1. 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

2. Clustering

3.  Clustering : grouping of data based on similarity measures

4.  Classification: assign a class label to new unseen data Classification

5. 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

6. 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

7. Linear dimension reduction Maximize class separability in the reduced dimensional space

8. Linear dimension reduction Maximize class separability in the reduced dimensional space

9. What if data is not linear separable? Nonlinear Dimension Reduction

10. Contents Linear Discriminant Analysis Nonlinear Dimension Reduction based on Kernel Methods - Nonlinear Discriminant Analysis Application to Fingerprint Classification

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

12. 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

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

14. Face Recognition … … 92 x 112 10304 … GTGT … ? dimension reduction to maximize the distances among classes.

15. 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

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

17. Nonlinear Dimension Reduction based on Kernel Methods

18. Nonlinear Dimension Reduction GTGT nonlinear mapping linear dimension reduction

19. 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. 

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

21. Positive Definite Kernel Functions Gaussian kernel Polynomial kernel

22. 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) 

23. 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 

24. 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 )

25. 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

26. 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 )

27. 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

28. 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

29. Each color represents different data sets methods Prediction accuracies

30. Linear and Nonlinear Discriminant Analysis Data sets

31. Face Recognition

32. Application of Nonlinear Discriminant Analysis to Fingerprint Classification

33. Left Loop Right Loop Whorl Arch Tented Arch Fingerprint Classification From NIST Fingerprint database 4

34. 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

35. Construction of Core Directional Images Left Loop Right Loop Whorl

36. Construction of Core Directional Images Core Point

37. Discrete Fourier transform (DFT)

39. 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

40.  Computation of local dominant directions by DFT and directional filtering

41. Construction of Directional Images 105 x 105 512 x 512

42. Nonlinear discriminant Analysis … … 105 x 105 11025- dim. space GTGT Left loop WhorlRight loop Tented arch Arch Maximizing class separability in the reduced dimensional space 4- dim. space

43. 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 95.3 PCASYS +  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 (%)

44. 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

45. 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

46. 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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