Kernel feature extraction methods observed from the viewpoint of generating-kernels

Abstract

This paper introduces an idea of generating a kernel from an arbitrary function by embedding the training samples into the function. Based on this idea, we present two nonlinear feature extraction methods: generating kernel principal component analysis (GKPCA) and generating kernel Fisher discriminant (GKFD). These two methods are shown to be equivalent to the function-mapping-space PCA (FMS-PCA) and the function-mapping-space linear discriminant analysis (FMS-LDA) methods, respectively. This equivalence reveals that the generating kernel is actually determined by the corresponding function map. From the generating kernel point of view, we can classify the current kernel Fisher discriminant (KFD) algorithms into two categories: KPCA+ LDA based algorithms and straightforward KFD (SKFD) algorithms. The KPCA+ LDA based algorithms directly work on the given kernel and are not suitable for non-kernel functions, while the SKFD algorithms essentially work on the generating kernel from a given symmetric function and are therefore suitable for non-kernels as well as kernels. Finally, we outline the tensor-based feature extraction methods and discuss ways of extending tensor-based methods to their generating kernel versions.

Key words： kernel methods; feature extraction; principal component analysis (PCA); Fisher linear discriminant analysis (FLD or LDA); tensor-based methods

References

1 Vapnik V. The Nature of Statistical Learning Theory. New York: Springer, 1995

2 Müller K R, Mika S, Rätsch G, Tsuda K, Schölkopf B. An introduction to kernel-based learning algorithms. IEEE Transactions on Neural Networks, 2001, 12(2): 181-201

3 Schölkopf B, Smola A, Muller K R. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 1998, 10(5): 1299-1319

4 Mika S, Rätsch G, Weston J, Schölkopf B, Müller K R. Fisher discriminant analysis with kernels. In: Proceedings of IEEE International Workshop on Neural Networks for Signal Processing IX. 1999, 41-48

5 Mika S, Rätsch G, Schölkopf B, Smola A, Weston J, Müller K R. Invariant feature extraction and classification in kernel spaces. Advances in Neural Information Processing Systems, 1999, 12: 526-532

6 Baudat G, Anouar F. Generalized discriminant analysis using a kernel approach. Neural Computation, 2000, 12(10): 2385-2404

7 Roth V, Steinhage V. Nonlinear discriminant analysis using kernel functions. In: Solla S A, Leen T K, Mueller K R, eds. Advances in Neural Information Processing Systems. 2000, 12: 568-574

8 Mika S, Rätsch G, Weston J, Schölkopf B, Smola A, Müller K R. Constructing descriptive and discriminative non-linear features: Rayleigh coefficients in kernel feature spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2003, 25(5): 623-628

9 Yang M H. Kernel Eigenfaces vs. kernel Fisherfaces: face recognition using kernel methods. In: Proceedings of the 5th IEEE International Conference on Automatic Face and Gesture Recognition. 2002, 215-220

10 Lu J, Plataniotis K N, Venetsanopoulos A N. Face recognition using kernel direct discriminant analysis algorithms. IEEE Transactions on Neural Networks, 2003, 14(1): 117-126

11 Schölkopf B, Smola A. Learning with Kernels. Cambridge: MIT Press, 2002

12 Shawe-Taylor J, Cristianini N. Kernel Methods for Pattern Analysis. Cambridge: Cambridge University Press, 2004

13 Xu J, Zhang X, Li Y. Kernel MSE algorithm: a unified framework for KFD, LS-SVM, and KRR. In: Proceedings of the International Joint Conference on Neural Networks. 2001, 1486-1491

14 Billings S A, Lee K L. Nonlinear fisher discriminant analysis using a minimum squared error cost function and the orthogonal least squares algorithm. Neural Networks, 2002, 15(2): 263-270

15 Zheng W, Zhao L, Zou C. Foley-Sammon optimal discriminant vectors using kernel approach. IEEE Transactions on Neural Networks, 2005, 16(1): 1-9

16 Yang J, Jin Z, Yang J Y, Zhang D, Frangi A F. Essence of kernel Fisher discriminant: KPCA plus LDA. Pattern Recognition, 2004, 37(10): 2097-2100

