the reduced dimensional space 4- dim. space

Comparison of Experimental Results NIST Database 4 Rejection rate (%) Nonlinear LDA/GSVD PCASYS +  Jain et.al. [1999,TPAMI] Yao et al. [2003,PR] 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

θ i,j : dominant direction on the neighborhood centered at (i, j) Measure consistency of local dominant directions | ΣΣ i,j=-1,0,1 [cos(2θ i,j ), sin(2θ i,j )] | : distance from the starting point to finishing point the lowest value -> Core point  Core point detection

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