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1 Introduction of Self-Organizing Map * 1 Ver ( ) *1 furukawa@brain.kyutech.ac.jp

2

3 i SOM SOM SOM SOM SOM

4

5 1 1 (Self-Organizing Map: SOM) T. Kohonen SOM (topology preserving mapping) [1, 2, 3] SOM 2 (topographic map) SOM SOM (self-organization) Kohonen SOM SOM (1) Winner-Take-All (2) (3) Hebb [4] EM E M [5] SOM (Principal Component Analysis: PCA) (Multidimensional Scaling: MDS) (minifold learning) SOM (Vector Quantization: VQ) SOM (latent variable model) (Generative Topographic Mapping: GTM)[6] (Gaussian Process Latent Variable Model: GPLVM)[7] SOM

6 SOM / *1 SOM topographic map, *1 [3] p.170 /

7 MDS SOM *2 1.1 U-matrix U-matrix U-matix?? (MDS) 1.2 *2 magnification factor

8 4 1 MDS SOM MDS 1.2 SOM 1980 [1] SOM (V1) V1 (retinotopy) [8] V1 [9] Kohonen Type-1 Type-2 [10] Kohonen SOM Type-2 SOM Self-organizing feature map (SOFM) Kohonen s map SOM SOM Winner-Take-All Hebb SOM SOM SOM SOM SOM [11]

9 1.3 5 [12, 13] 1990 SOM [14, 15, 5] SOM SOM SOM Self-Organizing Map *3 Self-Organizing Feature Map (SOFM) Kohonen map/net SOM 1.3 SOM SOM (observable space) (latent space) *4 (node) SOM SOM SOM X Z SOM f : Z X X = (x 1,..., x N ) x n f (z n ) f Z = (z 1,..., z N ) SOM k ζ k Z y k X {ζ 1,..., ζ K } f y k = f (ζ k ) f SOM f y k *3 Map (topographic map) (mathematical map) *4 SOM

10 SOM x n (Best Matching Unit: BMU) Heykin SOM (Competitive process), (Cooperative process), (Adaptive process) [4] Step 0 ( ): y k (0) t = 1 Step 1 Step 1 ( ): x n (n = 1,..., N) (Best Matching Unit: BMU) BMU kn x n n, kn(t) = arg min x n y k (t 1) 2 (1.1) k z n (t) BMU ζ k n (t) Step 2 ( ): R kn n, k R kn (t) = h(z n (t), ζ k ; σ(t)) (1.2) h(z, ζ) BMU [ h(z, ζ; σ) = exp 1 ] z ζ 2 2σ2 (1.3) σ(t) σ(t) t

11 1.3 7 Step 3 ( ): k, y k (t) = N n=1 R kn(t)x n N n =1 R kn (t) (1.4) t := t + 1 Step 1 z n SOM EM (Expectation Maximization algorithm: EM algorithm) [5] E M E, M X = (x 1,..., x N ) R D N *5 Y = (y 1,..., y K ) R D K B = ( ) δ k,k n δ kk H = (H kk ) R K K H kk = h(ζ k, ζ k ; σ(t)) (1.5) R R = (R kn ) = HB (1.6) G k = N n=1 R kn G 1 0 G = diag(g k ) =... 0 G K (1.7) Y T = G 1 RX = G 1 HBX T (1.8) SOM PCA z n (Step 2) PCA *5 D N N D X T

12 σ(t) t l σ 0 l/2 σ(t) σ min σ min σ min σ min l/10 τ τ σ(t) = max [σ 0 (1 t/τ), σ min ] (1.9) σ(t) = max [ σ 0 exp ( t/τ), σ min ] (1.10) max[a, b] a, b τ 1.4 [16] component plane U-matrix Component plane D d y kd component plane U-matrix [17] 1.1 U-matrix 1.5 SOM F = N n=1 k=1 K h(z n, ζ k ) x n y k 2 (1.11) F Y = (y k ) U = (u n ) EM [5] Y = (y k ) F z n z n = arg min ζ k K h(ζ k, ζ k ) x n y k 2 k =1

13 1.6 SOM 9 K h(ζ k, ζ k ) x n y k 2 x n y k 2 k =1 SOM (1.1) Z = (z n ) F y k F y k = N h(z n, ζ k )(x n y k ) = 0 (1.12) n=1 N n=1 y k = h(z n, ζ k )x n N n =1 h(z n, ζ k ) (1.13) (1.4) 1.6 SOM PCA MDS SOM (Principal curve) (Elastic net) Isometric feature mapping: ISOMAP Locally Linear Embedding (LLE) SOM SOM k-means Fuzzy k-means (Neural Gas: NG) SOM SOM Growing NG Evolving SOM (Gaussian Mixture Model: GMM) (Factor Analysis: FA) (Generative Topographic Mapping: GTM) [6] (Gaussian Process Latent Variable Model: GPLVM) [7, 18] SOM

14 SOM SOM PAK [19] MATLAB SOM Toolbox [20] SOM U-matrix [2] [3] [11] SOM [12, 13]

15 11 2 SOM SOM (1) SOM (2) (3) (4) SOM (5) (6) SOM (7) SOM SOM 2 SOM SOM

16 12 2 SOM 1 SOM SOM SOM 1 SOM 2 2 SOM SOM SOM BMI SOM kg SOM 1

17 SOM 100m SOM SOM SOM SOM (outlier) t SOM σ 0 σ mi τ

18 14 2 SOM σ 0 σ min σ min τ τ σ 0, σ min, τ SOM (1.11) (1) (2) τ (3) (1), (2) (1) (2) τ (3) (1.11) F F SOM

