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PRML 5.5.6~5.6
5501 酒井⼀徳
3/22/18
1/31
目次
5.5.6 たたみ込みニューラルネットワーク
5.5.7 ソフト重み共有
5.6 混合密度ネットワーク
2/31
5.5.6 たたみ込みニューラルネットワーク
3/31
たたみ込みニューラルネットワーク
CNN : Convolutional Neural Network
畳み込み層とプーリング層,全結合層などから構成されるニューラルネットワーク.
タスク例: ⼿書き⽂字認識
⼊⼒: 画像の画素の輝度値の集合
出⼒: 10個の数字のクラスの事後分布
平⾏移動や拡⼤縮⼩,(⼩さい)
回転などにも
不変であるべき
これは6だ
どちらも6だと認識してほしい
4/31
CNN
完全結合ネットワーク
⼊⼒同⼠の相関性を無視
⼊⼒同⼠が相関性を持つネットワーク
画像であれば,隣り合う画素同⼠は相関性が⾼い等
inputs
outputs
⼊⼒について相関性などの知⾒があるなら
ネットワークをあらかじめ変形させよう
5/31
CNNの3つの機構
局所的受容野
部分サンプリング
重み共有
• 局所的な特徴の抽出
• その特徴の情報を統合することによりさらに⾼次の特徴の検出
6/31
局所受容野と重み共有(畳み込み層)
局所的受容野
全結合じゃない
重み共有
⼀部の⼊⼒の影響しか
受けない
7/31
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死ぬほど分かりやすい畳み込み演算の様子
⼊⼒画像の画素表記
(簡略のため2値)
フィルター(重み)
(簡略のためバイアス0を想定)
⼊⼒画像のあらゆる位置で同じパターン(特徴)を
抽出している
⼊⼒画像の平⾏移動は特徴マップの平⾏移動と
して現れる
(たたみ込み層)
複数の特徴に対応して複数の特徴マップ,重
みとバイアスが存在する
8/31
部分サンプリング(プーリング操作)
Max-pooling
多少の数値の⼊れ替わりがあっても影響がない
先ほどの畳み込みと合わせて平⾏移動や歪みに対して頑健になる
畳み込み層の出⼒(特徴マップ)
9/31
特徴同⼠がどのような関係であるかが希薄になる問題もある。(⇒カプセルネットワーク)
CNNの概観
𝑛個の重み
(25次元+バイアス)
5×5
15×15
10×10×𝑛
…
⼊⼒ conv層 pooling層
2×2
2×2 からサンプリング
…
5×5×𝑛
conv層
𝑚個の重み
(2×2次元+バイアス)
3×3×𝑛𝑚
出⼒全結合層
層を重ねるごとに空間解像度は減るが特徴数を増やすことで補償する
層を重ねるごとに不変性は強固になる
Conv層とpooling層を合わせて畳み込み層と呼ぶ場合もある
最後に集めた特徴を集約する,多クラス分類問題ならソフトマックス関数など
…
…
10/31
逆伝播法による誤差最小化(演習5.28)
順伝播
逆伝播
順伝播
逆伝播
…
11/31
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5.5.7 ソフト重み共有
12/31
ソフト重み共有
重み共有はネットワークの複雑さを削減する有効なテクニック
しかし,制限の形が特定される特別な問題にしか適⽤できない
ソフト重み共有
重みが同じグループで似た値を取りやすくなるように正則化項の導⼊
つまり,緑のグループや⾚のグループごとに
重みの平均や分散を設定しよう
さらにそれらを学習の過程で得よう
13/31
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ソフト重み共有ー荷重減衰の拡張 14/31
単純な荷重減衰正則化項(5.112)が, 重みのガウス事前分布の負の対数尤度とみなせる(5.5節,3.1.1〜3.1.4節)
つまり,このときの重みはガウス分布に従う⼀つのグループであるとみなせる
複数のグループを構成したい
混合ガウス分布を代わりに⽤いよう(2.3.9)
確率密度関数:
ただし,
負の対数尤度を取る
誤差関数:
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ソフト重み共有-誤差関数の最小化 15/31
調整可能なパラメータ
中⼼:
分散:
混合係数:
2.3.9節に従って負担率の導⼊(2.192),
を事前分布とみなすとベイズの定理から
事後分布:
パラメータである平均 ,分散,混合係数についても最⼩化し,決定する
• 重みが定数なら,EMアルゴリズム
• 重みと同時最適化が必要な場合,共役勾配法,準ニュートン法
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誤差関数の最小化-重み, 中心について 16/31
重み(演習5.29)
中⼼(演習5.30)
事後分布:
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=
X
j
j(wi)
(wi µj)
2
j
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誤差関数の最小化-分散について 17/31
分散(演習5.31)
事後分布:
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混合ガウスの対数尤度の注意 18/31
実際の実装では と定義される変数 を導⼊し,この変数について最⼩化を⾏う
なぜか
• パラメータ の正値性の保証
• がゼロに近づくことで,ガウス要素が重みのパラメータの値の⼀つに収縮するという
病的な解を避けるため
対数尤度:
例えば… ある について となる重みが存在した時,その重みは対数尤度に対して
の形で寄与する,この項は の極限を考えると無限⼤に発散する.