17 Yang J, Frangi A F, Yang J Y, Zhang D, Jin Z. KPCA plus LDA: a complete kernel Fisher discriminant framework for feature extraction and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(2): 230-244

18 Xu Y, Zhang D, Jin Z, Li M, Yang J Y. A fast kernelbased nonlinear discriminant analysis for multi-class problems. Pattern Recognition, 2006, 39(6): 1026-1033

19 Zhao J, Wang H, Ren H, Kee S C. LBP discriminant analysis for face verification. In: Proceeding of the 2005 IEEE Conference on Computer Vision and Pattern Recognition. 2005, 3: 167

20 Liu C. Capitalize on dimensionality increasing techniques for improving face recognition grand challenge performance. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2006, 28(5): 725-737

21 Cevikalp H, Neamtu M, Wilkes M, Barkana A. Discriminative common vector method with kernels. IEEE Transactions on Neural Networks, 2006, 17(6): 1550-1565

22 Bach F R, Jordan MI. Kernel independent component analysis. Journal of Machine Learning Research, 2002, 3(1): 1-48

23 Yang J, Gao X, Zhang D, Yang J Y. Kernel ICA: an alternative formulation and its application to face recognition. Pattern Recognition, 2005, 38(10): 1784-1787

24 Ma J. Function replacement vs. kernel trick. Neurocomputing, 2003, 50: 479-483

25 Ma J, Theiler J, Perkins S. Two realizations of a general feature extraction framework. Pattern Recognition, 2004, 37(5): 875-887

26 Lodhi H, Saunders C, Shawe-Taylor J, Cristianini N, Watkins C. Text classification using string kernels. Journal of Machine Learning Research, 2002, 2(2): 419-444

27 Chen W S, Yuen P C, Huang J, Lai J. Wavelet kernel construction for kernel discriminant analysis on face recognition. In: Proceeding of the 2006 Conference on Computer Vision and Pattern Recognition Workshop. 2006, 47-52

28 Schölkopf B. Support vector learning. Dissertation for the Doctoral Degree. Berlin: Berlin Technical University, 1997

29 Camps-Valls G, Martin-Guerrero J, Rojo-Alvarez J, Soria-Olivas E. Fuzzy sigmoid kernel for support vector classifiers. Neurocomputing, 2004, 62: 501-506

30 Ahonen T, Hadid A, Pietikäinen M. Face description with local binary patterns: application to face recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2006, 28(12): 2037-2041

31 Mangasarian L. Generalized support vector machines. In: Smola A J, Bartlett P, Schokopf B, Schuurmans D, eds. Advances in Large Margin Classifiers. 2000, 135-146

32 Golub G H, Van Loan C F. Matrix Computations. 3rd ed. Baltimore: The Johns Hopkins University Press, 1996

33 Swets D L,Weng J. Using discriminant eigenfeatures for image retrieval. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1996, 18(8): 831-836

34 Belhumeur P N, Hespanha J P, Kriengman D J. Eigenfaces vs. Fisherfaces: Recognition using class specific linear projection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1997, 19(7): 711-720

35 Yang J, Yang J Y. Why can LDA be performed in PCA transformed space? Pattern Recognition, 2003, 36(2): 563-566

36 Liu C J, Wechsler H. Robust coding schemes for indexing and retrieval from large face databases. IEEE Transactions on Image Processing, 2000, 9(1): 132-137

37 Turk M, Pentland A. Eigenfaces for recognition. Journal of Cognitive Neuroscience, 1991, 3(1): 71-86

38 Liu K, Cheng Y Q, Yang J Y. Algebraic feature extraction for image recognition based on an optimal discriminant criterion. Pattern Recognition, 1993, 26(6): 903-911

39 Yang J, Zhang D, Frangi A F, Yang J Y. Two-dimensional PCA: A new approach to appearance-based face representation and recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2004, 26(1): 131-137

40 Visani M, Garcia C, Laurent C. Comparing robustness of two-dimensional PCA and eigenfaces for face recognition. Lecture Notes in Computer Science, 2004, 3212: 717-724

