PRML 12-12.1.4 主成分分析 (PCA) / Principal Component Analysis (PCA)

Pattern Recognition and Machine Learning
12 – 12.1.4
新田晃大
関西学院大学理工学部
http://www.akihironitta.com
2018 年 10 月 26 日

2本日の内容
p12 連続潜在変数
p12.1 主成分分析
n12.1.1 分散最大化による定式化
n12.1.2 誤差最小化による定式化
n12.1.3 主成分分析の応用
n12.1.4 高次元データに対する主成分分析

4離散潜在変数じゃなくて連続潜在変数 GMM との比較
9章
混合ガウス分布
z はクラスタ割当 (離散)
12章
PCA
z は低次元表現 (連続)

5連続潜在変数を考える理由 1/2
もともとデータが入っていた空間よりはるかに低い次元の多様体に
データ点がまとまっていることがよくあるため

例: ずらし数字データ自由度: 3
1. 水平方向の移動
2. 垂直方向の移動
3. 回転
実効次元: 3
(本質的には3次元空間中の点)
intrinsic: 本質的な
ある画像
平行移動と回転
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連続!
連続!!
連続!!!

例: 送油データ
先ほどと同様の話なのでスキップ．さよなら．

8多様体
もともとデータが入っていた空間よりはるかに低い次元の多様体に
データ点がまとまっていることがよくあるため
実際，滑らかな低次元多様体に
完全に閉じ込められることはないはず．
よし，多様体からのデータ点の隔たりを「ノイズ」として解釈しよう．
2次元空間中の
非線形1次元多様体
◯ : データ点
ー : 多様体
生成モデルを考える．

912章の内容
p12.1: 非確率的 PCA <- 本日の内容
p12.2: 確率的 PCA
n利点たくさん
p12.3: カーネル PCA
nPCA の非線形化
p12.4: 非線形潜在変数モデル
n線形ガウスから非線形，非ガウスへ
n独立成分分析

11主成分分析
p英語: PCA; principal component analysis
p別名: Karhunen-Loève 変換
p用途: 次元削減，非可逆データ圧縮，特徴抽出，データの可視化
p定義方法:
n12.1.1 分散最大化
n12.1.2 誤差最小化

12表記
データ数
データセット (観測値)
観測変数の次元数
潜在変数の次元数
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12.1.1 分散最大化による定式化
データマイニングの授業と同じ

14分散最大化による定式化
射影されたデータ点の分散を最大化しながら，
データを次元 M < D を持つ空間の上に射影すること
「この章の後の方で，適切な M の値をデータから決める方法について触れる」らしい
●: データ点
●: 主部分空間の上への直交射影
分散を最大化する u を見つける
目的
具体的に

15定式化の流れ
1. u を未知として，データを射影する
2. 射影されたデータの分散を考える
3. 分散を最大化する u を求める

16分散の定義
●: データ点
●: 主部分空間の上への直交射影
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各データ点 <latexit sha1_base64="kWet215mws0gc0PAFPezHIQjejI=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2QtpRsOtuGJtklyYpl6a/w4kERr/4cb/4b03YP2vpC4OGdGTLzhongxvr+t7e2vrG5tV3YKe7u7R8clo6OmyZONcMGi0Ws2yE1KLjChuVWYDvRSGUosBWOb2b11iNqw2N1bycJ9iQdKh5xRq2zHrJuGJGnaV/1S2W/4s9FViHIoQy56v3SV3cQs1SiskxQYzqBn9heRrXlTOC02E0NJpSN6RA7DhWVaHrZfOEpOXfOgESxdk9ZMnd/T2RUGjORoeuU1I7Mcm1m/lfrpDaq9jKuktSiYouPolQQG5PZ9WTANTIrJg4o09ztStiIasqsy6joQgiWT16F5mUlcHx3Va5V8zgKcApncAEBXEMNbqEODWAg4Rle4c3T3ov37n0sWte8fOYE/sj7/AGydpBL</latexit>
平均
分散
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<latexit sha1_base64="5hEWmGdPmRQ+KyydmSGR33mKY7s=">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</latexit>
分散の最大化するような u を求める

17分散最大化
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分散の最大化するような u を求める
●: データ点
●: 主部分空間の上への直交射影の条件下で分散最大化<latexit sha1_base64="Oczh3riJpxjd9AewxFe65mU4JSo=">AAACLHicbVDLSgMxFM3UV62vUZdugkVwVWZEsBuh0I3LCn1BO5ZMmmlDk8yQZIQyzAe58VcEcWERt36HmekstO2BcA/n3EvuPX7EqNKOs7BKW9s7u3vl/crB4dHxiX161lVhLDHp4JCFsu8jRRgVpKOpZqQfSYK4z0jPnzUzv/dMpKKhaOt5RDyOJoIGFCNtpJHdHHKkp34AExjDFI5MdU19MjV3JDesbZQUbu68h+7Irjo1JwdcJ25BqqBAa2S/D8chjjkRGjOk1MB1Iu0lSGqKGUkrw1iRCOEZmpCBoQJxorwkPzaFV0YZwyCU5gkNc/XvRIK4UnPum85sYbXqZeImbxDroO4lVESxJgIvPwpiBnUIs+TgmEqCNZsbgrCkZleIp0girE2+FROCu3ryOune1FzDH2+rjXoRRxlcgEtwDVxwBxrgAbRAB2DwAt7AJ1hYr9aH9WV9L1tLVjFzDv7B+vkFDr+jug==</latexit>
<latexit sha1_base64="kFh/dnsGXT470yE6SF1ddJlkyjk=">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</latexit>
u に関する微分を 0 とおくと，
<latexit sha1_base64="R3SnFpK3C1gW2EIZANeEFyJVXoI=">AAACKHicbVDLSgMxFL3js9bXqEs3wSK4KjMi2I1YcOOyon1AOwyZTKYNzTxIMkIZ+jlu/BU3Iop065eYaQetrRdCDuecm9x7vIQzqSxrYqysrq1vbJa2yts7u3v75sFhS8apILRJYh6Ljocl5SyiTcUUp51EUBx6nLa94U2utx+pkCyOHtQooU6I+xELGMFKU6553QuxGngBytA9StEYuRrZ+r5CPa6f8fEP8+uc87lmxapa00LLwC5ABYpquOZbz49JGtJIEY6l7NpWopwMC8UIp+NyL5U0wWSI+7SrYYRDKp1suugYnWrGR0Es9IkUmrLzHRkOpRyFnnbmw8pFLSf/07qpCmpOxqIkVTQis4+ClCMVozw15DNBieIjDTARTM+KyAALTJTOtqxDsBdXXgat86qt8d1FpV4r4ijBMZzAGdhwCXW4hQY0gcATvMA7fBjPxqvxaUxm1hWj6DmCP2V8fQP7s6Iy</latexit>
つまり λ，u が S の固有値・固有ベクトル！
ここまでのまとめ
射影されたデータの分散の最大化をしたければ，
元のデータ共分散行列の固有値分解をすればよい :D

18分散最大化の解
<latexit sha1_base64="R3SnFpK3C1gW2EIZANeEFyJVXoI=">AAACKHicbVDLSgMxFL3js9bXqEs3wSK4KjMi2I1YcOOyon1AOwyZTKYNzTxIMkIZ+jlu/BU3Iop065eYaQetrRdCDuecm9x7vIQzqSxrYqysrq1vbJa2yts7u3v75sFhS8apILRJYh6Ljocl5SyiTcUUp51EUBx6nLa94U2utx+pkCyOHtQooU6I+xELGMFKU6553QuxGngBytA9StEYuRrZ+r5CPa6f8fEP8+uc87lmxapa00LLwC5ABYpquOZbz49JGtJIEY6l7NpWopwMC8UIp+NyL5U0wWSI+7SrYYRDKp1suugYnWrGR0Es9IkUmrLzHRkOpRyFnnbmw8pFLSf/07qpCmpOxqIkVTQis4+ClCMVozw15DNBieIjDTARTM+KyAALTJTOtqxDsBdXXgat86qt8d1FpV4r4ijBMZzAGdhwCXW4hQY0gcATvMA7fBjPxqvxaUxm1hWj6DmCP2V8fQP7s6Iy</latexit>
<latexit sha1_base64="VBHxbLwu692IBIzqdm3GKm6yxDk=">AAACPHicbVDLSsNAFJ3UV62vqEs3g0VwVRIR7EYouHFZ6ROaGCaTSTt08mBmIpSQD3PjR7hz5caFIm5dO0kj2uqF4R7OOZc797gxo0IaxpNWWVldW9+obta2tnd29/T9g76IEo5JD0cs4kMXCcJoSHqSSkaGMScocBkZuNOrXB/cES5oFHblLCZ2gMYh9SlGUlGO3rECJCeuD1OYwAw6qpuq36peKDxQqKuYDP44OwveS2gxtdBD34yj142GURT8C8wS1EFZbUd/tLwIJwEJJWZIiJFpxNJOEZcUM5LVrESQGOEpGpORgiEKiLDT4vgMnijGg37E1QslLNjfEykKhJgFrnLmB4hlLSf/00aJ9Jt2SsM4kSTE80V+wqCMYJ4k9CgnWLKZAghzqv4K8QRxhKXKu6ZCMJdP/gv6Zw1T4ZvzeqtZxlEFR+AYnAITXIAWuAZt0AMY3INn8AretAftRXvXPubWilbOHIKF0j6/AHbTqWE=</latexit>
分散の最大化するような u は次を満たす．
S の固有値・固有ベクトルなら全て上を満たす．
<latexit sha1_base64="TxcrJn1mQytFiAeF6vmAArBwy3E=">AAACBXicbVDNS8MwHE3n15xfVY96CA7Bg4xWBHcc6MHjBPcBaylpmm5haVKSVBhlFy/+K148KOLV/8Gb/43p1oNuPgh5vPd+JL8Xpowq7TjfVmVldW19o7pZ29re2d2z9w+6SmQSkw4WTMh+iBRhlJOOppqRfioJSkJGeuH4uvB7D0QqKvi9nqTET9CQ05hipI0U2MceM+EIBe459FgktCruuXQT2HWn4cwAl4lbkjoo0Q7sLy8SOEsI15ghpQauk2o/R1JTzMi05mWKpAiP0ZAMDOUoIcrPZ1tM4alRIhgLaQ7XcKb+nshRotQkCU0yQXqkFr1C/M8bZDpu+jnlaaYJx/OH4oxBLWBRCYyoJFiziSEIS2r+CvEISYS1Ka5mSnAXV14m3YuGa/jdZb3VLOuogiNwAs6AC65AC9yCNugADB7BM3gFb9aT9WK9Wx/zaMUqZw7BH1ifP+Sbl4A=</latexit>
<latexit sha1_base64="WPnrHgltkVLp51TniNgJfnN5Zyc=">AAACBXicbZDLSsNAFIYn9VbrLepSF4NFcFFKIoJdFnThsoK9QBPCZDJph05mwsxEKKEbN76KGxeKuPUd3Pk2TtsI2vrDwMd/zuHM+cOUUaUd58sqrayurW+UNytb2zu7e/b+QUeJTGLSxoIJ2QuRIoxy0tZUM9JLJUFJyEg3HF1N6917IhUV/E6PU+InaMBpTDHSxgrs49wLY5hNArcGPRYJrWrwx7oO7KpTd2aCy+AWUAWFWoH96UUCZwnhGjOkVN91Uu3nSGqKGZlUvEyRFOERGpC+QY4Sovx8dsUEnhongrGQ5nENZ+7viRwlSo2T0HQmSA/VYm1q/lfrZzpu+DnlaaYJx/NFccagFnAaCYyoJFizsQGEJTV/hXiIJMLaBFcxIbiLJy9D57zuGr69qDYbRRxlcAROwBlwwSVoghvQAm2AwQN4Ai/g1Xq0nq03633eWrKKmUPwR9bHN77zl2g=</latexit>
最大固有値に属する固有ベクトルを選んだ時，分散最大．
ここまで M = 1 の話．
第 1 主成分