つまり,対数尤度も無限⼤に発散する.
混合要素の分布が特定の点に収束すれば対数尤度は常に増⼤する⽅向へ
働いてしまう.
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誤差関数の最小化-混合係数について 19/31
復習(2.3.9節-混合ガウス分布)
前述の通り, 𝜋* を事前分布として解釈するため
制約を考慮し,右記の補助変数 𝜂* を導⼊
𝜂* に関する微分(演習5.32)
制約
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5.6 混合密度ネットワーク
20/31
混合密度ネットワーク 21/31
教師あり学習の⽬標: のモデル化
単純な回帰問題ではガウス分布を仮定することが多い
例えば,PRML1章の⼆乗和誤差の最⼩化が
最尤推定と等価であった話もノイズがガウス
分布に従う仮定があった
もし,分布が多峰性を持つならば?
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順問題と逆問題 22/31
順問題 逆問題
関節⾓から終端位置 終端位置から関節⾓
特定の疾患から特定の症状 ある症状の集合から疾患
⾼次元の機械学習では多峰性の存在は明⽩ではない
順問題と逆問題 23/31
: ⼀様分布 に従う確率変数𝑥からサンプリングしたもの
: 関数 に から⽣成される乱数を加えたもの
順問題 逆問題
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混合密度ネットワーク 24/31
任意の条件付き密度関数をモデル化するための枠組み
もしガウス分布を要素に持つならば,
異分散(heteroscedastic)のモデルの⼀例
Wikiより
データに対するノイズの分散が𝐱の関数になっている
等⽅性共分散を持つことを仮定しない
Tの要素に分解できることを仮定しない
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混合密度ネットワーク 25/31
パラメータをNNの出⼒に
⽀配されるようにとる.
出⼒するパラメータ
平均:
分散:
混合係数:
例えば, のモデルは要素がK個であり,𝐭の要素がL個ならば
出⼒ 決定するもの 個数
混合係数 K
カーネルの幅 K
カーネルの中⼼ の要素 K×L
図5.19の例であれば
K=3, L=1でいい感じであることが
後に書かれている
つまり出⼒の総数は9個となる
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<latexit sha1_base64="ZFyf6/X4jGanzxWIc/TM9Wrmr2A=">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</latexit>
混合密度ネットワーク 26/31
制約パラメータについての注意
ソフト重み共有の時と同じように,混合係数は制約を満たすために
上記のソフトマックス関数を導⼊.同じく分散も制約から,
を導⼊する.平均は実数の値を取るので,
制約
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<latexit sha1_base64="i30dFJCORN5rHKDnGuVypS/yL5U=">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</latexit>
<latexit sha1_base64="b5BVzJ9XHllA/GejBK90yf4lZBE=">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</latexit>
<latexit sha1_base64="JpmtjMERbhxwkjeIlagqTcb58Jg=">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</latexit>
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PRML Chapter 5.5-5.6 Convolutional Neural Networks and Mixture Density Networks

  • 4. たたみ込みニューラルネットワーク CNN : Convolutional Neural Network 畳み込み層とプーリング層,全結合層などから構成されるニューラルネットワーク. タスク例: ⼿書き⽂字認識 ⼊⼒: 画像の画素の輝度値の集合 出⼒: 10個の数字のクラスの事後分布 平⾏移動や拡⼤縮⼩,(⼩さい) 回転などにも 不変であるべき これは6だ どちらも6だと認識してほしい 4/31