19 2.2 SOM PCA 3 PCA X R D N (Singular Value Decomposition: SVD) X = UΣV T (2.1) U R D I, V R N I Σ R I I σ 1,..., σ I σ i σ 1 σ 2 σ I U = (u 1,..., u I ) u i i U 123 = (u 1, u 2, u 3 ) 3 x 123 = U T 123 x (2.2) U 124 U Ẑ ΣV T, Ẑ = (ẑ 1,..., ẑ N ) ẑ n z n PCA 2.2.3

20 16 2 SOM SOM 1 PCA SOM (1.11) F PCA MDS 2.3 SOM SOM SOM

21 17 3 SOM SOM 3.1 SOM SOM N K D N K D exp N K SOM O(NKD) N D N 3.2 N 1 ND O(NKD) Algorithm 1 O(NKD) N K Algorithm 2 O(K 2 D) *1 Y T = G 1 (HB)X T Y T = G 1 H(BX T ) B 0 1 BX T 0 3σ 0 2σ *1 N > K O(K 2 D) N < K O(NKD) N K

22 18 3 SOM Algorithm 1 Cooperative and Adaptive process (1) for n = 1 to N do for k = 1 to K do R kn neighborhood(k n, k, σ) end for end for for k = 1 to K do G k N n=1 R kn for d = 1 to D do Y kd N n=1 R knx kd Y kd Y kd /G k end for end for Algorithm 2 Cooperative and Adaptive process (2) for k = 1 to K do for k = 1 to K do H kk neighborhood(k, k, σ) end for end for k, ŷ k 0, N k 0, G k 0 for n = 1 to N do ŷ k n ŷ k n + x n N k n N k n + 1 end for for k = 1 to K do for k = 1 to K do if N k 0 then G k G k + N k R kk y k y k + H kk ŷ k end if end for y k y k /G k end for 1

23 D SOM O(NKD) (1) K (2) N (3) D K SOM K σ 2σ * D D L 2 L 1 D *3 SOM N N < N *2 *3 Tensor SOM D

24 20 3 SOM N Y kd (t + 1) = R kn X n d (3.1) X n d N Y = G 1 HBX N Y kd (t + 1) = (1 ε)y kd (t) + ε R kn X n d (3.2) N n =1 n =1 3.4 τ PCA PCA N 3.5 PC Tensor SOM PC

25 21 [1] T. Kohonen, Self-organized formation of topologically correct feature maps, Biological Cybernetics 43 (1) (1982) [2] T. Kohonen, Self-Organizing Maps, Springer-Verlag, Berlin Heidelberg, [3] T.,,, [4] S. Haykin, Neural Networks and Learning Machines, Prentice Hall, 2009, Ch. 9, pp [5] J. Verbeek, N. Vlassis, B. Krose, Self-organizing mixture models, Neurocomputing 63 (2005) doi: /j.neucom [6] C. M. Bishop, Pattern Recognition and Machine Learning, Springer, 2006, Ch. 12, pp [7] N. D. Lawrence, Gaussian process latent variable models for visualisation of high dimensional data, in: Advances in Neural Information Processing Systems 16 [Neural Information Processing Systems, NIPS 2003, December 8-13, 2003, Vancouver and Whistler, British Columbia, Canada], 2003, pp URL [8] S.-I. Amari, Topographic organization of nerve fields, Bulletin of Mathematical Biology 42 (3) (1980) doi: /bf [9] C. von der Malsburg, Self-organization of orientation sensitive cells in the striate cortex, Kybernetik 14 (2) (1973) doi: /bf [10] T. Kohonen, Self-organizing neural projections, Neural Networks 19 (6 7) (2006) [11] T. Kohonen, Essentials of the self-organizing map, Neural Networks 37 (2013) [12] S. Kaski, J. Kangas, T. Kohonen, Bibliopgraphy of self-organizing map (SOM) papers: , Neural Computing Surveys 1 (1998) [13] M. Oja, S. Kaski, T. Kohonen, Bibliopgraphy of self-organizing map (SOM) papers: addendum, Neural Computing Surveys 3 (2003) [14] C. M. Bishop, M. Svensen, C. K. I. Williams, GTM: The generative topographic mapping, Neural Computation 10 (1998) [15] C. M. Bishop, M. Svensen, C. K. I. Williams, Developments of the generative topographic mapping., Neurocomputing 21 (1-3) (1998) [16] P. Stefanovic, O. Kurasova, Visual analysis of self-organzing maps, Nonlinear Analysis: Modelling and Control 16 (4) (2011) [17] A. Ultsch, H. P. Siemon, Kohonen s self organizing feature maps for exploratory data

26 22 analysis., in: Proc. INNC 90, Int. Neural Network Conf., 1990, pp [18] N. Lawrence, Probabilistic non-linear principal component analysis with gaussian process latent variable models, Journal of Machine Learning Research 6 (2005) 1783? [19] SOM PAK. URL lvq pak.shtml [20] SOM Toolbox. URL

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