19分散最大化の一般解
M 次元に一般化 [Ex. 12.1]
データ分散行列 S の，
大きい順に M 個の固有値に対応する M 個の固有ベクトルにより，
射影されたデータ分散最大の最適な線形射影が定義される．
<latexit sha1_base64="FqRbPFIeHY/kDyejblWHYTGu2Vs=">AAACBXicbZDLSsNAFIYn9VbrLepSF4NFcFFKIoJdFty4ESrYCzQhTCaTduhkJsxMhBK6ceOruHGhiFvfwZ1v47SNoK0/DHz85xzOnD9MGVXacb6s0srq2vpGebOytb2zu2fvH3SUyCQmbSyYkL0QKcIoJ21NNSO9VBKUhIx0w9HVtN69J1JRwe/0OCV+ggacxhQjbazAPs69MIbZJHBr0GOR0KoGf6ybwK46dWcmuAxuAVVQqBXYn14kcJYQrjFDSvVdJ9V+jqSmmJFJxcsUSREeoQHpG+QoIcrPZ1dM4KlxIhgLaR7XcOb+nshRotQ4CU1ngvRQLdam5n+1fqbjhp9TnmaacDxfFGcMagGnkcCISoI1GxtAWFLzV4iHSCKsTXAVE4K7ePIydM7rruHbi2qzUcRRBkfgBJwBF1yCJrgGLdAGGDyAJ/ACXq1H69l6s97nrSWrmDkEf2R9fAPMl5dx</latexit>
<latexit sha1_base64="HC9hJ1JoVXgFAec8vOUj4Upq4wc=">AAACBXicbVDNS8MwHE3n15xfVY96CA7Bg4xWBHccePEiTHAfsJaSpukWliYlSYVRdvHiv+LFgyJe/R+8+d+Ybj3o5oOQx3vvR/J7Ycqo0o7zbVVWVtfWN6qbta3tnd09e/+gq0QmMelgwYTsh0gRRjnpaKoZ6aeSoCRkpBeOrwu/90CkooLf60lK/AQNOY0pRtpIgX3sMROOUOCeQ49FQqvinku3gV13Gs4McJm4JamDEu3A/vIigbOEcI0ZUmrgOqn2cyQ1xYxMa16mSIrwGA3JwFCOEqL8fLbFFJ4aJYKxkOZwDWfq74kcJUpNktAkE6RHatErxP+8Qabjpp9TnmaacDx/KM4Y1AIWlcCISoI1mxiCsKTmrxCPkERYm+JqpgR3ceVl0r1ouIbfXdZbzbKOKjgCJ+AMuOAKtMANaIMOwOARPINX8GY9WS/Wu/Uxj1ascuYQ/IH1+QPyP5eJ</latexit>

20ここまでのまとめ
PCA に必要なもの
平均
共分散行列
<latexit sha1_base64="UYnI85iJToT9DKupKHj1QtFSS4Y=">AAAB+XicbVBNSwMxFHxbv2r9WvXoJVgET2VXBHssePFYwbZCdynZNNuGZpMlyRbL0n/ixYMiXv0n3vw3Zts9aOtAYJh5w3uZKOVMG8/7diobm1vbO9Xd2t7+weGRe3zS1TJThHaI5FI9RlhTzgTtGGY4fUwVxUnEaS+a3BZ+b0qVZlI8mFlKwwSPBIsZwcZKA9cNpLWLdB5EMXqaD9y61/AWQOvEL0kdSrQH7lcwlCRLqDCEY637vpeaMMfKMMLpvBZkmqaYTPCI9i0VOKE6zBeXz9GFVYYolso+YdBC/Z3IcaL1LInsZILNWK96hfif189M3AxzJtLMUEGWi+KMIyNRUQMaMkWJ4TNLMFHM3orIGCtMjC2rZkvwV7+8TrpXDd/y++t6q1nWUYUzOIdL8OEGWnAHbegAgSk8wyu8Obnz4rw7H8vRilNmTuEPnM8fyTGTtQ==</latexit>
<latexit sha1_base64="tfI8DNxB0bX/ih/OtavytV9YH1Y=">AAAB7nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeyx4MVjRfsBbSib7aRdutmE3Y1QQn+EFw+KePX3ePPfuE1z0NYXFh7emWFn3iARXBvX/XZKG5tb2zvl3cre/sHhUfX4pKPjVDFss1jEqhdQjYJLbBtuBPYShTQKBHaD6e2i3n1CpXksH80sQT+iY8lDzqixVjcbBCF5mA+rNbfu5iLr4BVQg0KtYfVrMIpZGqE0TFCt+56bGD+jynAmcF4ZpBoTyqZ0jH2Lkkao/Sxfd04urDMiYazsk4bk7u+JjEZaz6LAdkbUTPRqbWH+V+unJmz4GZdJalCy5UdhKoiJyeJ2MuIKmREzC5QpbnclbEIVZcYmVLEheKsnr0Pnqu5Zvr+uNRtFHGU4g3O4BA9uoAl30II2MJjCM7zCm5M4L86787FsLTnFzCn8kfP5A/PCj0U=</latexit>
べき乗法を使えば
固有値
固有ベクトル <latexit sha1_base64="FqRbPFIeHY/kDyejblWHYTGu2Vs=">AAACBXicbZDLSsNAFIYn9VbrLepSF4NFcFFKIoJdFty4ESrYCzQhTCaTduhkJsxMhBK6ceOruHGhiFvfwZ1v47SNoK0/DHz85xzOnD9MGVXacb6s0srq2vpGebOytb2zu2fvH3SUyCQmbSyYkL0QKcIoJ21NNSO9VBKUhIx0w9HVtN69J1JRwe/0OCV+ggacxhQjbazAPs69MIbZJHBr0GOR0KoGf6ybwK46dWcmuAxuAVVQqBXYn14kcJYQrjFDSvVdJ9V+jqSmmJFJxcsUSREeoQHpG+QoIcrPZ1dM4KlxIhgLaR7XcOb+nshRotQ4CU1ngvRQLdam5n+1fqbjhp9TnmaacDxfFGcMagGnkcCISoI1GxtAWFLzV4iHSCKsTXAVE4K7ePIydM7rruHbi2qzUcRRBkfgBJwBF1yCJrgGLdAGGDyAJ/ACXq1H69l6s97nrSWrmDkEf2R9fAPMl5dx</latexit>
PCA の計算コスト
= 固有値分解の計算コスト <latexit sha1_base64="tpN15+iw5+FZ2JO/x254mhsgBVM=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8Ei1E2ZUcEuC7pwZwV7gXYsmTTTxmaSIckIZeg7uHGhiFvfx51vY9rOQlt/CHz85xxyzh/EnGnjut9ObmV1bX0jv1nY2t7Z3SvuHzS1TBShDSK5VO0Aa8qZoA3DDKftWFEcBZy2gtHVtN56okozKe7NOKZ+hAeChYxgY63mbfn64fy0Vyy5FXcmtAxeBiXIVO8Vv7p9SZKICkM41rrjubHxU6wMI5xOCt1E0xiTER7QjkWBI6r9dLbtBJ1Yp49CqewTBs3c3xMpjrQeR4HtjLAZ6sXa1Pyv1klMWPVTJuLEUEHmH4UJR0ai6emozxQlho8tYKKY3RWRIVaYGBtQwYbgLZ68DM2zimf57qJUq2Zx5OEIjqEMHlxCDW6gDg0g8AjP8ApvjnRenHfnY96ac7KZQ/gj5/MHH0iOIQ==</latexit>
<latexit sha1_base64="fBOgiXKROo/lhX9LrIzSE490hKc=">AAAB7nicbZDLSgMxFIbP1Futt6pLN8Ei1E2ZKYJdFnThRqxgL9COJZNm2tBMJiQZoQx9CDcuFHHr87jzbUzbWWjrD4GP/5xDzvkDyZk2rvvt5NbWNza38tuFnd29/YPi4VFLx4kitEliHqtOgDXlTNCmYYbTjlQURwGn7WB8Nau3n6jSLBYPZiKpH+GhYCEj2FirfVe+vX6snveLJbfizoVWwcugBJka/eJXbxCTJKLCEI617nquNH6KlWGE02mhl2gqMRnjIe1aFDii2k/n607RmXUGKIyVfcKguft7IsWR1pMosJ0RNiO9XJuZ/9W6iQlrfsqETAwVZPFRmHBkYjS7HQ2YosTwiQVMFLO7IjLCChNjEyrYELzlk1ehVa14lu8vSvVaFkceTuAUyuDBJdThBhrQBAJjeIZXeHOk8+K8Ox+L1pyTzRzDHzmfP7kQjnc=</latexit>
<latexit sha1_base64="TxcrJn1mQytFiAeF6vmAArBwy3E=">AAACBXicbVDNS8MwHE3n15xfVY96CA7Bg4xWBHcc6MHjBPcBaylpmm5haVKSVBhlFy/+K148KOLV/8Gb/43p1oNuPgh5vPd+JL8Xpowq7TjfVmVldW19o7pZ29re2d2z9w+6SmQSkw4WTMh+iBRhlJOOppqRfioJSkJGeuH4uvB7D0QqKvi9nqTET9CQ05hipI0U2MceM+EIBe459FgktCruuXQT2HWn4cwAl4lbkjoo0Q7sLy8SOEsI15ghpQauk2o/R1JTzMi05mWKpAiP0ZAMDOUoIcrPZ1tM4alRIhgLaQ7XcKb+nshRotQkCU0yQXqkFr1C/M8bZDpu+jnlaaYJx/OH4oxBLWBRCYyoJFiziSEIS2r+CvEISYS1Ka5mSnAXV14m3YuGa/jdZb3VLOuogiNwAs6AC65AC9yCNugADB7BM3gFb9aT9WK9Wx/zaMUqZw7BH1ifP+Sbl4A=</latexit>
<latexit sha1_base64="WPnrHgltkVLp51TniNgJfnN5Zyc=">AAACBXicbZDLSsNAFIYn9VbrLepSF4NFcFFKIoJdFnThsoK9QBPCZDJph05mwsxEKKEbN76KGxeKuPUd3Pk2TtsI2vrDwMd/zuHM+cOUUaUd58sqrayurW+UNytb2zu7e/b+QUeJTGLSxoIJ2QuRIoxy0tZUM9JLJUFJyEg3HF1N6917IhUV/E6PU+InaMBpTDHSxgrs49wLY5hNArcGPRYJrWrwx7oO7KpTd2aCy+AWUAWFWoH96UUCZwnhGjOkVN91Uu3nSGqKGZlUvEyRFOERGpC+QY4Sovx8dsUEnhongrGQ5nENZ+7viRwlSo2T0HQmSA/VYm1q/lfrZzpu+DnlaaYJx/NFccagFnAaCYyoJFizsQGEJTV/hXiIJMLaBFcxIbiLJy9D57zuGr69qDYbRRxlcAROwBlwwSVoghvQAm2AwQN4Ai/g1Xq0nq03633eWrKKmUPwR9bHN77zl2g=</latexit>
<latexit sha1_base64="FqRbPFIeHY/kDyejblWHYTGu2Vs=">AAACBXicbZDLSsNAFIYn9VbrLepSF4NFcFFKIoJdFty4ESrYCzQhTCaTduhkJsxMhBK6ceOruHGhiFvfwZ1v47SNoK0/DHz85xzOnD9MGVXacb6s0srq2vpGebOytb2zu2fvH3SUyCQmbSyYkL0QKcIoJ21NNSO9VBKUhIx0w9HVtN69J1JRwe/0OCV+ggacxhQjbazAPs69MIbZJHBr0GOR0KoGf6ybwK46dWcmuAxuAVVQqBXYn14kcJYQrjFDSvVdJ9V+jqSmmJFJxcsUSREeoQHpG+QoIcrPZ1dM4KlxIhgLaR7XcOb+nshRotQ4CU1ngvRQLdam5n+1fqbjhp9TnmaacDxfFGcMagGnkcCISoI1GxtAWFLzV4iHSCKsTXAVE4K7ePIydM7rruHbi2qzUcRRBkfgBJwBF1yCJrgGLdAGGDyAJ/ACXq1H69l6s97nrSWrmDkEf2R9fAPMl5dx</latexit>
EM も利用できる [12.2.2 節]