  • 7. 局所受容野と重み共有(畳み込み層) 局所的受容野 全結合じゃない 重み共有 ⼀部の⼊⼒の影響しか 受けない 7/31 <latexit sha1_base64="8FPn0zBku4VEeXwRy5jz+zzrUD8=">AAACanichVG5TsNAEH0xVwhXIA2IJiKAqKIxDUcVQUMZjkAkQMg2C1jxJdsJAis/QElDERqQEEJ8Bg0/QJFPQIgKJBoKxo4lBAgYa71v386bfTurOobu+UTNhNTW3tHZlexO9fT29Q+kB4fWPbvqaqKk2YbtllXFE4ZuiZKv+4YoO65QTNUQG2plMdzfqAnX021rzT9yxLap7Fv6nq4pPlPlLdU2doPD+k46R3mKIvsTyDHIIY6inb7GFnZhQ0MVJgQs+IwNKPD424QMgsPcNgLmXEZ6tC9QR4q1Vc4SnKEwW+H/Pq82Y9bidVjTi9Qan2LwcFmZxQQ90A290D3d0iO9/1oriGqEXo54Vlta4ewMnAyvvv2rMnn2cfCp+tOzjz3MRl519u5ETHgLraWvHZ+9rM6vTASTdElP7P+CmnTHN7Bqr9rVslhpIMUPIH9v909Qms7P5eVlyhUW4pdIYhRjmOJ2z6CAJRRRitp8igbOE89SRhqRRlupUiLWZPAlpPEPW2KNAg==</latexit> <latexit sha1_base64="8FPn0zBku4VEeXwRy5jz+zzrUD8=">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</latexit><latexit sha1_base64="8FPn0zBku4VEeXwRy5jz+zzrUD8=">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</latexit><latexit sha1_base64="8FPn0zBku4VEeXwRy5jz+zzrUD8=">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</latexit>
  • 10. CNNの概観 𝑛個の重み (25次元+バイアス) 5×5 15×15 10×10×𝑛 … ⼊⼒ conv層 pooling層 2×2 2×2 からサンプリング … 5×5×𝑛 conv層 𝑚個の重み (2×2次元+バイアス) 3×3×𝑛𝑚 出⼒全結合層 層を重ねるごとに空間解像度は減るが特徴数を増やすことで補償する 層を重ねるごとに不変性は強固になる Conv層とpooling層を合わせて畳み込み層と呼ぶ場合もある 最後に集めた特徴を集約する,多クラス分類問題ならソフトマックス関数など … … 10/31
  • 11. 逆伝播法による誤差最小化(演習5.28) 順伝播 逆伝播 順伝播 逆伝播 … 11/31 <latexit sha1_base64="q4KSEB+TAT8TkrTqYz/f6HGFa7A=">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</latexit> <latexit 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  • 13. ソフト重み共有 重み共有はネットワークの複雑さを削減する有効なテクニック しかし,制限の形が特定される特別な問題にしか適⽤できない ソフト重み共有 重みが同じグループで似た値を取りやすくなるように正則化項の導⼊ つまり,緑のグループや⾚のグループごとに 重みの平均や分散を設定しよう さらにそれらを学習の過程で得よう 13/31 <latexit sha1_base64="9GgjT8RFvP1BUY7xqBxPn+V8XhU=">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</latexit> <latexit sha1_base64="Pqs7LeVl1yK5y0XRDs3fsi0be+0=">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</latexit> <latexit sha1_base64="jNvCA8pj6fn/Rj3h1dJEq0ls8FI=">AAACd3ichVHLSgMxFD0d3/VVdSO4GSyKq3JHBB8r0Y1LX9VCFZkZo4bOi5m0UIs/4A+4cCEKouJnuPEHXPgJ4lJBEBfeTgdFRb0hycnJPTcniRU4MlJE9ymtqbmlta29I93Z1d3Tm+nrX4v8cmiLvO07fliwzEg40hN5JZUjCkEoTNdyxLpVmq/vr1dEGEnfW1XVQGy65q4nd6RtKqa2MgMGbSjpikj/AMxmKUdx6D+BkYAsklj0MxfYwDZ82CjDhYAHxdiBiYhbEQYIAXObqDEXMpLxvsAB0qwtc5bgDJPZEo+7vComrMfres0oVtt8isM9ZKWOEbqjK3qiW7qmB3r7tVYtrlH3UuXZamhFsNV7OLjy8q/K5Vlh71P1p2eFHUzFXiV7D2Kmfgu7oa/sHz2tzCyP1EbpjB7Z/ynd0w3fwKs82+dLYvkYaf4A4/tz/wT58dx0zliayM7OJT/RjiEMY4yfexKzWMAi8nxsFSe4xFXqVdO1UW2skaqlEs0AvoRmvAOp1pCl</latexit>