41 Zuo W M, Zhang D. K. Wang K. An assembled matrix distance metric for 2DPCA-based image recognition. Pattern Recognition Letters, 2006, 27(3): 210-216

42 Zhang D Q, Zhou Z H. (2D)(2)PCA: Two-directional twodimensional PCA for efficient face representation and recognition. Neurocomputing, 2005, 69(1-3): 224-231

43 Kong H,Wang L, Teoh E K, Li X, Wang J G, Venkateswarlu R. Generalized 2D principal component analysis for face image representation and recognition. Neural Networks, 2005, 18(5-6): 585-594

44 Ye J. Generalized low rank approximations of matrices. Machine Learning, 2005, 61(1-3): 167-191

45 Hyvärinen A, Oja E. Independent component analysis: algorithms and applications. Neural Networks, 2000, 13(4-5): 411-430

46 He X, Yan S, Hu Y, Niyogi P, Zhang H J. Face recognition using Laplacianfaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005, 27(3): 328-340

47 Ye J, Janardan R, Li Q. Two-dimensional linear discriminant analysis. Advances in Neural Information Processing Systems, 2004, 17: 1569-1576

48 Li M, Yuan B. 2D-LDA: a novel statistical linear discriminant analysis for image matrix. Pattern Recognition Letters, 2005, 26(5): 527-532

49 Xiong H, Swamy M, Ahmad M. Two-dimensional FLD for face recognition. Pattern Recognition, 2005, 38(7): 1121-1124

50 Yang J, Zhang D, Yong X, Yang J. Two-dimensional linear discriminant transform for face recognition. Pattern Recognition, 2005, 38(7): 1125-1129

51 Zuo W, Zhang D, Yang J, Wang K. BDPCA plus LDA: A novel fast feature extraction technique for face recognition. IEEE Transactions on Systems, Man, and Cybernetics, Part B, 2006, 36(4): 946-953

52 Zou C, Sun N, Ji Z, Zhao L. 2DCCA: a novel method for small sample size face recognition. In: Proceedings of IEEE Workshop on Applications of Computer Vision. 2007, 43

53 Lee S H, Choi S. Two-dimensional canonical correlation analysis. IEEE Signal Processing Letters, 2007, 14(10): 735-738

54 Gao Q, Zhang L, Zhang D, Xu H. Independent components extraction from image matrix. Pattern Recognition Letters, 2010, 31(3): 171-178

55 Chen S B, Zhao H F, Kong M, Luo B. 2d-lpp: a twodimensional extension of locality preserving projections. Neurocomputing, 2007, 70(4-6): 912-921

56 Hu D W, Feng G Y, Zhou Z T. Two-dimensional locality preserving projections (2DLPP) with its application to palmprint recognition. Pattern Recognition, 2007, 40(1): 339-342

57 Xu D, Yan S, Zhang L, Lin S, Zhang H J, Huang T S. Reconstruction and recognition of tensor-based objects with concurrent subspaces analysis. IEEE Transactions on Circuits and Systems for Video Technology, 2008, 18(1): 36-47

58 Wang H, Ahuja N. A tensor approximation approach to dimensionality reduction. Journal of Computer Vision, 2008, 76(3): 217-229

59 Tao D, Li X, Wu X, Maybank S J. General tensor discriminant analysis and gabor features for gait recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2007, 29(10): 1700-1715

60 Lu H, Plataniotis K N, Venetsanopoulos A N. MPCA: multilinear principal component analysis of tensor objects. IEEE Transactions on Neural Networks, 2008, 19(1): 18-39

61 Zhang L, Gao Q, Zhang D. Directional independent component analysis with tensor representation. In: Proceedings of Computer Vision and Pattern Recognition. 2008, 1-7

62 Sun N, Wang H, Ji Z, Zou C, Zhao L. An efficient algorithm for kernel two-dimensional principal component analysis. Neural Computing & Applications, 2008, 17(1): 59-64

63 Liu J, Chen S, Zhou Z H, Tan X. Generalized low-rank approximations of matrices revisited. IEEE Transactions on Neural Networks, 2010, 21(4): 621-632