12.1.2 誤差最小化による定式化

22誤差最小化による定式化
<latexit sha1_base64="C1NPH1bFJ97eeCQcbn3CeXWOpm4=">AAACO3icbVA9SwNBEN2LXzF+nVraLAbBKtyJYBohYGMZJV+QxLC3t5es2b07dveEcNz/svFP2NnYWChia+8kuUITB5b3eG+GnXleLLg2jvNiFVZW19Y3ipulre2d3T17/6Clo0RR1qSRiFTHI5oJHrKm4UawTqwYkZ5gbW98NfXbD0xpHoUNM4lZX5JhyANOiQFpYN/2JDEjL8ApTnCGB4Ac8A5w5igJrAFKhpc77wEvcc9nwpB8FKSBXXYqzqzwMnFzUkZ51Qf2c8+PaCJZaKggWnddJzb9lCjDqWBZqZdoFhM6JkPWBRoSyXQ/nd2e4RNQfBxECl5o8Ez9PZESqfVEetA53V8velPxP6+bmKDaT3kYJ4aFdP5RkAhsIjwNEvtcMWrEBAihisOumI6IItRA3CUIwV08eZm0ziou8Jvzcq2ax1FER+gYnSIXXaAaukZ11EQUPaJX9I4+rCfrzfq0vuatBSufOUR/yvr+AezvqcA=</latexit>
D次元の基底ベクトル <latexit sha1_base64="OQ2A3SAe13sAdayiyUVf1v9ssjU=">AAACAHicbZC7SgNBFIbPxluMt1ULC5vBIFiFXRFMIwS0sIxgLpCNy+xkNhkye2FmVgjDNr6KjYUitj6GnW/jJNlCE38Y+PjPOZw5f5ByJpXjfFulldW19Y3yZmVre2d3z94/aMskE4S2SMIT0Q2wpJzFtKWY4rSbCoqjgNNOML6e1juPVEiWxPdqktJ+hIcxCxnByli+feRp7QUhynKfebmv2ZWbP+ib3LerTs2ZCS2DW0AVCjV9+8sbJCSLaKwIx1L2XCdVfY2FYoTTvOJlkqaYjPGQ9gzGOKKyr2cH5OjUOAMUJsK8WKGZ+3tC40jKSRSYzgirkVysTc3/ar1MhfW+ZnGaKRqT+aIw40glaJoGGjBBieITA5gIZv6KyAgLTJTJrGJCcBdPXob2ec01fHdRbdSLOMpwDCdwBi5cQgNuoQktIJDDM7zCm/VkvVjv1se8tWQVM4fwR9bnD8THlng=</latexit>
完全正規直交系の定義より
各データ点は基底ベクトルの線形結合
係数は各データ点で異なる値
<latexit sha1_base64="5MISuVoWCeJVUxCMHkQz+XRThsQ=">AAACQHicbVDLSsNAFJ3UV62vqEs3g0VwVRIR7EYo6MJlBfuAppbJdNIOnUzCzEQsoZ/mxk9w59qNC0XcuvKmDaKtF4Z77rnPOX4suDaO82wVlpZXVteK66WNza3tHXt3r6mjRFHWoJGIVNsnmgkuWcNwI1g7VoyEvmAtf3SR5Vt3TGkeyRszjlk3JAPJA06JAapnt7yQmKEf4BTf4wnugZfgz7Gnk3AacghcoG4BX4L3iIiHJK/kGfEzIcknANuzy07FmRpeBG4Oyii3es9+8voRTUImDRVE647rxKabEmU4FWxS8hLNYkJHZMA6ACUJme6mUwEm+AiYPg4iBU8aPGV/d6Qk1Hoc+lCZHavncxn5X66TmKDaTbmME8MknS0KEoFNhDM1cZ8rRo0YAyBUcbgV0yFRhBrQvAQiuPNfXgTNk4oL+Pq0XKvmchTRATpEx8hFZ6iGrlAdNRBFD+gFvaF369F6tT6sz1lpwcp79tEfs76+AaTDqms=</latexit>
<latexit sha1_base64="kWet215mws0gc0PAFPezHIQjejI=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2QtpRsOtuGJtklyYpl6a/w4kERr/4cb/4b03YP2vpC4OGdGTLzhongxvr+t7e2vrG5tV3YKe7u7R8clo6OmyZONcMGi0Ws2yE1KLjChuVWYDvRSGUosBWOb2b11iNqw2N1bycJ9iQdKh5xRq2zHrJuGJGnaV/1S2W/4s9FViHIoQy56v3SV3cQs1SiskxQYzqBn9heRrXlTOC02E0NJpSN6RA7DhWVaHrZfOEpOXfOgESxdk9ZMnd/T2RUGjORoeuU1I7Mcm1m/lfrpDaq9jKuktSiYouPolQQG5PZ9WTANTIrJg4o09ztStiIasqsy6joQgiWT16F5mUlcHx3Va5V8zgKcApncAEBXEMNbqEODWAg4Rle4c3T3ov37n0sWte8fOYE/sj7/AGydpBL</latexit>
<latexit sha1_base64="YPwKtLQgtE3voBl+sHzc6tu9sSc=">AAAB8nicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2AdimzabYNzSZLkhXK0p/hxYMiXv013vw3pu0etPWFwMM7M2TmjVLBjfX9b29jc2t7Z7e0V94/ODw6rpycto3KNGUtqoTS3QgNE1yyluVWsG6qGSaRYJ1ocjevd56YNlzJRztNWZjgSPKYU7TO6vVRpGMc5DKYDSpVv+YvRNYhKKAKhZqDyld/qGiWMGmpQGN6gZ/aMEdtORVsVu5nhqVIJzhiPYcSE2bCfLHyjFw6Z0hipd2Tlizc3xM5JsZMk8h1JmjHZrU2N/+r9TIb18OcyzSzTNLlR3EmiFVkfj8Zcs2oFVMHSDV3uxI6Ro3UupTKLoRg9eR1aF/XAscPN9VGvYijBOdwAVcQwC004B6a0AIKCp7hFd486714797HsnXDK2bO4I+8zx9KM5E2</latexit>
<latexit sha1_base64="RCtQ7H/qW2HEFP57dtz3hYCTYCA=">AAAB8nicbZBNS8NAEIYn9avWr6pHL4tF8FSSIthjwYvHCvYD0lA22027dLMJuxOhhP4MLx4U8eqv8ea/cdvmoK0vLDy8M8POvGEqhUHX/XZKW9s7u3vl/crB4dHxSfX0rGuSTDPeYYlMdD+khkuheAcFSt5PNadxKHkvnN4t6r0nro1I1CPOUh7EdKxEJBhFa/kDKtMJHeaqMR9Wa27dXYpsgldADQq1h9WvwShhWcwVMkmN8T03xSCnGgWTfF4ZZIanlE3pmPsWFY25CfLlynNyZZ0RiRJtn0KydH9P5DQ2ZhaHtjOmODHrtYX5X83PMGoGuVBphlyx1UdRJgkmZHE/GQnNGcqZBcq0sLsSNqGaMrQpVWwI3vrJm9Bt1D3LDze1VrOIowwXcAnX4MEttOAe2tABBgk8wyu8Oei8OO/Ox6q15BQz5/BHzucPS7iRNw==</latexit>
<latexit sha1_base64="Mkw12ox66wLBlhtmBMcc/KEZa8Y=">AAAB7XicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2AtpRsOtvGZpMlyYpl6X/w4kERr/4fb/4b03YP2vpC4OGdGTLzhongxvr+t7e2vrG5tV3YKe7u7R8clo6Om0almmGDKaF0O6QGBZfYsNwKbCcaaRwKbIXjm1m99YjacCXv7STBXkyHkkecUeus5lM/k8G0Xyr7FX8usgpBDmXIVe+XvroDxdIYpWWCGtMJ/MT2MqotZwKnxW5qMKFsTIfYcShpjKaXzbedknPnDEiktHvSkrn7eyKjsTGTOHSdMbUjs1ybmf/VOqmNqr2MyyS1KNnioygVxCoyO50MuEZmxcQBZZq7XQkbUU2ZdQEVXQjB8smr0LysBI7vrsq1ah5HAU7hDC4ggGuowS3UoQEMHuAZXuHNU96L9+59LFrXvHzmBP7I+/wBmyePGg==</latexit>
<latexit sha1_base64="OeaawEh8OijfvDVYOEiXywFrOXU=">AAAB7XicbZDLSgMxFIZP6q3WW9Wlm2ARXJWZIthlwY3LCvYC7VAyaaaNzSRDkhHL0Hdw40IRt76PO9/GtJ2Ftv4Q+PjPOeScP0wEN9bzvlFhY3Nre6e4W9rbPzg8Kh+ftI1KNWUtqoTS3ZAYJrhkLcutYN1EMxKHgnXCyc283nlk2nAl7+00YUFMRpJHnBLrrPbTIJO12aBc8areQngd/BwqkKs5KH/1h4qmMZOWCmJMz/cSG2REW04Fm5X6qWEJoRMyYj2HksTMBNli2xm+cM4QR0q7Jy1euL8nMhIbM41D1xkTOzartbn5X62X2qgeZFwmqWWSLj+KUoGtwvPT8ZBrRq2YOiBUc7crpmOiCbUuoJILwV89eR3atarv+O6q0qjncRThDM7hEny4hgbcQhNaQOEBnuEV3pBCL+gdfSxbCyifOYU/Qp8/nKyPGw==</latexit>
<latexit sha1_base64="dmkWiF/WObZg/13ZckxC73RvUjA=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RbDHghePFeyHtKVk09k2NMkuSVYoS3+FFw+KePXnePPfmLZ70NYXAg/vzJCZN0wEN9b3v73CxubW9k5xt7S3f3B4VD4+aZk41QybLBax7oTUoOAKm5ZbgZ1EI5WhwHY4uZ3X20+oDY/Vg50m2Jd0pHjEGbXOesx6YUTS2SAYlCt+1V+IrEOQQwVyNQblr94wZqlEZZmgxnQDP7H9jGrLmcBZqZcaTCib0BF2HSoq0fSzxcIzcuGcIYli7Z6yZOH+nsioNGYqQ9cpqR2b1drc/K/WTW1U62dcJalFxZYfRakgNibz68mQa2RWTB1QprnblbAx1ZRZl1HJhRCsnrwOratq4Pj+ulKv5XEU4QzO4RICuIE63EEDmsBAwjO8wpunvRfv3ftYtha8fOYU/sj7/AFRbZAL</latexit>
<latexit sha1_base64="hB3eIEvdn9cnPB/q9xjPbGG4vU8=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInspuEeyx4MVjBfshbSnZNNuGJtklmRXK0l/hxYMiXv053vw3pu0etPWFwMM7M2TmDRMpLPr+t7exubW9s1vYK+4fHB4dl05OWzZODeNNFsvYdEJquRSaN1Gg5J3EcKpCydvh5HZebz9xY0WsH3Ca8L6iIy0iwSg66zHrhRFJZ4PqoFT2K/5CZB2CHMqQqzEoffWGMUsV18gktbYb+An2M2pQMMlnxV5qeULZhI5416Gmitt+tlh4Ri6dMyRRbNzTSBbu74mMKmunKnSdiuLYrtbm5n+1bopRrZ8JnaTINVt+FKWSYEzm15OhMJyhnDqgzAi3K2FjaihDl1HRhRCsnrwOrWolcHx/Xa7X8jgKcA4XcAUB3EAd7qABTWCg4Ble4c0z3ov37n0sWze8fOYM/sj7/AFS8ZAM</latexit>
新しい座標系への回転を意味している．
<latexit sha1_base64="Zd8BBI7tVkbql1SHsI/MafJUNTI=">AAACG3icbVDLSgMxFM3UV62vqks3wSK4kDJTBLss6MJlBfuAzjBk0kwbmskMyR2xDPMfbvwVNy4UcSW48G9MH4K2Hgg5nHPvTe4JEsE12PaXVVhZXVvfKG6WtrZ3dvfK+wdtHaeKshaNRay6AdFMcMlawEGwbqIYiQLBOsHocuJ37pjSPJa3ME6YF5GB5CGnBIzkl2uuYCG4Gb7HPs6wxA7O8Rl2+zFoc/+oV0Z1FR8Mwc39csWu2lPgZeLMSQXN0fTLH2YcTSMmgQqidc+xE/AyooBTwfKSm2qWEDoiA9YzVJKIaS+b7pbjE6P0cRgrcyTgqfq7IyOR1uMoMJURgaFe9Cbif14vhbDuZVwmKTBJZw+FqcAQ40lQuM8VoyDGhhCquPkrpkOiCAUTZ8mE4CyuvEzatapj+M15pVGfx1FER+gYnSIHXaAGukZN1EIUPaAn9IJerUfr2Xqz3melBWvec4j+wPr8Bs/tniM=</latexit>
もともとの D 個の成分 <latexit sha1_base64="nI5GP9PVkhFLAHqTc4LjyH8NVw0=">AAACJHicbVDLSsNAFJ34tr6qLt1cLIILKYkIFtwIunBZwarQhDCZTtqhk0mYuRFKyMe48VfcuPCBCzd+i5O2C18Hhjmcc++duSfKpDDouh/OzOzc/MLi0nJtZXVtfaO+uXVt0lwz3mGpTPVtRA2XQvEOCpT8NtOcJpHkN9HwrPJv7rg2IlVXOMp4kNC+ErFgFK0U1k98yWP0C/CpzAYUQihAgQclHIDfS9FU93frHEpfi/4A/TKsN9ymOwb8Jd6UNMgU7bD+akeyPOEKmaTGdD03w6CgGgWTvKz5ueEZZUPa511LFU24CYrxkiXsWaUHcartUQhj9XtHQRNjRklkKxOKA/Pbq8T/vG6OcSsohMpy5IpNHopzCZhClRj0hOYM5cgSyrSwfwU2oJoytLnWbAje75X/kuvDpmf55VHjtDWNY4nskF2yTzxyTE7JBWmTDmHknjySZ/LiPDhPzpvzPimdcaY92+QHnM8vcRWiMQ==</latexit>
新たな座標系での成分