  • 14. ソフト重み共有ー荷重減衰の拡張 14/31 単純な荷重減衰正則化項(5.112)が, 重みのガウス事前分布の負の対数尤度とみなせる(5.5節,3.1.1〜3.1.4節) つまり,このときの重みはガウス分布に従う⼀つのグループであるとみなせる 複数のグループを構成したい 混合ガウス分布を代わりに⽤いよう(2.3.9) 確率密度関数: ただし, 負の対数尤度を取る 誤差関数: <latexit sha1_base64="/HXLvBGwi9BoGEtnnjYQiLA3Mmc=">AAAClXicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsYFsTkpKaVaMQk5eekVJfXxhRlpmeUaNrGFBTlp8RnVkOly+MzoTK18QLKBnoGYKCAyTCEMpQZoCAgX2A5QwxDCkM+QzJDKUMuQypDHkMJkJ3DkMhQDITRDIYMBgwFQLFYhmqgWBGQlQmWT2WoZeAC6i0FqkoFqkgEimYDyXQgLxoqmgfkg8wsButOBtqSA8RFQJ0KDKoGVw1WGnw2OGGw2uClwR+cZlWDzQC5pRJIJ0H0phbE83dJBH8nqCsXSJcwZCB04XVzCUMagwXYrZlAtxeARUC+SIboL6ua/jnYKki1Ws1gkcFroPsXGtw0OAz0QV7Zl+SlgalBsxm4gBFgiB7cmIxQIz1LPcNAE2UHJ2hMcDBIMygxaACD25zBgcGDIYAhFGjtdIa9DMcYjjNJMNkxuTC5QZQyMUL1CDOgACZ/AB3dnaM=</latexit> <latexit sha1_base64="Yi9FIE+eXSyxbkuOVWPmLUngbhE=">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</latexit> <latexit sha1_base64="w59PcUiizfSp1p8i7guXWxtVlFA=">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</latexit> <latexit sha1_base64="BbThxDLc3lNmrT/fxQ2um/0INa4=">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</latexit>
  • 15. ソフト重み共有-誤差関数の最小化 15/31 調整可能なパラメータ 中⼼: 分散: 混合係数: 2.3.9節に従って負担率の導⼊(2.192), を事前分布とみなすとベイズの定理から 事後分布: パラメータである平均 ,分散,混合係数についても最⼩化し,決定する • 重みが定数なら,EMアルゴリズム • 重みと同時最適化が必要な場合,共役勾配法,準ニュートン法 <latexit sha1_base64="K4JAeKL9OPxgHzDK58oHwQi36uI=">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</latexit> <latexit sha1_base64="lP3HrNSDSqRxo5fXvMAFX0S524k=">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</latexit> <latexit sha1_base64="bzv7MlNiMChAq0liQDAhjPKmkHo=">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</latexit> <latexit sha1_base64="Gvy5nI9rQEoSDXI4IHgHCwASvGE=">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</latexit> <latexit sha1_base64="8E/1yBWl5MVQAy1c7f/HwwGixEM=">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</latexit> <latexit sha1_base64="OeKLSn7Yuzw8sfevG0KW3nrtGnY=">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</latexit>
  • 16. 誤差関数の最小化-重み, 中心について 16/31 重み(演習5.29) 中⼼(演習5.30) 事後分布: <latexit sha1_base64="S1XX5wIcOJnDUxVXidKwAi0iYWE=">AAAC4HichVHPS9xAFH5JrW7XqqteCl4WF4tSXGZFUAuCtAge/dFVwdgwGWfXWSc/SCYrmuYPaC/Fk6AnC6UU/wwv/gM9ePUmHhW8eOhLNiBdsb6QmW++931v3sxYnhSBIuRC0190vOzsyr3Kd7/u6e0r9A+sBm7oM15lrnT9dYsGXAqHV5VQkq97Pqe2JfmatfMxya81uR8I1/mk9jy+adO6I2qCUYWUWfhi1HzKIsOjvhJUFg0l5BaP5uP4gds1RTzbpptvy78zgtA2G5FRp7ZNzUZxFNmxlimB44Ydmo0xdAWingg+T8SxWSiRMkmj+BhUMlCCLBbdwi8wYAtcYBCCDRwcUIglUAjw24AKEPCQ24QIOR+RSPMcYsijN0QVRwVFdgfHOq42MtbBdVIzSN0Md5H4++gswgj5Q36TG3JOTskVuX+yVpTWSHrZw9lqebln9n17s3L3rMvGWcH2g+u/PSuowXTaq8DevZRJTsFa/ub+4c3K++WR6C35Qa6x/xNyQc7wBE7zlv1c4svHkMcHqLRf92NQnSjPlCtLk6W5D9lL5GAIhmEUr3sK5mABFqGK215qOa1fG9CZ/lX/rh+0pLqWeQbhn9CP/gIj/bpd</latexit> <latexit 