Fuzzy Multiple Discrimminant Analysis (FMDA) 5) (SOM) 6) SOM 3 6) SOM SOM SOM SOM SOM SOM 7) 8) SOM SOM SOM GPU 2. n k f(x) m g(x) (1) 12) { min(max)
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1 SOM (SOM: Self-Organizing Maps) 3 SOM SOM SOM SOM GPU A Study on Visualization of Pareto Solutions by Spherical Self-Organizing Maps MASATO YOSHIMI, 1 KANAME NISHIMOTO, 2 LUYI WANG, 2 TOMOYUKI HIROYASU 3 and MITSUNORI MIKI 1 In this paper, Self-Organizing Maps (SOM) which is one of neural network systems was applied to illustrate the Pareto solution set which is derived by multi-objective optimization. Especially, this paper described that Spherical SOM is effective for illustrating the Pareto solution set. These discussions have been performed through the real world problem. At the same time, the implementation and parallel method of Spherical SOM by GPUs were also discussed. 1 Faculty of Science and Engineering, Doshisha University 2 Faculty of Engineering, Doshisha University 3 Faculty of department of life and medical science, Doshisha University 1. Computer Aided Design(CAD) Finite Element Method(FEM) 1) 2) 4) c2010 Information
2 Fuzzy Multiple Discrimminant Analysis (FMDA) 5) (SOM) 6) SOM 3 6) SOM SOM SOM SOM SOM SOM 7) 8) SOM SOM SOM GPU 2. n k f(x) m g(x) (1) 12) { min(max) f i(x 1, x 2,..., x n) (i = 1, 2,..., k) subject to g j (x 1, x 2,..., x n ) 0 (j = 1, 2,..., m) f i (x) 1 (1) 13) x 1, x 2 I(x= (x 1, x 2,..., x n)) (a) f i (x 1 ) f i (x 2 )( i = 1, 2,..., k) x 1 x 2 (b) f i (x 1 ) < f i (x 2 )( i = 1, 2,..., k) x 1 x 2 x 1 x 2 x 1 x 2 x 0 I(x= (x 1, x 2,..., x n )) (a) x 0 x I x 0 Weak Pareto-optimal solution (b) x 0 x I x 0 Pareto-optimal solution (1) 2 k = 2 Feasible Region 3. SOM 3.1 SOM SOM SOM SOM SOM SOM i n t m i (t) x(t) 2 c2010 Information
3 f (x) 2 Feasible region Weak pareto-optimal solutions Pareto-optimal solutions 1 f (x) 1 α(t) h ci(t) σ(t) SOM ( 1 ) t 0 m i (0) ( 2 ) x(t) m i (t) m i (t) x(t) ( 3 ) c x(t) = min m i (t) x(t) c ( 4 ) c d ci (2) (2) h ci(t) α(t) (3) α(t) σ(t) (4) (5) m i (t + 1) = m i (t) + h ci (t)(x(t) m i (t)) (2) { 0 (d ci > σ(t)) h ci(t) = (3) α(t) (d ci σ(t)) α(t) = α(0)(t t)/t (4) σ(t) = σ(0)(t t)/t (5) ( 5 ) t < T t t + 1 (2) t = T 4. SOM 4.1 SOM SOM SOM 3 SOM 3 SOM SOM SOM c2010 Information
4 SOM SOM a spherical surface U development V U' 2 3D V' U' rectangular coordinates put neurons on the apices 3D SOM SOM V' 2 2 SOM 3 4 c2010 Information
5 h d i map e j f k m n o p q r s a b c g l d 3 e tree j f k o n i a b c g l p s r not-learned neurons learn and add tree pre-learned neurons adjacent neurons e HSV (Saturation) (Value) (Hue) [0, 360] [0, 280] 5. SOM GPU 5.1 SOM SOM 1 3 SOM 1987 Carpenter SOM 1992 Speckmann SOM SIMD COKOS(COprocessor for KOhonen s Self-organizing map) COKOS 8 SOM SOM FPGA Porrmann j k p s r n f k o n i 11) Porrmann 6 SOM Intel Xeon 2.80GHz CPU GPU SOM 10) NVIDIA GeForce GTX280 CPU 150 SOM GPU GPU SOM GPU 5.2 SOM SOM (2) SOM GPU ) SOM SOM GPU 5.3 SOM SOM GPU 18) C++ NVIDIA CUDA Opteron 1210 HE(2.50GHz, Linux amd64, 5 c2010 Information