23誤差最小化による定式化
新たな座標系での成分
<latexit sha1_base64="kWet215mws0gc0PAFPezHIQjejI=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2QtpRsOtuGJtklyYpl6a/w4kERr/4cb/4b03YP2vpC4OGdGTLzhongxvr+t7e2vrG5tV3YKe7u7R8clo6OmyZONcMGi0Ws2yE1KLjChuVWYDvRSGUosBWOb2b11iNqw2N1bycJ9iQdKh5xRq2zHrJuGJGnaV/1S2W/4s9FViHIoQy56v3SV3cQs1SiskxQYzqBn9heRrXlTOC02E0NJpSN6RA7DhWVaHrZfOEpOXfOgESxdk9ZMnd/T2RUGjORoeuU1I7Mcm1m/lfrpDaq9jKuktSiYouPolQQG5PZ9WTANTIrJg4o09ztStiIasqsy6joQgiWT16F5mUlcHx3Va5V8zgKcApncAEBXEMNbqEODWAg4Rle4c3T3ov37n0sWte8fOYE/sj7/AGydpBL</latexit>
<latexit sha1_base64="YPwKtLQgtE3voBl+sHzc6tu9sSc=">AAAB8nicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2AdimzabYNzSZLkhXK0p/hxYMiXv013vw3pu0etPWFwMM7M2TmjVLBjfX9b29jc2t7Z7e0V94/ODw6rpycto3KNGUtqoTS3QgNE1yyluVWsG6qGSaRYJ1ocjevd56YNlzJRztNWZjgSPKYU7TO6vVRpGMc5DKYDSpVv+YvRNYhKKAKhZqDyld/qGiWMGmpQGN6gZ/aMEdtORVsVu5nhqVIJzhiPYcSE2bCfLHyjFw6Z0hipd2Tlizc3xM5JsZMk8h1JmjHZrU2N/+r9TIb18OcyzSzTNLlR3EmiFVkfj8Zcs2oFVMHSDV3uxI6Ro3UupTKLoRg9eR1aF/XAscPN9VGvYijBOdwAVcQwC004B6a0AIKCp7hFd486714797HsnXDK2bO4I+8zx9KM5E2</latexit>
<latexit sha1_base64="RCtQ7H/qW2HEFP57dtz3hYCTYCA=">AAAB8nicbZBNS8NAEIYn9avWr6pHL4tF8FSSIthjwYvHCvYD0lA22027dLMJuxOhhP4MLx4U8eqv8ea/cdvmoK0vLDy8M8POvGEqhUHX/XZKW9s7u3vl/crB4dHxSfX0rGuSTDPeYYlMdD+khkuheAcFSt5PNadxKHkvnN4t6r0nro1I1CPOUh7EdKxEJBhFa/kDKtMJHeaqMR9Wa27dXYpsgldADQq1h9WvwShhWcwVMkmN8T03xSCnGgWTfF4ZZIanlE3pmPsWFY25CfLlynNyZZ0RiRJtn0KydH9P5DQ2ZhaHtjOmODHrtYX5X83PMGoGuVBphlyx1UdRJgkmZHE/GQnNGcqZBcq0sLsSNqGaMrQpVWwI3vrJm9Bt1D3LDze1VrOIowwXcAnX4MEttOAe2tABBgk8wyu8Oei8OO/Ox6q15BQz5/BHzucPS7iRNw==</latexit>
<latexit sha1_base64="Mkw12ox66wLBlhtmBMcc/KEZa8Y=">AAAB7XicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2AtpRsOtvGZpMlyYpl6X/w4kERr/4fb/4b03YP2vpC4OGdGTLzhongxvr+t7e2vrG5tV3YKe7u7R8clo6Om0almmGDKaF0O6QGBZfYsNwKbCcaaRwKbIXjm1m99YjacCXv7STBXkyHkkecUeus5lM/k8G0Xyr7FX8usgpBDmXIVe+XvroDxdIYpWWCGtMJ/MT2MqotZwKnxW5qMKFsTIfYcShpjKaXzbedknPnDEiktHvSkrn7eyKjsTGTOHSdMbUjs1ybmf/VOqmNqr2MyyS1KNnioygVxCoyO50MuEZmxcQBZZq7XQkbUU2ZdQEVXQjB8smr0LysBI7vrsq1ah5HAU7hDC4ggGuowS3UoQEMHuAZXuHNU96L9+59LFrXvHzmBP7I+/wBmyePGg==</latexit>
<latexit sha1_base64="OeaawEh8OijfvDVYOEiXywFrOXU=">AAAB7XicbZDLSgMxFIZP6q3WW9Wlm2ARXJWZIthlwY3LCvYC7VAyaaaNzSRDkhHL0Hdw40IRt76PO9/GtJ2Ftv4Q+PjPOeScP0wEN9bzvlFhY3Nre6e4W9rbPzg8Kh+ftI1KNWUtqoTS3ZAYJrhkLcutYN1EMxKHgnXCyc283nlk2nAl7+00YUFMRpJHnBLrrPbTIJO12aBc8areQngd/BwqkKs5KH/1h4qmMZOWCmJMz/cSG2REW04Fm5X6qWEJoRMyYj2HksTMBNli2xm+cM4QR0q7Jy1euL8nMhIbM41D1xkTOzartbn5X62X2qgeZFwmqWWSLj+KUoGtwvPT8ZBrRq2YOiBUc7crpmOiCbUuoJILwV89eR3atarv+O6q0qjncRThDM7hEny4hgbcQhNaQOEBnuEV3pBCL+gdfSxbCyifOYU/Qp8/nKyPGw==</latexit>
<latexit sha1_base64="dmkWiF/WObZg/13ZckxC73RvUjA=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RbDHghePFeyHtKVk09k2NMkuSVYoS3+FFw+KePXnePPfmLZ70NYXAg/vzJCZN0wEN9b3v73CxubW9k5xt7S3f3B4VD4+aZk41QybLBax7oTUoOAKm5ZbgZ1EI5WhwHY4uZ3X20+oDY/Vg50m2Jd0pHjEGbXOesx6YUTS2SAYlCt+1V+IrEOQQwVyNQblr94wZqlEZZmgxnQDP7H9jGrLmcBZqZcaTCib0BF2HSoq0fSzxcIzcuGcIYli7Z6yZOH+nsioNGYqQ9cpqR2b1drc/K/WTW1U62dcJalFxZYfRakgNibz68mQa2RWTB1QprnblbAx1ZRZl1HJhRCsnrwOratq4Pj+ulKv5XEU4QzO4RICuIE63EEDmsBAwjO8wpunvRfv3ftYtha8fOYU/sj7/AFRbZAL</latexit>
<latexit sha1_base64="hB3eIEvdn9cnPB/q9xjPbGG4vU8=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInspuEeyx4MVjBfshbSnZNNuGJtklmRXK0l/hxYMiXv053vw3pu0etPWFwMM7M2TmDRMpLPr+t7exubW9s1vYK+4fHB4dl05OWzZODeNNFsvYdEJquRSaN1Gg5J3EcKpCydvh5HZebz9xY0WsH3Ca8L6iIy0iwSg66zHrhRFJZ4PqoFT2K/5CZB2CHMqQqzEoffWGMUsV18gktbYb+An2M2pQMMlnxV5qeULZhI5416Gmitt+tlh4Ri6dMyRRbNzTSBbu74mMKmunKnSdiuLYrtbm5n+1bopRrZ8JnaTINVt+FKWSYEzm15OhMJyhnDqgzAi3K2FjaihDl1HRhRCsnrwOrWolcHx/Xa7X8jgKcA4XcAUB3EAd7qABTWCg4Ble4c0z3ov37n0sWze8fOYM/sj7/AFS8ZAM</latexit>
<latexit sha1_base64="nI5GP9PVkhFLAHqTc4LjyH8NVw0=">AAACJHicbVDLSsNAFJ34tr6qLt1cLIILKYkIFtwIunBZwarQhDCZTtqhk0mYuRFKyMe48VfcuPCBCzd+i5O2C18Hhjmcc++duSfKpDDouh/OzOzc/MLi0nJtZXVtfaO+uXVt0lwz3mGpTPVtRA2XQvEOCpT8NtOcJpHkN9HwrPJv7rg2IlVXOMp4kNC+ErFgFK0U1k98yWP0C/CpzAYUQihAgQclHIDfS9FU93frHEpfi/4A/TKsN9ymOwb8Jd6UNMgU7bD+akeyPOEKmaTGdD03w6CgGgWTvKz5ueEZZUPa511LFU24CYrxkiXsWaUHcartUQhj9XtHQRNjRklkKxOKA/Pbq8T/vG6OcSsohMpy5IpNHopzCZhClRj0hOYM5cgSyrSwfwU2oJoytLnWbAje75X/kuvDpmf55VHjtDWNY4nskF2yTzxyTE7JBWmTDmHknjySZ/LiPDhPzpvzPimdcaY92+QHnM8vcRWiMQ==</latexit>
Q. これってどういう値？
A. <latexit sha1_base64="696JsJ585x+VDv0wkUM8nUA2rdk=">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</latexit>
さっきの線形結合を書き換えると
<latexit sha1_base64="f7g+ECwcS2WLdEWbuHkybkohiI4=">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</latexit>