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sha1_base64="2fLBxjMSVX/RcLRtUQeEHcddahw=">AAACwnichVHLThRBFD00PnBQGWBj4mbiBDIunNQQE4WEhGhIXPJwhISGTnVTMxRT/aCregi0/QP+gAtXmhhj/Axd+AMu+ARxOSRuXHC7pxMfRL2Vqrp16pxbp6rcSEltGDsZsUYvXb5ydexaZfz6jZsT1cmpZzpMYk+0vVCF8abLtVAyEG0jjRKbUSy47yqx4fYe5/sbfRFrGQZPzVEktn3eDWRHetwQ5FR37E7MvdSOeGwkVzXbSLUr0uUs+wXzE2c/W7R14jsytbvc97mzX2scOvLuUN4oKPdygHRadnPCzlyWOdU6a7IiaheTVpnUUcZKWH0HG7sI4SGBD4EAhnIFDk1tCy0wRIRtIyUspkwW+wIZKqRNiCWIwQnt0dil1VaJBrTOa+pC7dEpinpMyhpm2Bf2ng3YZ/aBfWU//lorLWrkXo5ododaETkTL26tf/+vyqfZYO+n6p+eDTp4WHiV5D0qkPwW3lDfP345WF9Ym0ln2Rv2jfy/ZifsI90g6J95b1fF2itU6ANafz73xaQ915xvtlbv15celT8xhtu4gwY99wMs4QlW0KZjP+EUA5xZy1bPOrD0kGqNlJpp/BbW83PvBbCM</latexit><latexit sha1_base64="2fLBxjMSVX/RcLRtUQeEHcddahw=">AAACwnichVHLThRBFD00PnBQGWBj4mbiBDIunNQQE4WEhGhIXPJwhISGTnVTMxRT/aCregi0/QP+gAtXmhhj/Axd+AMu+ARxOSRuXHC7pxMfRL2Vqrp16pxbp6rcSEltGDsZsUYvXb5ydexaZfz6jZsT1cmpZzpMYk+0vVCF8abLtVAyEG0jjRKbUSy47yqx4fYe5/sbfRFrGQZPzVEktn3eDWRHetwQ5FR37E7MvdSOeGwkVzXbSLUr0uUs+wXzE2c/W7R14jsytbvc97mzX2scOvLuUN4oKPdygHRadnPCzlyWOdU6a7IiaheTVpnUUcZKWH0HG7sI4SGBD4EAhnIFDk1tCy0wRIRtIyUspkwW+wIZKqRNiCWIwQnt0dil1VaJBrTOa+pC7dEpinpMyhpm2Bf2ng3YZ/aBfWU//lorLWrkXo5ododaETkTL26tf/+vyqfZYO+n6p+eDTp4WHiV5D0qkPwW3lDfP345WF9Ym0ln2Rv2jfy/ZifsI90g6J95b1fF2itU6ANafz73xaQ915xvtlbv15celT8xhtu4gwY99wMs4QlW0KZjP+EUA5xZy1bPOrD0kGqNlJpp/BbW83PvBbCM</latexit><latexit sha1_base64="2fLBxjMSVX/RcLRtUQeEHcddahw=">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</latexit><latexit sha1_base64="2fLBxjMSVX/RcLRtUQeEHcddahw=">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</latexit> <latexit 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sha1_base64="luBRBxOZS+g9PXDGGx+7ZUbnNXg=">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</latexit><latexit 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sha1_base64="w59PcUiizfSp1p8i7guXWxtVlFA=">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</latexit> @N(x|µ, 2 ) @x = (x µ) 2 N x|µ, 2 <latexit sha1_base64="1wRrVcSiSE5VENVvqcbaskg7lwU=">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</latexit><latexit sha1_base64="1wRrVcSiSE5VENVvqcbaskg7lwU=">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</latexit><latexit 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  • 17. 