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1 SVM Boosting c /(19)

2 AD c /(19)

3 1-4 c /(19)

4 Principal Component Analysis PCA 2) KL Karhunen-Lo ève expansion PCA PCA d x = (x 1, x 2,, x d ) T T c (< d) y = (y 1, y 2,, y c ) T y = A T x d c A A A T A = I y Ay x 2 A n x 1, x 2,, x n y 1, y 2,, y n σ 2 σ 2 = 1 n n (y i 1 n n y j ) T (y i 1 n j=1 n y j ) j=1 (1 1) y i = A T x i m = 1 nj=1 n x j σ 2 = = 1 n 1 n n {A T (x i m)} T {A T (x i m)} (1 2) n Tr[{A T (x i m)}{a T (x i m)} T ] = Tr[A T { 1 n (1 3) n (x i m)(x i m) T }A] (1 4) = Tr(A T SA) (1 5) Tr( ) S x i 1 n n (x i m)(x i m) T A T A = I L(A, Λ) = σ 2 Tr{(A T A I)Λ} = Tr(A T SA) Tr{(A T A I)Λ} (1 6) Λ c (1 6) A 2SA 2AΛ 0 SA = AΛ d d S A Λ A Λ σ 2 Tr(A T SA ) = Tr(A T A Λ ) = Tr(Λ ) Λ S {λ 1, λ 2,, λ c } c A {u 1, u 2,, u c } A T A = I c /(19)

5 1 c Tr(Λ ) = c λ i ( c λ i )/( d λ i) 0.9 c PCA 1 2 PCA 1 u A PCA B 1 2 PCA d n S n 1 d d PCA X = (x 1 m, x 2 m,..., x n m) T S = 1 n XT X SA = AΛ nx (XX T )(XA) = (XA)(nΛ) n n XX T n < d d d S 2) XX T A = XA Λ = nλ X T A = X T XA = nsa = naλ = AΛ A = Λ 1 X T A A A T A = I A = Λ 1 2 X T A A A T A = I PCA 2, PCA 35) 3-4 SVM φ(x) φ(x) PCA K(x, y) = φ(x) T φ(y) φ(x) c /(19)

6 Rosenblatt 29) S A R 3 S A A R A R (x 1, x 2,..., x d ) T T z = f (w 0 + d w ix i ) {w 1, w 2,..., w d } w 0 f ( ) f ( ) a 0 f (a) = 1 f (a) = 0 f (a) = 1/(1 + e a ) f (a) = a x = (1, x 1, x 2,..., x d ) T w = (w 0, w 1, w 2,..., w d ) T z = f (w T x) x 1 x 2 x d w 1 w 2 w d w 0 f ( ) z n {x 1, x 2,..., x n } {z 1, z 2,..., z n } 2 J = 1 n 2 {z i f (w T x i )} 2 w f ( ) f (a) = a w ρ w w + ρ J n w = w + ρ (z i w T x i )x i (1 7) Widrow-Hoff f ( ) (1 7) Rosenblatt X = (x 1, x 2,..., x n ) T z = (z 1, z 2,..., z n ) T f ( ) J = 1 2 (z wt X T ) T (z w T X T ) J/ w = 0 J w w = (X T X) 1 X T z (1 8) c /(19)

7 Linear Discriminant Analysis LDA 2 LDA d x = (x 1, x 2,..., x d ) T d A = (a 1, a 2,..., a d ) T z = A T x 2 A z σ 2 W σ2 B σ 2 B /σ2 W A 7) i i = 1, 2 C i n i z = A T x m i = 1 n i x C i A T x m = (n 1 m 1 + n 2 m 2 )/(n 1 + n 2 ) σ 2 W = { 1 (A T x m i ) 2 }, σ 2 B n = (m i m) 2 (1 9) i x C i,2,2 x m i = 1 n i x C i x m = (n 1 m 1 + n 2 m 2 )/(n 1 + n 2 ) S W S B S W = { 1 (x m i )(x m i ) T }, S B = (m i m)(m i m) T (1 10) n i x C i,2,2 m i = A T m i m = A T m A T S W A = σ 2 W AT S B A = σ 2 B σ 2 B /σ2 W A σ 2 W = AT S W A = 1 σ 2 B = AT S B A L(A, λ) = A T S B A λ(a T S W A 1) (1 11) A 0 (S B + S T B )A λ(s W + S T W )A = 0 S B A = λs W A S W (S W 1 S B )A = λa (1 12) λ S 1 W S B σ 2 B /σ2 W = λ A K ( 2) A d (K 1) z = A T x 2 z S ZW S ZB Tr(S 1 ZW S ZB ) A 2 S 1 W S B K 1 λ i u i (i = 1, 2,..., K 1) Tr(S 1 ZW S ZB ) K 1 λ i A = (u 1, u 2,..., u K 1 ) c /(19)