24定式化: 目的の設定
元のデータ点
M < D という限られた個数の変数を用いてこのデータを近似すること
目的
<latexit sha1_base64="VXtDWVAem+J8qXmCgxuaaaTVonM=">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</latexit>
<latexit sha1_base64="9ZtRb/zsSMmi/2ZoDTWzA3WUUME=">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</latexit>
近似
すべてのデータ点に共通各データ点に依存
もとのデータ点と近似が似るように，を最適化．<latexit sha1_base64="UAr+jQdqFyBuFEXNExzAabu0KfU=">AAACC3icbZDLSsNAFIYn9VbrLerSzdAiuJCSiGCXBTcuK9gLNCFMppN26GQSZiZCHbJ346u4caGIW1/AnW/jpM1CW38Y+PjPOZw5f5gyKpXjfFuVtfWNza3qdm1nd2//wD486skkE5h0ccISMQiRJIxy0lVUMTJIBUFxyEg/nF4X9f49EZIm/E7NUuLHaMxpRDFSxgrsuqe1F0YwywPq5efQ0w+B5jRfcFiYgd1wms5ccBXcEhqgVCewv7xRgrOYcIUZknLoOqnyNRKKYkbympdJkiI8RWMyNMhRTKSv57fk8NQ4Ixglwjyu4Nz9PaFRLOUsDk1njNRELtcK87/aMFNRy9eUp5kiHC8WRRmDKoFFMHBEBcGKzQwgLKj5K8QTJBBWJr6aCcFdPnkVehdN1/DtZaPdKuOoghNQB2fABVegDW5AB3QBBo/gGbyCN+vJerHerY9Fa8UqZ47BH1mfP+Simug=</latexit>
ここからやること

25
歪み尺度 J をに関して最適化
定式化: 歪み尺度の定義
歪み尺度
<latexit sha1_base64="/YKDkkb9WYX8El7ufn8ozT6HG8A=">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</latexit>
<latexit sha1_base64="UAr+jQdqFyBuFEXNExzAabu0KfU=">AAACC3icbZDLSsNAFIYn9VbrLerSzdAiuJCSiGCXBTcuK9gLNCFMppN26GQSZiZCHbJ346u4caGIW1/AnW/jpM1CW38Y+PjPOZw5f5gyKpXjfFuVtfWNza3qdm1nd2//wD486skkE5h0ccISMQiRJIxy0lVUMTJIBUFxyEg/nF4X9f49EZIm/E7NUuLHaMxpRDFSxgrsuqe1F0YwywPq5efQ0w+B5jRfcFiYgd1wms5ccBXcEhqgVCewv7xRgrOYcIUZknLoOqnyNRKKYkbympdJkiI8RWMyNMhRTKSv57fk8NQ4Ixglwjyu4Nz9PaFRLOUsDk1njNRELtcK87/aMFNRy9eUp5kiHC8WRRmDKoFFMHBEBcGKzQwgLKj5K8QTJBBWJr6aCcFdPnkVehdN1/DtZaPdKuOoghNQB2fABVegDW5AB3QBBo/gGbyCN+vJerHerY9Fa8UqZ47BH1mfP+Simug=</latexit>
目的

26定式化: z に関する最小化
に関して最小化<latexit sha1_base64="2btvX1v7Dww6eJDZI4lsnds2GqY=">AAACBXicbZDLSsNAFIYnXmu9RV3qYrAIrkoigl0W3LisYC/QhDCZTtqhk0mYORFqyMaNr+LGhSJufQd3vo3TNgtt/WHg4z/ncOb8YSq4Bsf5tlZW19Y3Nitb1e2d3b19++Cwo5NMUdamiUhULySaCS5ZGzgI1ksVI3EoWDccX0/r3XumNE/kHUxS5sdkKHnEKQFjBfaJJ1gEXo4fcIBzLDHHBfYUH47AKwK75tSdmfAyuCXUUKlWYH95g4RmMZNABdG67zop+DlRwKlgRdXLNEsJHZMh6xuUJGbaz2dXFPjMOAMcJco8CXjm/p7ISaz1JA5NZ0xgpBdrU/O/Wj+DqOHnXKYZMEnni6JMYEjwNBI84IpREBMDhCpu/orpiChCwQRXNSG4iycvQ+ei7hq+vaw1G2UcFXSMTtE5ctEVaqIb1EJtRNEjekav6M16sl6sd+tj3rpilTNH6I+szx/MFZdy</latexit>
<latexit sha1_base64="vCoIhPj627861zb7K3BKVPDvq8o=">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</latexit>
に関する微分を 0 とおくと，<latexit sha1_base64="2btvX1v7Dww6eJDZI4lsnds2GqY=">AAACBXicbZDLSsNAFIYnXmu9RV3qYrAIrkoigl0W3LisYC/QhDCZTtqhk0mYORFqyMaNr+LGhSJufQd3vo3TNgtt/WHg4z/ncOb8YSq4Bsf5tlZW19Y3Nitb1e2d3b19++Cwo5NMUdamiUhULySaCS5ZGzgI1ksVI3EoWDccX0/r3XumNE/kHUxS5sdkKHnEKQFjBfaJJ1gEXo4fcIBzLDHHBfYUH47AKwK75tSdmfAyuCXUUKlWYH95g4RmMZNABdG67zop+DlRwKlgRdXLNEsJHZMh6xuUJGbaz2dXFPjMOAMcJco8CXjm/p7ISaz1JA5NZ0xgpBdrU/O/Wj+DqOHnXKYZMEnni6JMYEjwNBI84IpREBMDhCpu/orpiChCwQRXNSG4iycvQ+ei7hq+vaw1G2UcFXSMTtE5ctEVaqIb1EJtRNEjekav6M16sl6sd+tj3rpilTNH6I+szx/MFZdy</latexit>
<latexit sha1_base64="Dunr+h2Q8U5hpUNTW2QqDjFYbs0=">AAACNnicbVBNS8MwGE79nPOr6tFLcAieRiuCuwgDL16ECfuCrZY0S7e4NC1JKs7SX+XF3+FtFw+KePUnmHY9zM0XQp48z/Mmbx4vYlQqy5oaK6tr6xubpa3y9s7u3r55cNiWYSwwaeGQhaLrIUkY5aSlqGKkGwmCAo+Rjje+zvTOIxGShrypJhFxAjTk1KcYKU255u0zdGECOXyAKbyC/QCpkedr5kmfZ0oK7/WeKyLQqKmZdM4ZF059g2tWrKqVF1wGdgEqoKiGa771ByGOA8IVZkjKnm1FykmQUBQzkpb7sSQRwmM0JD0NOQqIdJL82yk81cwA+qHQiyuYs/MdCQqknASedmbDykUtI//TerHya05CeRQrwvHsIT9mUIUwyxAOqCBYsYkGCAuqZ4V4hATCSidd1iHYi19eBu3zqq3x3UWlXiviKIFjcALOgA0uQR3cgAZoAQxewBR8gE/j1Xg3vozvmXXFKHqOwJ8yfn4BskSnrw==</latexit>

27定式化: b に関する最小化
に関して最小化<latexit sha1_base64="91vafD0NYl+dK0o1PP9KGOCdkZM=">AAACA3icbZDLSsNAFIYn9VbrLepON4NFcFUSEeyy4MZlBXuBJoTJdNIOnUzCzIlQQsCNr+LGhSJufQl3vo3TNgtt/WHg4z/ncOb8YSq4Bsf5tipr6xubW9Xt2s7u3v6BfXjU1UmmKOvQRCSqHxLNBJesAxwE66eKkTgUrBdObmb13gNTmifyHqYp82MykjzilICxAvvEEywCL8chDnCOOS6wp/hoDF4R2HWn4cyFV8EtoY5KtQP7yxsmNIuZBCqI1gPXScHPiQJOBStqXqZZSuiEjNjAoCQx034+v6HA58YZ4ihR5knAc/f3RE5iradxaDpjAmO9XJuZ/9UGGURNP+cyzYBJulgUZQJDgmeB4CFXjIKYGiBUcfNXTMdEEQomtpoJwV0+eRW6lw3X8N1VvdUs46iiU3SGLpCLrlEL3aI26iCKHtEzekVv1pP1Yr1bH4vWilXOHKM/sj5/AHPwlrg=</latexit>
<latexit sha1_base64="vCoIhPj627861zb7K3BKVPDvq8o=">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</latexit>
に関する微分を 0 とおくと，
<latexit sha1_base64="y1Ojjw7PP/u2weGhgvxYwV6Aeb0=">AAACOnicbVBNS8MwGE79nPOr6tFLcAieRiuCuwgDLx432BesdaRZusUlbUlScZT+Li/+Cm8evHhQxKs/wLSrqJsvBJ48z/O+efN4EaNSWdaTsbS8srq2Xtoob25t7+yae/sdGcYCkzYOWSh6HpKE0YC0FVWM9CJBEPcY6XqTy0zv3hIhaRi01DQiLkejgPoUI6Wpgdn04AAm8Aam8AI6obZmkzTjcKTGnq/RndZSeP3NCa5RK+d+PLG+FXMGZsWqWnnBRWAXoAKKagzMR2cY4piTQGGGpOzbVqTcBAlFMSNp2YkliRCeoBHpaxggTqSb5F9P4bFmhtAPhT6Bgjn7uyNBXMop97QzW1bOaxn5n9aPlV9zExpEsSIBnj3kxwyqEGY5wiEVBCs21QBhQfWuEI+RQFjptMs6BHv+y4ugc1q1NW6eVeq1Io4SOARH4ATY4BzUwRVogDbA4B48g1fwZjwYL8a78TGzLhlFzwH4U8bnFykZqgQ=</latexit>
<latexit sha1_base64="91vafD0NYl+dK0o1PP9KGOCdkZM=">AAACA3icbZDLSsNAFIYn9VbrLepON4NFcFUSEeyy4MZlBXuBJoTJdNIOnUzCzIlQQsCNr+LGhSJufQl3vo3TNgtt/WHg4z/ncOb8YSq4Bsf5tipr6xubW9Xt2s7u3v6BfXjU1UmmKOvQRCSqHxLNBJesAxwE66eKkTgUrBdObmb13gNTmifyHqYp82MykjzilICxAvvEEywCL8chDnCOOS6wp/hoDF4R2HWn4cyFV8EtoY5KtQP7yxsmNIuZBCqI1gPXScHPiQJOBStqXqZZSuiEjNjAoCQx034+v6HA58YZ4ihR5knAc/f3RE5iradxaDpjAmO9XJuZ/9UGGURNP+cyzYBJulgUZQJDgmeB4CFXjIKYGiBUcfNXTMdEEQomtpoJwV0+eRW6lw3X8N1VvdUs46iiU3SGLpCLrlEL3aI26iCKHtEzekVv1pP1Yr1bH4vWilXOHKM/sj5/AHPwlrg=</latexit>

28定式化: 変位ベクトルはどんな向き?
変位を表すベクトル (12.9, 12.10, 12.12, 12.13 より)
歪み尺度
<latexit sha1_base64="/YKDkkb9WYX8El7ufn8ozT6HG8A=">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</latexit>
<latexit sha1_base64="a61x5kRuwAk3ZZa32yt1PBFdOGc=">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</latexit>
「変位を表すベクトルは主部分空間に直交する空間にある」