6 G , -O3) 1 3 GPU ( ) [0, 9] CPU SP ( ) (GHz) (GHz) GeForce8400GS GeForce GTX Tesla C GPU SOM SOM 10) SOM GPU 6. SOM 4 4 (SP) GPU 10 SP 5 SOM SP 8400GS 240 SP GPU 10 SP SP 240 GPU SOM. SOM SOM SOM 3) 3) SOM SOM 6 c2010 Information
7 6.1 SOM, 4. SOM )9) SOM 6.2 CO2 14) 16) (SFC) (NOX) NOX (Soot) SFC NOX Soot NOX SFC 6 2 Start Angle Exhaust gas recirculation Rate EGR RateSwirl Ratio Boost Pressure ) NOX (Soot) SFC NOX SFC f 1 NOX f 2 Soot f 3 x 1 x 2 x 3 x 4 x 5 x 6 x 7 ( ) x SOM SOM SOM SOM SOM PAK SOM PAK SOM SOM SOM c2010 Information
8 情報処理学会研究報告 となる ニューロン数 (個) 平面 SOM 球面 SOM 900 (30 30) 1000 (20 50) 表 3 SOM の設定 学習回数 (回) 学習率係数の初期値 p2: f1: f3: 近傍半径の初期値 p1: f1: f3: p3: f1: f3: 提案する球面 SOM によって得られた結果を図 7 に示す ここでは 90 度ずつ回転させ た様子を示している 球面上には 目的関数 f1 (燃料消費量) の値を表示している すなわ ち f1 の値が高ければ赤く 低ければ紫となる 従って 球面 SOM で解の分布を学習さ せると f1 の値が高いものが島状に配置される p5: f1: f3: p1 p4: f1: f3: 図8 平面 SOM における SFC の空間の可視化 p2 p3 わかる 従って大林が平面 SOM での設計空間の分類 構造の把握を行えると示したことと p4 0 同様に 球面 SOM でも可能であると言え パレート解表示は有効であることが再確認され た9) 平面 SOM と球面 SOM におけるデータ配置の違い 平面 SOM が パレート解表示する上で有効であることは再確認されたが 境界において は 各競合層に配置されたデータにゆがみが生じる可能性がある そのため 図 8 において 示した平面 SOM に配置された各データが 球面 SOM ではどのように配置されているかを 検討する p5 図 8 における点 p1 から p5 までの各目的関数の値および設計変数の値を表 4 にまとめて 示す まず 図 8 における点 p2 および p4 に着目する これらの目的関数値および設計変数の 値は比較的類似しており SOM 上では 近傍に配置されることが期待される しかしなが ら 実際には平面 SOM 上では 右端に対峙して配置されている これは 類似したデータ 図 7 球面 SOM における SFC の空間の可視化 がすべて右端に偏って配置されたため 実際には近いデータにもかかわらず 距離をおいて 平面 SOM での学習結果を図 8 に示す f1 の値が高いものが左上に集まって表示され 右 配置されてしまったためであると考えられる 一方で 図 7 からもわかるように球面 SOM 上 右下 左下という順番で値が小さくなりながら配置されていることがわかる においては p2 および p4 は近傍に配置されている よって 球面 SOM によるパレート解 球面 SOM の結果も平面 SOM と同様に値を徐々に低下させながら配置されていることが 表示では境界のゆがみの影響を受けていないことが分かる Information
9 4 data f 1 f 3 f 3 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 p p p p p p1 p3 8 f SOM p1 p3 7 SOM 4 x 4 x 8 SOM p1 p3 SOM SOM SOM SOM SOM SOM 7. (Self-Organizing Maps, SOM) SOM SOM SOM SOM 2 3 SOM SOM SOM 1 SOM SOM SOM SOM SOM SOM SOM SOM SOM GPU SOM 1) (TOM) Vol.2, No.3 pp (2009) 2) N ) (TOM) Vol.1 No.1 pp.27-42, (2008) 4), Vol. 46, No. SIG 17(TOM13) pp (2005) 5) 9 c2010 Information
10 Vol.20, No.6, pp , (2008) 6), : Vol.58, pp (2005/5) 7) Ying Xin WU SOM Vol.19, No.6, pp , (2007) 8),,, SOM Vol.8, No.1, pp ), 14, pp (2001) 10) HOKKE-2009 IPSJ SIG Notes pp (2009) 11) H. Tamukoh, T. Aso, K. Horio, T. Yamakawa : Self-organizing map hardware accelerator system and its application to realtime image enlargement, Neural Networks, Proceedings IEEE International Joint Conference on, Vol. 4, (2004) 12), 2 (1995) 13),,, :,, 17, (1998) 14), Vol.55, No.9, pp.17-22, (2001) 15) :, : Vol.55, No.9, pp.10-16, (2001) 16), :, : Vol.55, No.9, pp.41-45, (2001) 17), :, : Vol.55, No.9, pp.46-52, (2001) 18) GPU SOM HOKKE-17 IPSJ SIG Note (2009) 10 c2010 Information
11 ( 1 ) ( 2 ) 1 CPU GPU MASATO YOSHIMI KANAME NISHIMOTO LUYI WANG TOMOYUKI HIROYASU MITSUNORI MIKI KANAME NISHIMOTO MASATO YOSHIMI LUYI WANG TOMOYUKI HIROYASU MITSUNORI MIKI
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