8 McCulloch-Pitts 28) Hebb 17) 30) error backpropagation algorithm, BP 31) Self Organizing Maps, SOM 23) SOM c /(19)

9 2 14) Hopfield 19) Hopfield 18) Hopfield T T T simulated annealing Hopfield 20) 15) SVM Support Vector Machines, SVM Vapnik 37) SVM 3) SVM 2 SVM d (x 1, x 2,..., x d ) T T y { 1, 1} f (x) d w b f (x) = w T x + b n x i y i y i f (x i ) = y i (w T x i + b) 1, i = 1, 2,..., n (1 13) w b f (x) SVM 1 4 x f (x) = 0 f (x) / w T w (1 13) f (x) 1 1/ w T w c /(19)

10 w T w (1 13) w T w α i 0, i = 1, 2,..., n L(w, b, α) = 1 n 2 wt w α i {y i f (x i ) 1} (1 14) α i α i > 0 α i 0 w = n y i α i x i (1 13) ξ i 0 y i f (x i ) 1 ξ i, i = 1, 2,..., n (1 15) 1 2 wt w + c n ξ i c ξ i = 0 α i < c SVM SVM φ(x) φ( ) SVM K(x, z) = φ(x) T φ(z) K(x, z) = (x T z + β) γ RBF Radial Basis Function K(x, z) = e β(x z)t (x z) K(x, z) = tanh(βx T z + γ) c /(19)

11 1 5 (a), (b) (a) X T X 1 X T T h i (x), (i = 1 T) T 2 1 5(b) i D(i) AdaBoost 8) X 1 h 1 (x) h 1 (x) X 1 X 2 X 2 h 2 (x) X 2 X 3 c /(19)

12 T T m (x 1, y 1 ),..., (x m, y m ) AdaBoost x i i y i {+1, 1} x i step0 D 1 (i) D 1 (i) = 1/m step1 For t = 1,..., T : 1-1 X t ɛ t = i:y i h t(x i) D t (i) h t (x) 1-2 h t (x) α t = 1 1 ɛt 2 ln( ) ɛ t 1-3 i D t (i) D t+1 (i) = D ( t(i) exp( αt ) if h t (x i ) = y i Z t exp( α t ) if h t (x i ) y i = D t(i)exp( α t y i h t (x i )) Z t Z t Z t = m D t (i)exp( α t y i h t (x i )) Step2 Step1 T α t g(x) = T t=1 α t h t (x) H(x) = sign(g(x)) {+1, 1} AdaBoost 8) T T AdaBoost Real AdaBoost 33) MadaBoost 4) LogitBoost 5) AdaBoost 2 AdaBoost.M1 AdaBoost.M2 9) AdaBoost 38) 2 AdaBoost Haar-like Haar-like Haar-like Haar-like AdaBoost c /(19)

13 Haar-like T Haar-like AdaBoost , 30 13) d d d d c /(19)

14 27) 2 1 7(a). (a) (b) (b) M P N Q M N N. 1 θ 1 2 θ 2 θ 1, 3 θ 3 2 θ 2. N. 11) 4 c /(19)

15 39) 24) 21, 22) 11) 24) 26, 36) 32) 11) 3 24, 13, 11) 24, 13) 2006 S ubspace c /(19)

16 ) Bayes Resubstitution method Hold-out method Hold-out method, H 6) Cross validation method CV 1. K 2. K K 1 3. K K K K-Fold Cross- Validation 1 Leaveone-outmethod 25) CV c /(19)

17 CV K Bootstrap method BS 6) 1. n X 0 n X B(n) 2. X B(n) h B(n) (x) X B(n) e B(n) 3. h B(n) (x) X 0 e B 4. e B(n) e B δ = e B e B(n) 5. B B δ δ 6. R B R B = R 0 + δ R 0 X 0 X 0 B 200 1),,,, 6,, ) C.M. Bishop Pattern Recognition and Machine Learning, Springer, C.M.,, ) N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernelbased Learning Methods, Cambridge University Press, N. J.S.,,, ) C. Domingo, O. Watanabe, MadaBoost: A modification of AdaBoost, COLT 00, pp , ) Jerome Friedman, Trevor Hastie, Robert Tibshirani, Aditive logistic regression: A statistical view of boosting, The Annals of Statistic, vol.28, no.2, pp , ) B. Efron, Bootstrap Methods: Another Look at the Jackknife, Annals of Statistics, vol.7, no.1, pp.1-26, ) R.A. Fisher, The use of multiple measurement in taxonmic problems, Annals of Eugenics 7, c /(19)