29定式化: u に関する最小化
に関して最小化<latexit sha1_base64="PiUzA6QhwTj0iaifBJpQCTIvlrw=">AAACD3icbVDLSsNAFJ3UV62vqEs3g0VxVRIR7LLgxmUF+4AmhMl00g6dTMLMjVBC/8CNv+LGhSJu3brzb5y0WWjrgeEezrmXufeEqeAaHOfbqqytb2xuVbdrO7t7+wf24VFXJ5mirEMTkah+SDQTXLIOcBCsnypG4lCwXji5KfzeA1OaJ/IepinzYzKSPOKUgJEC+9wTLAIvx15MYBxGOMcZnuHAVG6qp/hoDN4ssOtOw5kDrxK3JHVUoh3YX94woVnMJFBBtB64Tgp+ThRwKtis5mWapYROyIgNDJUkZtrP5/fM8JlRhjhKlHkS8Fz9PZGTWOtpHJrOYmu97BXif94gg6jp51ymGTBJFx9FmcCQ4CIcPOSKURBTQwhV3OyK6ZgoQsFEWDMhuMsnr5LuZcM1/O6q3mqWcVTRCTpFF8hF16iFblEbdRBFj+gZvaI368l6sd6tj0VrxSpnjtEfWJ8/IrGbaQ==</latexit>
<latexit sha1_base64="9R8vhdwu9PccSq7h/GdEKFUgC2c=">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</latexit>
無意味な結果を避けるため，制約を課して最小化<latexit sha1_base64="EH1HX0ZeSN1UtDQ04ksJKxfW+9c=">AAACAnicbVDLSsNAFL2pr1pfUVfiZrAIrkoiBbsRCm5cVrAPaEOYTCft0MkkzEyEEoobf8WNC0Xc+hXu/BsnbRbaemC4h3PuZe49QcKZ0o7zbZXW1jc2t8rblZ3dvf0D+/Coo+JUEtomMY9lL8CKciZoWzPNaS+RFEcBp91gcpP73QcqFYvFvZ4m1IvwSLCQEayN5NsngwjrcRCiDKVohnxTmanXyPHtqlNz5kCrxC1IFQq0fPtrMIxJGlGhCcdK9V0n0V6GpWaE01llkCqaYDLBI9o3VOCIKi+bnzBD50YZojCW5gmN5urviQxHSk2jwHTmC6tlLxf/8/qpDhtexkSSairI4qMw5UjHKM8DDZmkRPOpIZhIZnZFZIwlJtqkVjEhuMsnr5LOZc01/K5ebTaKOMpwCmdwAS5cQRNuoQVtIPAIz/AKb9aT9WK9Wx+L1pJVzBzDH1ifP+4olSM=</latexit>
数式的な解を求める前に．．．

30定式化: u に関する最小化
直感的理解のため，D = 2，M = 1 の場合を考える．
規格化条件 <latexit sha1_base64="pxM21BHUhNHATDOm8L62Y2grfEM=">AAACLHicbVDLSgMxFM34rPU16tJNsAiuykwR7EYodOOyQl/QjkMmzbShSWZIMkIZ5oPc+CuCuLCIW7/DdDoLbXsg3MM595J7TxAzqrTjzK2t7Z3dvf3SQfnw6Pjk1D4776ookZh0cMQi2Q+QIowK0tFUM9KPJUE8YKQXTJsLv/dMpKKRaOtZTDyOxoKGFCNtJN9uDjnSkyCEKUxgBn1Ta6Y+mZo7khvWNkoGN3feQ9e3K07VyQHXiVuQCijQ8u334SjCCSdCY4aUGrhOrL0USU0xI1l5mCgSIzxFYzIwVCBOlJfmx2bw2igjGEbSPKFhrv6dSBFXasYD07lYWK16C3GTN0h0WPdSKuJEE4GXH4UJgzqCi+TgiEqCNZsZgrCkZleIJ0girE2+ZROCu3ryOunWqq7hj7eVRr2IowQuwRW4AS64Aw3wAFqgAzB4AW/gE8ytV+vD+rK+l61bVjFzAf7B+vkFEf2jvA==</latexit>
歪み尺度を最小化 <latexit sha1_base64="cnjAbdBGEVN60J95EDMjnoHi6Jc=">AAACOnicbVDLSsNAFJ3UV62vqEs3g0VwVZIi2I1QcCOuWuwL2hgm00k7dDIJMxOhhHyXG7/CnQs3LhRx6wc4SbvowwvDPXPOvdx7jxcxKpVlvRmFjc2t7Z3ibmlv/+DwyDw+6cgwFpi0cchC0fOQJIxy0lZUMdKLBEGBx0jXm9xmeveJCElD3lLTiDgBGnHqU4yUplyzeQ9v4CBAauz5MIExTKGrc1XnR51zRQQatTSTLlQ+LP0W+lyzbFWsPOA6sOegDObRcM3XwTDEcUC4wgxJ2betSDkJEopiRtLSIJYkQniCRqSvIUcBkU6Sn57CC80MoR8K/biCObvYkaBAymng6cpsWbmqZeR/Wj9Wfs1JKI9iRTieDfJjBlUIMx/hkAqCFZtqgLCgeleIx0ggrLTbJW2CvXryOuhUK7bGzatyvTa3owjOwDm4BDa4BnVwBxqgDTB4Bu/gE3wZL8aH8W38zEoLxrznFCyF8fsHZbSo+g==</latexit>
<latexit sha1_base64="IziZwGr1c6oX2R32pc/OItprJIk=">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</latexit>
ラグランジュ未定乗数法より
に関して微分を 0 とおくと<latexit sha1_base64="khX90j16n0jbvK4nJ9g8nQSPIPs=">AAAB/nicbVDLSsNAFL2pr1pfUXHlZrAIrkpSBLssuHFZwT6gDWEynbRDJ5MwMxFKCPgrblwo4tbvcOffOGmz0NYDwxzOuZc5c4KEM6Ud59uqbGxube9Ud2t7+weHR/bxSU/FqSS0S2Iey0GAFeVM0K5mmtNBIimOAk77wey28PuPVCoWiwc9T6gX4YlgISNYG8m3z0YR1tMgRBlKUY58czdR7tt1p+EsgNaJW5I6lOj49tdoHJM0okITjpUauk6ivQxLzQineW2UKppgMsMTOjRU4IgqL1vEz9GlUcYojKU5QqOF+nsjw5FS8ygwk0VYteoV4n/eMNVhy8uYSFJNBVk+FKYc6RgVXaAxk5RoPjcEE8lMVkSmWGKiTWM1U4K7+uV10ms2XMPvr+vtVllHFc7hAq7AhRtowx10oAsEMniGV3iznqwX6936WI5WrHLnFP7A+vwB4u6UFw==</latexit>
<latexit sha1_base64="42E7EGK0gDNjFvItcsbB9t4sqsQ=">AAACKHicbVBdS8MwFE3n15xfVR99CQ7Bp9EOwb2IA198nOg+YC0lTdMtLE1Lkgqj7Of44l/xRUSRvfpLTLeic/NCyOGcc5N7j58wKpVlTY3S2vrG5lZ5u7Kzu7d/YB4edWScCkzaOGax6PlIEkY5aSuqGOklgqDIZ6Trj25yvftIhKQxf1DjhLgRGnAaUoyUpjzz2omQGvohzOA9TOEEehrV9X0FHaafCdAP8+tc8Hlm1apZs4KrwC5AFRTV8sw3J4hxGhGuMENS9m0rUW6GhKKYkUnFSSVJEB6hAelryFFEpJvNFp3AM80EMIyFPlzBGbvYkaFIynHka2c+rFzWcvI/rZ+qsOFmlCepIhzPPwpTBlUM89RgQAXBio01QFhQPSvEQyQQVjrbig7BXl55FXTqNVvju4tqs1HEUQYn4BScAxtcgia4BS3QBhg8gRfwDj6MZ+PV+DSmc2vJKHqOwZ8yvr4BAJKiNQ==</latexit>
S の 2 つの固有値のうち，
小さい方に属する固有ベクトルのとき最小．

31誤差最小化問題の一般解
任意の D と任意の M < D に対する一般化 [Ex. 12.2]
<latexit sha1_base64="woyhNiUFYlkKpEQUVGK/BuPLsEI=">AAACGXicbVDLSgMxFM34rPU16tLNxSIIQpkRwW6Egi5EECrYB3TGIZNJ29DMgyQjlGF+w42/4saFIi515d+YtrPQ1gsh555z703u8RPOpLKsb2NhcWl5ZbW0Vl7f2NzaNnd2WzJOBaFNEvNYdHwsKWcRbSqmOO0kguLQ57TtDy/GevuBCsni6E6NEuqGuB+xHiNYacozrWs4B0emIXiQAdPJDRyDDTnc6/xS3w7X0wJc6LlnVqyqNQmYB3YBKqiIhmd+OkFM0pBGinAsZde2EuVmWChGOM3LTippgskQ92lXwwiHVLrZZLMcDjUTQC8W+kQKJuzvjgyHUo5CX1eGWA3krDYm/9O6qerV3IxFSapoRKYP9VIOKoaxTRAwQYniIw0wEUz/FcgAC0yUNrOsTbBnV54HrZOqrfHtaaVeK+wooX10gI6Qjc5QHV2hBmoigh7RM3pFb8aT8WK8Gx/T0gWj6NlDf8L4+gHaHJvW</latexit>
歪み尺度の値
一般解 S の固有値の小さいものから D-M 個に対応する固有ベクトル
<latexit sha1_base64="fvwaCMVHGzCWYqHIJxZMLXvfPvs=">AAACC3icbVDNS8MwHE3n15xfVY9ewoYgKKMVwR0HevAiTHAfsJaSpukWljYlSYVRevfiv+LFgyJe/Qe8+d+Ybj3o5oOQx3vvR/J7fsKoVJb1bVRWVtfWN6qbta3tnd09c/+gJ3kqMOlizrgY+EgSRmPSVVQxMkgEQZHPSN+fXBV+/4EISXl8r6YJcSM0imlIMVJa8sy6w3Q4QF52e2rnZ9BhAVeyuEv5OvfMhtW0ZoDLxC5JA5ToeOaXE3CcRiRWmCEph7aVKDdDQlHMSF5zUkkShCdoRIaaxigi0s1mu+TwWCsBDLnQJ1Zwpv6eyFAk5TTydTJCaiwXvUL8zxumKmy5GY2TVJEYzx8KUwYVh0UxMKCCYMWmmiAsqP4rxGMkEFa6vpouwV5ceZn0zpu25ncXjXarrKMKjkAdnAAbXII2uAEd0AUYPIJn8ArejCfjxXg3PubRilHOHII/MD5/AKw0miQ=</latexit>
<latexit sha1_base64="j8Yhysltsrd6mU5Eg7r5x44HDKw=">AAACC3icbZDLSsNAFIYn9VbrLerSzdAiCJaSiGCXBV24ESrYCzQhTCaTdugkE2YmQgnZu/FV3LhQxK0v4M63cdoG0dYfBj7+cw5nzu8njEplWV9GaWV1bX2jvFnZ2t7Z3TP3D7qSpwKTDuaMi76PJGE0Jh1FFSP9RBAU+Yz0/PHltN67J0JSHt+pSULcCA1jGlKMlLY8s5o5fgjT3MtuTu28Dh0WcCXr8Me+yj2zZjWsmeAy2AXUQKG2Z346AcdpRGKFGZJyYFuJcjMkFMWM5BUnlSRBeIyGZKAxRhGRbja7JYfH2glgyIV+sYIz9/dEhiIpJ5GvOyOkRnKxNjX/qw1SFTbdjMZJqkiM54vClEHF4TQYGFBBsGITDQgLqv8K8QgJhJWOr6JDsBdPXobuWcPWfHteazWLOMrgCFTBCbDBBWiBa9AGHYDBA3gCL+DVeDSejTfjfd5aMoqZQ/BHxsc3hiyaDA==</latexit>
M < D という場合を考えてきたが，M = D の場合でも PCA 適用可．
主成分に沿うように座標軸の回転がするだけ．