誤差関数の最小化-分散について 17/31 分散(演習5.31) 事後分布: <latexit sha1_base64="Oa0lGOzKY3Q+f3SWV080D7Zesfs=">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</latexit> <latexit sha1_base64="kMiqWR82tjyn0Wx2S75jjlIU19E=">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</latexit> <latexit sha1_base64="ZVux7NguZk2J22I9pRG6Alcnqtg=">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</latexit> <latexit sha1_base64="BbThxDLc3lNmrT/fxQ2um/0INa4=">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</latexit> <latexit sha1_base64="w59PcUiizfSp1p8i7guXWxtVlFA=">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</latexit> <latexit sha1_base64="bdGipDBIhtxrSoA9UShEFpAQU68=">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</latexit> <latexit sha1_base64="OeKLSn7Yuzw8sfevG0KW3nrtGnY=">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</latexit>
  • 18. 混合ガウスの対数尤度の注意 18/31 実際の実装では と定義される変数 を導⼊し,この変数について最⼩化を⾏う なぜか • パラメータ の正値性の保証 • がゼロに近づくことで,ガウス要素が重みのパラメータの値の⼀つに収縮するという 病的な解を避けるため 対数尤度: 例えば… ある について となる重みが存在した時,その重みは対数尤度に対して の形で寄与する,この項は の極限を考えると無限⼤に発散する. つまり,対数尤度も無限⼤に発散する. 混合要素の分布が特定の点に収束すれば対数尤度は常に増⼤する⽅向へ 働いてしまう. 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  • 19. 誤差関数の最小化-混合係数について 19/31 復習(2.3.9節-混合ガウス分布) 前述の通り, 𝜋* を事前分布として解釈するため 制約を考慮し,右記の補助変数 𝜂* を導⼊ 𝜂* に関する微分(演習5.32) 制約 <latexit sha1_base64="b5BVzJ9XHllA/GejBK90yf4lZBE=">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</latexit> <latexit sha1_base64="JpmtjMERbhxwkjeIlagqTcb58Jg=">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</latexit> <latexit sha1_base64="Lsk9+vutgIqlcKlX21iZJRN5/Qs=">AAACmXichVFNSxtBGH6yrVVTram9CPYQGiL2EiaiWIWCpBfxZBLTBLJh2d1OdMx+sTsJ1SV/wD/gIacKItJfUXopeu4hP6F4TKGXHnyzWdBW1HeYmWeeeZ93npkxPEsEkrF+QnnydOzZ+MRk8vnU9IuZ1MvZj4Hb9k1eMV3L9WuGHnBLOLwihbR4zfO5bhsWrxqtD8P9aof7gXCdHXng8Yat7zqiKUxdEqWlVlRPaPvp90m16etmqPLPXriocqlr+2+73VAN2rbWuk23iO5qqQzLsSjSd0E+BhnEse2mzqDiE1yYaMMGhwNJ2IKOgFodeTB4xDUQEucTEtE+RxdJ0rYpi1OGTmyLxl1a1WPWofWwZhCpTTrFou6TMo0s+8nO2YD9YF/ZL/b33lphVGPo5YBmY6TlnjZzNFf+86jKplli70b1oGeJJt5FXgV59yJmeAtzpO8cHg/K66VsuMBO2BX5/8L67DvdwOn8Nk+LvNRDkj4g//9z3wWVpdxaLl9czmwU4p+YwDzeYJGeexUb2MQ2KnRsD99wgUvltVJQNpWtUaqSiDWv8E8o5WsyJJ7x</latexit> <latexit sha1_base64="dEkW1poJdqp4INZJsBqTiaL/H0I=">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</latexit> <latexit sha1_base64="ot3yGhtt0BOR7/rBdZgnS9pUgFk=">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</latexit> <latexit sha1_base64="GKAbWnGgozRLB/Zu15fyJclsU+4=">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</latexit> <latexit sha1_base64="xRDYs+80UDj9/PeS5tUOsEZRkZQ=">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</latexit>
