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1 2008 :

2 () /24

3 [1] *1 1 [5] *1 () 2

4 l(x), ψ a,t (x), f(x) a, t (translation) (dilation) x y W (1), W (2) I, J, K x = {x i 1 < i < I} (1) x i = l(x i ) (2) h = W (1) x (3) ( ) h hj t j j = ψ a,t (h j ) = ψ (4) a j h = {h j 1 < j < J} (5) 3

5 y = W (2) h (6) y k = f(y k ) (7) y = {y k 1 < k < K} (8) * () (1) (2) a,t 2 (1) *2 0 MexicanHat ψ a,t (x) = (1 2X 2 )e X2, X = (x t)/a 5 a = 1 8 4

6 * XOR 10-bit tight encorders *4 (-1,1) (BP) (0.0,5.0) 5000epochs () Microsoft Windows XP on Intel R Mac with Bootcamp Intel R Core T M 2 Duo CPU 2.40GHz, 2GB RAM Microsoft Visual Studio 2008, C++ 1 ( s) (%) Sigmoid Haar MexicanHat XOR * i-j-k *4 N log 2 N encorders() N-bit tight encorders 5

7 ( s) (%) Sigmoid Haar MexicanHat bit tight encorders *5 ([1][3] *6 ) *7 () ([1][3] ) XOR * / * / () *7 6

8 3.2.2 *8 ( ) (-1,1) 1 () (encorders ) BP ( *9 ) ( * 10 ) () *8 *9 (4) a j *10 (4) t j 7

9 ( W (1) ) 2. (a j, t j ) 3. ( W (2) ) W (1) W (1) W (1) N N K-means a j t j / 1 1 0,1,2,3 4 {0,1,2,3} {0,1},{2,3} 8

10 2 {0},{1},{2},{3} W (2) (-0.5,0.5) W (2) W (1) W (2) 4.3 XOR 10-bit tight encorders () ( s) ( s) (%) Sigmoid Haar MexicanHat XOR ( s) ( s) (%) Haar MexicanHat XOR 9

11 ( s) ( s) (%) Sigmoid Haar MexicanHat bit tight encorders ( s) ( s) (%) Haar MexicanHat encorders 4.4 XOR Haar 7 K-means XOR MexicanHat encorders BP W (1) K-means K-means XOR encorders [0.0,1.0] W (1) [0.0,1.0] K-means 10

12 (1)W (1) (2) (1) ( ) (2) K-means tanh Gabor MexicanHat tanh [, ] 0 [S 0, S 1 ] tanh (9) * 11 ψ(x)dx = 0 (9) *11 ψ(ω) 2 < ω (9) 11

13 5.1.3 (10) W f ψ (t, a) = 1 ( ) i t ψ f(i) (10) a a a t 1 a a i 6 1 [5]-[7] [8] x, y w, v θ = (w 1,..., w m, v 1,..., v n ) (11) y = f(x, θ) = v j ψ(w i x i ) (12) i,j ε y = f(x, θ) + ε (13) 12

14 2 [8] ε x, θ y x q(x) p(y, x, θ) = q(x) 1 2 e 1 2 (y f(x,θ))2 (14) θ N θ E[(ˆθ θ)(ˆθ θ)] T 1 N G 1 (θ) (15) [ l(y, x, θ) l(y, x, θ) T ] G(θ) = E (16) θ θ 13

15 = plateau 3 [8] 4 [8] G(θ) 1 l η θ θ t+1 = θ t ηg 1 (θ t ) l(y t, x t, θ t ) θ t (17) G 1 (1) 0 (2) * 12 G 1 7 (1) (2) Haar 5 MexicanHat 6 *12 tanh 14

16 1 0 Haar 0 * 13 * 14 5 Haar 6 MexicanHat xor * 15 2 * 16 xor [0.0, 3.0] 30 * *13 *14 *15 z t+1 = f(x t, y t, z t ) z t+1 [0.1] *16 y = (x2 1 x2 2 )sin(10x 1x 2 )+1 2, x 1, x 2 [ 1.1] *

17 Amari [8] (1)1 0 (2)1 2 (1) (2) Haar MexicanHat 16

18 Haar (2) MexicanHat (2) xor / % (26/30) XOR * *

19 1 (1) 0 (1) 12 Haar / % (1/30) - - Haar XOR Haar [0,1] * *

20 13 14 MexicanHat % (30/30) MexicanHat XOR % (26/30) /2 (2)

21 MexicanHat % (27/30) MexicanHat 20

22

23 % (0/30) MexicanHat % (30/30) MexicanHat

24 % 23

25 [9] [1] Qinghua Zhang, Using Wavelet Network in Nonparametric Estimation, IEEE Transaction on Neural Networks, vol.8, Issue 2, 1997, pp [2] Kang Li, Jian-Xun Peng, Neural input selection A fast model-based approach, Neurocomputing 70, 2007, pp [3] Cheng-Jian Lin, Wavelet Neural Networks with a hybrid Learning Approach, JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 22, 2006, pp [4] Rui Xu, Survey of Clustering Algorithms, IEEE Transaction on Neural Networks, vol.16, Issue 3, 2005, pp [5] Shun-ichi Amari, Natural Gradient Works Efficiently in Learning, Neural Computation 10, 1998 [6] Shun-ishi Amari, Hyeyoung Park, Tomoko Ozeki, Singularities Affect Dynamics of Learning in Neuromanifolds, Neural Computation 18, 2006 [7] Florent Cousseau, Tomoko Ozeki, Shun-ichi Amari, Dynamics of Learning in Multilayer PerceptronsNear Singularities, IEEE Transactions on Neural Networks, vol.19, no.8, 2008 [8], I, //, vol.49, no.8, 2005 [9] A. Waibel, T. Hanazawa, G. Hinton, K. Shikano, and K. Lang, Phoneme recognition using time-delay neural networks, IEEE Trans. Acoustics, Speech, Signal Processing, vol. 37, pp ,

4. C i k = 2 k-means C 1 i, C 2 i 5. C i x i p [ f(θ i ; x) = (2π) p 2 Vi 1 2 exp (x µ ] i) t V 1 i (x µ i ) 2 BIC BIC = 2 log L( ˆθ i ; x i C i ) + q

4. C i k = 2 k-means C 1 i, C 2 i 5. C i x i p [ f(θ i ; x) = (2π) p 2 Vi 1 2 exp (x µ ] i) t V 1 i (x µ i ) 2 BIC BIC = 2 log L( ˆθ i ; x i C i ) + q x-means 1 2 2 x-means, x-means k-means Bayesian Information Criterion BIC Watershed x-means Moving Object Extraction Using the Number of Clusters Determined by X-means Clustering Naoki Kubo, 1 Kousuke

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