32定式化のまとめ
分散最大化: 射影されたデータの分散が最大になる方向を選ぶ
誤差最大化: 射影されたデータが元のデータと離れない方向を選ぶ
<latexit sha1_base64="j8Yhysltsrd6mU5Eg7r5x44HDKw=">AAACC3icbZDLSsNAFIYn9VbrLerSzdAiCJaSiGCXBV24ESrYCzQhTCaTdugkE2YmQgnZu/FV3LhQxK0v4M63cdoG0dYfBj7+cw5nzu8njEplWV9GaWV1bX2jvFnZ2t7Z3TP3D7qSpwKTDuaMi76PJGE0Jh1FFSP9RBAU+Yz0/PHltN67J0JSHt+pSULcCA1jGlKMlLY8s5o5fgjT3MtuTu28Dh0WcCXr8Me+yj2zZjWsmeAy2AXUQKG2Z346AcdpRGKFGZJyYFuJcjMkFMWM5BUnlSRBeIyGZKAxRhGRbja7JYfH2glgyIV+sYIz9/dEhiIpJ5GvOyOkRnKxNjX/qw1SFTbdjMZJqkiM54vClEHF4TQYGFBBsGITDQgLqv8K8QgJhJWOr6JDsBdPXobuWcPWfHteazWLOMrgCFTBCbDBBWiBa9AGHYDBA3gCL+DVeDSejTfjfd5aMoqZQ/BHxsc3hiyaDA==</latexit>
<latexit sha1_base64="AyNwNw9TSX3ViaO8StJ9HsFqPdI=">AAACCXicbZDLSsNAFIYnXmu9RV26GSyCi1ISEeyy4MaNUMFeoAlhMpm0QyczYWYilJCtG1/FjQtF3PoG7nwbp21Abf1h4OM/53Dm/GHKqNKO82WtrK6tb2xWtqrbO7t7+/bBYVeJTGLSwYIJ2Q+RIoxy0tFUM9JPJUFJyEgvHF9N6717IhUV/E5PUuInaMhpTDHSxgpsmHthDLMiyN2iDj0WCa3qP+ZNEdg1p+HMBJfBLaEGSrUD+9OLBM4SwjVmSKmB66Taz5HUFDNSVL1MkRThMRqSgUGOEqL8fHZJAU+NE8FYSPO4hjP390SOEqUmSWg6E6RHarE2Nf+rDTIdN/2c8jTThOP5ojhjUAs4jQVGVBKs2cQAwpKav0I8QhJhbcKrmhDcxZOXoXvecA3fXtRazTKOCjgGJ+AMuOAStMA1aIMOwOABPIEX8Go9Ws/Wm/U+b12xypkj8EfWxzeBO5mJ</latexit>

33正準相関分析 (CCA; canonical correlation analysis)
主成分分析
1つの確率変数を扱う．
正準相関分析
複数の確率変数を考え，高い交差相関を持つ線形の部分空間の対を見出す．
結果，一方の部分空間におけるひとつの成分は，
別の部分空間のあるひとつの成分と相関を持つ．
一般化固有値問題の買いとして表現できる．

12.1.3 主成分分析の応用

35主成分の可視化
例ずらし数字データ
青: 正値
白: 0
黄: 負値
…
<latexit sha1_base64="dmkWiF/WObZg/13ZckxC73RvUjA=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RbDHghePFeyHtKVk09k2NMkuSVYoS3+FFw+KePXnePPfmLZ70NYXAg/vzJCZN0wEN9b3v73CxubW9k5xt7S3f3B4VD4+aZk41QybLBax7oTUoOAKm5ZbgZ1EI5WhwHY4uZ3X20+oDY/Vg50m2Jd0pHjEGbXOesx6YUTS2SAYlCt+1V+IrEOQQwVyNQblr94wZqlEZZmgxnQDP7H9jGrLmcBZqZcaTCib0BF2HSoq0fSzxcIzcuGcIYli7Z6yZOH+nsioNGYqQ9cpqR2b1drc/K/WTW1U62dcJalFxZYfRakgNibz68mQa2RWTB1QprnblbAx1ZRZl1HJhRCsnrwOratq4Pj+ulKv5XEU4QzO4RICuIE63EEDmsBAwjO8wpunvRfv3ftYtha8fOYU/sj7/AFRbZAL</latexit> <latexit sha1_base64="hB3eIEvdn9cnPB/q9xjPbGG4vU8=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInspuEeyx4MVjBfshbSnZNNuGJtklmRXK0l/hxYMiXv053vw3pu0etPWFwMM7M2TmDRMpLPr+t7exubW9s1vYK+4fHB4dl05OWzZODeNNFsvYdEJquRSaN1Gg5J3EcKpCydvh5HZebz9xY0WsH3Ca8L6iIy0iwSg66zHrhRFJZ4PqoFT2K/5CZB2CHMqQqzEoffWGMUsV18gktbYb+An2M2pQMMlnxV5qeULZhI5416Gmitt+tlh4Ri6dMyRRbNzTSBbu74mMKmunKnSdiuLYrtbm5n+1bopRrZ8JnaTINVt+FKWSYEzm15OhMJyhnDqgzAi3K2FjaihDl1HRhRCsnrwOrWolcHx/Xa7X8jgKcA4XcAUB3EAd7qABTWCg4Ble4c0z3ov37n0sWze8fOYM/sj7/AFS8ZAM</latexit>
<latexit sha1_base64="h5AWn8/fo3vhELQmaRGU5BDW4NY=">AAAB8HicbZBNSwMxEIZn/az1q+rRS7AInsquCvZY8OKxgv2QtpRsOtuGJtklyQpl6a/w4kERr/4cb/4b03YP2vpC4OGdGTLzhongxvr+t7e2vrG5tV3YKe7u7R8clo6OmyZONcMGi0Ws2yE1KLjChuVWYDvRSGUosBWOb2f11hNqw2P1YCcJ9iQdKh5xRq2zHrNuGJF02r/ql8p+xZ+LrEKQQxly1fulr+4gZqlEZZmgxnQCP7G9jGrLmcBpsZsaTCgb0yF2HCoq0fSy+cJTcu6cAYli7Z6yZO7+nsioNGYiQ9cpqR2Z5drM/K/WSW1U7WVcJalFxRYfRakgNiaz68mAa2RWTBxQprnblbAR1ZRZl1HRhRAsn7wKzctK4Pj+ulyr5nEU4BTO4AICuIEa3EEdGsBAwjO8wpunvRfv3ftYtK55+cwJ/JH3+QNUdZAN</latexit>
<latexit sha1_base64="a7sHWXlqpgepE0KPjEqsOGJrDa0=">AAAB8HicbZBNSwMxEIZn61etX1WPXoJF8FR2RbDHghePFeyHtKVk02wbmmSXZFYoS3+FFw+KePXnePPfmLZ70NYXAg/vzJCZN0yksOj7315hY3Nre6e4W9rbPzg8Kh+ftGycGsabLJax6YTUcik0b6JAyTuJ4VSFkrfDye283n7ixopYP+A04X1FR1pEglF01mPWCyOSzgbXg3LFr/oLkXUIcqhArsag/NUbxixVXCOT1Npu4CfYz6hBwSSflXqp5QllEzriXYeaKm772WLhGblwzpBEsXFPI1m4vycyqqydqtB1Kopju1qbm//VuilGtX4mdJIi12z5UZRKgjGZX0+GwnCGcuqAMiPcroSNqaEMXUYlF0KwevI6tK6qgeP760q9lsdRhDM4h0sI4AbqcAcNaAIDBc/wCm+e8V68d+9j2Vrw8plT+CPv8wdV+ZAO</latexit>
全て足せば，元どおり．
<latexit sha1_base64="5MISuVoWCeJVUxCMHkQz+XRThsQ=">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</latexit>

36主成分と歪み尺度
固有値スペクトル歪み尺度

37前処理としての PCA
<latexit sha1_base64="JL3LL5RiSsQq2pnEq18To1YgqLM=">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</latexit>
全データ点で共通の項
各データ点依存の項
主成分近似

標準化
平均を 0，共分散行列を単位行列に．
<latexit sha1_base64="0p9ywXkc8Ru4+u3p48012Fy7fh8=">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</latexit>
個々の変数に対して，平均が 0 分散が 1 となるように個別に線形変換
標準化されたデータの共分散行列の要素
…相関係数
白色化
(PCA)
<latexit sha1_base64="P7JZDxW5S2YUOU5LBeP8Y195sWs=">AAACCHicbZDLSsNAFIZP6q3WW9SlCweL4KokItiNUHDjwkVF0xbaUCbTSTt0cmFmIpSQpRtfxY0LRdz6CO58GydtQG39YeDjP+cw5/xezJlUlvVllJaWV1bXyuuVjc2t7R1zd68lo0QQ6pCIR6LjYUk5C6mjmOK0EwuKA4/Ttje+zOvteyoki8I7NYmpG+BhyHxGsNJW3zzsBViNPB+l6NZBGbpAP4ZzjbK+WbVq1lRoEewCqlCo2Tc/e4OIJAENFeFYyq5txcpNsVCMcJpVeomkMSZjPKRdjSEOqHTT6SEZOtbOAPmR0C9UaOr+nkhxIOUk8HRnvqWcr+Xmf7Vuovy6m7IwThQNyewjP+FIRShPBQ2YoETxiQZMBNO7IjLCAhOls6voEOz5kxehdVqzNd+cVRv1Io4yHMARnIAN59CAK2iCAwQe4Ale4NV4NJ6NN+N91loyipl9+CPj4xvdU5fp</latexit>
<latexit sha1_base64="ggMW/4nLQSq/NDknxgKcNlio8hU=">AAACNHicbVBLS8NAEJ7UV62vqEcvi0XwVBIR7EUoeBG8VLSt0Iaw2WzapZsHuxuhhP4oL/4QLyJ4UMSrv8FNG3y0HVjmm28eO/N5CWdSWdaLUVpaXlldK69XNja3tnfM3b22jFNBaIvEPBZ3HpaUs4i2FFOc3iWC4tDjtOMNL/J8554KyeLoVo0S6oS4H7GAEaw05ZpXvRCrgRegDN2gMfqNUh252jPtz1GP65E+/mEW1rlm1apZE0PzwC5AFQpruuZTz49JGtJIEY6l7NpWopwMC8UIp+NKL5U0wWSI+7SrYYRDKp1scvQYHWnGR0Es9IsUmrB/OzIcSjkKPV2ZLytnczm5KNdNVVB3MhYlqaIRmX4UpBypGOUKIp8JShQfaYCJYHpXRAZYYKK0zhUtgj178jxon9Rsja9Pq416IUcZDuAQjsGGM2jAJTShBQQe4Bne4N14NF6ND+NzWloyip59+GfG1zcxaad4</latexit>
をまとめて
<latexit sha1_base64="EJWdhnAujm3d9V599ElSlfntp5A=">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</latexit>
として変数変換

白色化・球状化 <latexit sha1_base64="EJWdhnAujm3d9V599ElSlfntp5A=">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</latexit>
Q. ほんとうに共分散行列が単位行列になっているか？
A.
<latexit sha1_base64="0PeOnRZT2P8QUjY1xA+Ar3z/WS8=">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</latexit>

オリジナル標準化白色化

41フィッシャーの線形判別と PCA の比較
フィッシャーの線形判別 (supervised)
クラス間分散大きく
クラス内分散小さく
PCA (unsupervised)