18 8) Yoav Freund, Robert E. Schapire, A decision-theoretic generalization of on-line learning and an application to boosting, Proceedings of the Second European Conference on Computational Learning Theory table of contents, Lecture Notes in Computer Science, vol.904, pp.23-37, ) Yoav Freund, Robert E. Schapire, Experiments with a New Boosting Algorithm, Proceedings of the Thirteenth International Conference on Machine Learning, pp , ),,, < >,, vol.14, no.5, pp , ),,,, vol.46, no.sig 15 CVIM 12, pp.21-34, ) Kazuhiro Fukui, Osamu Yamaguchi, The Kernel Orthogonal Mutual Subspace Method and Its Application to 3D Object Recognition, ACCV2007, pp , ), : :,, vol.49, no.6, pp , ) K. Fukushima, Neocognitron: A Self-Organizing Neural Network Model for a Mechanism of Pattern Recognition Unaffected by Shift in Position, Biol. Cybern., vol.36, no.4, pp , ) S. Geman and D. Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Trans. Pattern Analysis and Machine Intelligence, vol.6, no.6, pp , ),,, ) D.O. Hebb, The Organization of Behavior, Willey, ) J. Hinton and T. Sejnowski, Learning and Relearning in Boltzmann Machines, in D.E. Rumelhart, J.L. McClelland and the PDP Research Group eds., Parallel Distributed Processing, vol.1, pp , The MIT Press, ) J.J. Hopfield, Neural Networks and Physical Systems with Emergent Collective Computational Abilities, Proc. of the National Academy of Sciences of the USA, vol.79, no.8 pp , ) J.J. Hopfield and D.W. Tank, Neural Computation of Decisions in Optimization Problems, Biol. Cybern., vol.52, no.3, pp , ),,,, CVIM-151, pp.17-24, ) Tomokazu Kawahara, Masashi Nishiyama, Tatsuo Kozakaya, Oamu Yamaguchi, Face Recognition based on Whitening Transformation of Distribution of Subspaces, ACCV Workshop Subspace2007, pp , ) T. Kohonen, Self-Organizing Maps, Springer-Verlag, ), :,, vol.49, no.5, pp , ) Peter A. Lachenbruch and M. Ray Mickey, Estimation of Error Rates in Discriminant Analysis, Technometrics, vol.10, no.1, pp.1-11, ),, (D-II), vol.j82-d-ii, no.4, pp , ),, (D), vol.j68-d, no.3, pp , ) W.S. McCulloch and W. Pitts, A Logical Calculus of the Ideas Immanent in Nervous Activity, Bulletin of Mathematical Biophysics, vol.5, pp , ) F. Rosenblatt, The Perceptron : A Probabilistic Mode for Information Storage and Organization in the Brain, Psychological Review, vol.65, no.6, pp , ) D.E. Rumelhart, G.E. Hinton and R.J. Williams, Learning Internal Representations by Error Propagation, in D.E. Rumelhart, J.L. McClelland and the PDP Research Group eds., Parallel Distributed c /(19)

19 Processing, vol.1, pp , The MIT Press, ) D.E. Rumelhart, G.E. Hinton and R.J. Williams, Learning Representations by Back-Propagating Errors, Nature, vol.323, pp , ),, (D-II), vol.j84-d-ii, no.8, pp , ) R.E. Schapire, Y. Singer, Improved Boosting Algorithms Using Confidencerated Predictions, Machine Learning, pp , ) B. Schölkopf, J.C. Platt, J. Shawe-Taylor, A.J. Smola and R.C. Williamson, Estimating the Support of High-Dimensional Distribution, Neural Computation, vol.13, no.7, pp , ) B. Schölkopf, A.J. Smola and K.R. Müller, Nonlinear principal component analysys as a kernel eigenvalue problem, Neural Computation, vol.10, no.5, pp , ),, (D-II), vol.j82-d-ii, no.4, pp , ) V.N. Vapnik, The Nature of Statistical Learning Theory, Springer, ) P. Viola, Michael and J. Jones, Robust real-time face detection, IJCV Vol.57, no.2, pp , ) Y. Washizawa, K. Hikida, T. Tanaka, Y. Yamashita, Kernel Relative Principal Component Analysis for Pattern Recognition, Proc. of Joint IAPR Iinternational Workshops on Syntactical and Structural Pattern Recognition and Statistical Pattern Recognition SSPR/SPR 2004, pp , Lisbon, ), < 1 > Level Set,Graph Cut,Particle Filter,Tensor,AdaBoost CVIM, 5, c /(19)

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