12.1.4 高次元データに対する主成分分析

43高次元データに対する PCA の問題
FHD (1920x1080) で
約200万次元
例: 数百枚の画像データセット，各データが数百万次元
<latexit sha1_base64="awUoR7j22wSPEyE3a9cATRDn1sI=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeyx4MWTtGA/oA1ls520azebsLsRSugv8OJBEa/+JG/+G7dtDtr6wsLDOzPszBskgmvjut9OYWNza3unuFva2z84PCofn7R1nCqGLRaLWHUDqlFwiS3DjcBuopBGgcBOMLmd1ztPqDSP5YOZJuhHdCR5yBk11mreD8oVt+ouRNbBy6ECuRqD8ld/GLM0QmmYoFr3PDcxfkaV4UzgrNRPNSaUTegIexYljVD72WLRGbmwzpCEsbJPGrJwf09kNNJ6GgW2M6JmrFdrc/O/Wi81Yc3PuExSg5ItPwpTQUxM5leTIVfIjJhaoExxuythY6ooMzabkg3BWz15HdpXVc9y87pSr+VxFOEMzuESPLiBOtxBA1rAAOEZXuHNeXRenHfnY9lacPKZU/gj5/MHo5GMyA==</latexit> <latexit sha1_base64="ahsCYXUAldde9fVJe64ujOTGLDQ=">AAAB6HicbZBNS8NAEIYn9avWr6pHL4tF8FQSEeyxoAePLdgPaEPZbCft2s0m7G6EEvoLvHhQxKs/yZv/xm2bg7a+sPDwzgw78waJ4Nq47rdT2Njc2t4p7pb29g8Oj8rHJ20dp4phi8UiVt2AahRcYstwI7CbKKRRILATTG7n9c4TKs1j+WCmCfoRHUkeckaNtZp3g3LFrboLkXXwcqhArsag/NUfxiyNUBomqNY9z02Mn1FlOBM4K/VTjQllEzrCnkVJI9R+tlh0Ri6sMyRhrOyThizc3xMZjbSeRoHtjKgZ69Xa3Pyv1ktNWPMzLpPUoGTLj8JUEBOT+dVkyBUyI6YWKFPc7krYmCrKjM2mZEPwVk9eh/ZV1bPcvK7Ua3kcRTiDc7gED26gDvfQgBYwQHiGV3hzHp0X5935WLYWnHzmFP7I+fwBlGmMvg==</latexit>
DxD 行列の固有値分解 <latexit sha1_base64="tpN15+iw5+FZ2JO/x254mhsgBVM=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8Ei1E2ZUcEuC7pwZwV7gXYsmTTTxmaSIckIZeg7uHGhiFvfx51vY9rOQlt/CHz85xxyzh/EnGnjut9ObmV1bX0jv1nY2t7Z3SvuHzS1TBShDSK5VO0Aa8qZoA3DDKftWFEcBZy2gtHVtN56okozKe7NOKZ+hAeChYxgY63mbfn64fy0Vyy5FXcmtAxeBiXIVO8Vv7p9SZKICkM41rrjubHxU6wMI5xOCt1E0xiTER7QjkWBI6r9dLbtBJ1Yp49CqewTBs3c3xMpjrQeR4HtjLAZ6sXa1Pyv1klMWPVTJuLEUEHmH4UJR0ai6emozxQlho8tYKKY3RWRIVaYGBtQwYbgLZ68DM2zimf57qJUq2Zx5OEIjqEMHlxCDW6gDg0g8AjP8ApvjnRenHfnY96ac7KZQ/gj5/MHH0iOIQ==</latexit>
NxN 行列の固有値分解に置き換えて計算量を削減!!<latexit sha1_base64="sDm9/I3fPLylU8nVKD8ps/wgbuM=">AAAB7XicbZDLSgMxFIbP1Futt6pLN8Ei1E2ZUcEuC25caQV7gXYsmTTTxmaSIckIZeg7uHGhiFvfx51vY9rOQlt/CHz85xxyzh/EnGnjut9ObmV1bX0jv1nY2t7Z3SvuHzS1TBShDSK5VO0Aa8qZoA3DDKftWFEcBZy2gtHVtN56okozKe7NOKZ+hAeChYxgY63mbfnm4fy0Vyy5FXcmtAxeBiXIVO8Vv7p9SZKICkM41rrjubHxU6wMI5xOCt1E0xiTER7QjkWBI6r9dLbtBJ1Yp49CqewTBs3c3xMpjrQeR4HtjLAZ6sXa1Pyv1klMWPVTJuLEUEHmH4UJR0ai6emozxQlho8tYKKY3RWRIVaYGBtQwYbgLZ68DM2zimf57qJUq2Zx5OEIjqEMHlxCDa6hDg0g8AjP8ApvjnRenHfnY96ac7KZQ/gj5/MHLo6OKw==</latexit>
計算量的に実効不可能

44高次元データ PCA の注意
D 次元空間の N 点は N-1 次元の部分空間を定義するため
3 次元空間の 2 点は 1 次元の部分空間
N-1 よりも大きい M の値に対する PCA はほとんど意味がない．
例 D=3，N=2, M=2
3 次元空間に 2 点．
直線上 (1次元部分空間) に必ず存在
すでに 1 次元の部分空間に存在するのに，2 次元に射影は意味なし．
<latexit sha1_base64="nsGez1CvXf2ALHb/KLM2afmht9o=">AAAB7HicbZA9SwNBEIbn4leMX1FLm8UgWIU7EUxhEdDCSiJ4SSA5wt5mL1myt3fszgkh5DfYWChi6w+y89+4Sa7QxBcWHt6ZYWfeMJXCoOt+O4W19Y3NreJ2aWd3b/+gfHjUNEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wjm1m99cS1EYl6xHHKg5gOlIgEo2gt/55ck9teueJW3bnIKng5VCBXo1f+6vYTlsVcIZPUmI7nphhMqEbBJJ+WupnhKWUjOuAdi4rG3AST+bJTcmadPokSbZ9CMnd/T0xobMw4Dm1nTHFolmsz879aJ8OoFkyESjPkii0+ijJJMCGzy0lfaM5Qji1QpoXdlbAh1ZShzadkQ/CWT16F5kXVs/xwWanX8jiKcAKncA4eXEEd7qABPjAQ8Ayv8OYo58V5dz4WrQUnnzmGP3I+fwBce42w</latexit>

45NxN から DxD の固有値分解へ
データ行列 <latexit sha1_base64="p5iV1arBYreEJZ1WLFdFFwgFq9Y=">AAAB7nicbZBNSwMxEIZn/az1q+rRS7AInsquCPZY8OKxgv2AdinZNNuGJtklmRXK0h/hxYMiXv093vw3pu0etPWFwMM7M2TmjVIpLPr+t7exubW9s1vaK+8fHB4dV05O2zbJDOMtlsjEdCNquRSat1Cg5N3UcKoiyTvR5G5e7zxxY0WiH3Ga8lDRkRaxYBSd1cn7UUy6s0Gl6tf8hcg6BAVUoVBzUPnqDxOWKa6RSWptL/BTDHNqUDDJZ+V+ZnlK2YSOeM+hporbMF+sOyOXzhmSODHuaSQL9/dETpW1UxW5TkVxbFdrc/O/Wi/DuB7mQqcZcs2WH8WZJJiQ+e1kKAxnKKcOKDPC7UrYmBrK0CVUdiEEqyevQ/u6Fjh+uKk26kUcJTiHC7iCAG6hAffQhBYwmMAzvMKbl3ov3rv3sWzd8IqZM/gj7/MH+1uPSg==</latexit>
<latexit sha1_base64="1ImZBmv85r/hmfM2zgxlVhIKzZY=">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</latexit>
第n行
<latexit sha1_base64="wliKsg5peq0eLibf43waTAm5ULI=">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</latexit>
<latexit sha1_base64="eib8iymPlvWnOhkh02uTrEXzFPY=">AAACNHicbZBNSwMxEIazftb6terRS7AIXiy7ItiLUPAiCFKxX9CuJZtm29AkuyRZoSz9UV78IV5E8KCIV3+D2e2C2joQeOadGTLz+hGjSjvOi7WwuLS8slpYK65vbG5t2zu7TRXGEpMGDlko2z5ShFFBGppqRtqRJIj7jLT80UVab90TqWgo6nocEY+jgaABxUgbqWdfdTnSQz+ACbyFE3gOr+Gd4WPomuyn1jZZqmeK5IbqRpnp6Nklp+xkAefBzaEE8qj17KduP8QxJ0JjhpTquE6kvQRJTTEjk2I3ViRCeIQGpGNQIE6Ul2RHT+ChUfowCKV5QsNM/T2RIK7UmPumM11SzdZS8b9aJ9ZBxUuoiGJNBJ5+FMQM6hCmDsI+lQRrNjaAsKRmV4iHSCKsjc9FY4I7e/I8NE/KruGb01K1kttRAPvgABwBF5yBKrgENdAAGDyAZ/AG3q1H69X6sD6nrQtWPrMH/oT19Q3trabK</latexit>
<latexit sha1_base64="+3fB09w2SskEEpQHZXxNKyOs7Hk=">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</latexit>
<latexit sha1_base64="AuzxghpdK7eAXYSAtCOmrC9nbzg=">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</latexit>
<latexit sha1_base64="rq/ezm8kA6TleEJttyzWW65QiYM=">AAACJHicbVBLS8NAEJ7UV62vqEcvi0XwVBIRLIhQ8OKxgn1AG8pmu2mXbjZhd1MooT/Gi3/FiwcfePDib3HTBqqtA8t888037Mznx5wp7ThfVmFtfWNzq7hd2tnd2z+wD4+aKkokoQ0S8Ui2fawoZ4I2NNOctmNJcehz2vJHt1m/NaZSsUg86ElMvRAPBAsYwdpQPfu6G2I99AOUojGaop7JzOQbtODbpl5UyULVs8tOxZkFWgVuDsqQR71nv3f7EUlCKjThWKmO68TaS7HUjHA6LXUTRWNMRnhAOwYKHFLlpbMjp+jMMH0URNI8odGM/T2R4lCpSegbZbasWu5l5H+9TqKDqpcyESeaCjL/KEg40hHKHEN9JinRfGIAJpKZXREZYomJNr6WjAnu8smroHlRcQ2+vyzXqrkdRTiBUzgHF66gBndQhwYQeIRneIU368l6sT6sz7m0YOUzx/AnrO8fxhShwQ==</latexit>
共分散行列
固有ベクトル方程式
変数変換
固有ベクトル方程式*
<latexit sha1_base64="LkzN3ss2wZ/mdAtZTXpMuxWbmw8=">AAACGHicbZBNS8NAEIYnftb6FfXoZbEInmoigj0WvHis0C9oY9lsN+3SzSbsboQS+jO8+Fe8eFDEa2/+GzdpDrZ1YNmHd2aYmdePOVPacX6sjc2t7Z3d0l55/+Dw6Ng+OW2rKJGEtkjEI9n1saKcCdrSTHPajSXFoc9px5/cZ/nOM5WKRaKppzH1QjwSLGAEayMN7Ot+iPXYD1CKumiGnsyfKzI01DTKDC1VDOyKU3XyQOvgFlCBIhoDe94fRiQJqdCEY6V6rhNrL8VSM8LprNxPFI0xmeAR7RkUOKTKS/PDZujSKEMURNI8oVGu/u1IcajUNPRNZbakWs1l4n+5XqKDmpcyESeaCrIYFCQc6QhlLqEhk5RoPjWAiWRmV0TGWGKijZdlY4K7evI6tG+qruHH20q9VthRgnO4gCtw4Q7q8AANaAGBF3iDD/i0Xq1368v6XpRuWEXPGSyFNf8FP6ed+g==</latexit>
<latexit sha1_base64="adrmjma4H6XIDe+VpVVwdosbdJ0=">AAACGHicbZA9T8MwEIYvfJbyFWBksaiQmEqCkOhYiYWxSP2S2lA5rtNadZzIdpCqqD+Dhb/CwgBCrN34NzhpBtpykuXH793Jd68fc6a04/xYG5tb2zu7pb3y/sHh0bF9ctpWUSIJbZGIR7LrY0U5E7Slmea0G0uKQ5/Tjj+5z/KdZyoVi0RTT2PqhXgkWMAI1kYa2NeoH2I99gOUoi6aLT3Qk7lzRYaGmkaZDeyKU3XyQOvgFlCBIhoDe94fRiQJqdCEY6V6rhNrL8VSM8LprNxPFI0xmeAR7RkUOKTKS/PFZujSKEMURNIcoVGu/u1IcajUNPRNZTamWs1l4n+5XqKDmpcyESeaCrL4KEg40hHKXEJDJinRfGoAE8nMrIiMscREGy/LxgR3deV1aN9UXcOPt5V6rbCjBOdwAVfgwh3U4QEa0AICL/AGH/BpvVrv1pf1vSjdsIqeM1gKa/4LP3+d+g==</latexit>
: DxD
: NxN
<latexit sha1_base64="KSXU5ptEn3wBTXGxsv6ZosZrF6Q=">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</latexit>
DxD の固有値分解をせずに
固有値を得ることができた!

PRML 12-12.1.4 主成分分析 (PCA) / Principal Component Analysis (PCA)

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