  • 21. 混合密度ネットワーク 21/31 教師あり学習の⽬標: のモデル化 単純な回帰問題ではガウス分布を仮定することが多い 例えば,PRML1章の⼆乗和誤差の最⼩化が 最尤推定と等価であった話もノイズがガウス 分布に従う仮定があった もし,分布が多峰性を持つならば? <latexit sha1_base64="s2keATwNOF5NSYjYeOdb4PLoulg=">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</latexit>
  • 22. 順問題と逆問題 22/31 順問題 逆問題 関節⾓から終端位置 終端位置から関節⾓ 特定の疾患から特定の症状 ある症状の集合から疾患 ⾼次元の機械学習では多峰性の存在は明⽩ではない
  • 23. 順問題と逆問題 23/31 : ⼀様分布 に従う確率変数𝑥からサンプリングしたもの : 関数 に から⽣成される乱数を加えたもの 順問題 逆問題 <latexit sha1_base64="2SbCaxlT9KqMdzKm705HqP9wdJg=">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</latexit> <latexit 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  • 24. 混合密度ネットワーク 24/31 任意の条件付き密度関数をモデル化するための枠組み もしガウス分布を要素に持つならば, 異分散(heteroscedastic)のモデルの⼀例 Wikiより データに対するノイズの分散が𝐱の関数になっている 等⽅性共分散を持つことを仮定しない Tの要素に分解できることを仮定しない <latexit sha1_base64="MvUAcYE/CeUhZem5g4e66TrlrU4=">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</latexit>
  • 25. 混合密度ネットワーク 25/31 パラメータをNNの出⼒に ⽀配されるようにとる. 出⼒するパラメータ 平均: 分散: 混合係数: 例えば, のモデルは要素がK個であり,𝐭の要素がL個ならば 出⼒ 決定するもの 個数 混合係数 K カーネルの幅 K カーネルの中⼼ の要素 K×L 図5.19の例であれば K=3, L=1でいい感じであることが 後に書かれている つまり出⼒の総数は9個となる <latexit sha1_base64="MvUAcYE/CeUhZem5g4e66TrlrU4=">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</latexit> <latexit 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  • 26. 混合密度ネットワーク 26/31 制約パラメータについての注意 ソフト重み共有の時と同じように,混合係数は制約を満たすために 上記のソフトマックス関数を導⼊.同じく分散も制約から, を導⼊する.平均は実数の値を取るので, 制約 <latexit sha1_base64="i7DUjh9J261T3VPSBSf5x7kYBn8=">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</latexit> <latexit sha1_base64="G1LURrz7V486a8hLKIbKtZDUyQQ=">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</latexit> <latexit sha1_base64="7STqpEju5tUYMgjA4FNvF7c9/CQ=">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</latexit> <latexit sha1_base64="i30dFJCORN5rHKDnGuVypS/yL5U=">AAACg3ichVG7SgNBFD1ZXzG+ojaChWKIKEiYSMAHCEEbyyQaE0g07K6TuGZf7G6CuqS08QcsrBREJK1+gY0/YOEniKWCjYU3mwVRUe+yM2fOvefOmRnJVBXbYewxIHR0dnX3BHtDff0Dg0Ph4ZEt26hZMs/KhmpYeUm0uaroPOsojsrzpsVFTVJ5TqqutfK5OrdsxdA3nUOTb2tiRVfKiiw6RJXCE0WtVnKr+42ZomSou+5BY3ZF9Igdl1KNUjjCYsyLyZ8g7oMI/EgZ4SsUsQsDMmrQwKHDIaxChE1fAXEwmMRtwyXOIqR4eY4GQqStURWnCpHYKo0VWhV8Vqd1q6ftqWXaRaXfIuUkouyBXbMXds+a7Im9/9rL9Xq0vBzSLLW13CwNnYxtvP2r0mh2sPep+tOzgzIWPa8KeTc9pnUKua2vH52+bCxnou40u2DP5P+cPbI7OoFef5Uv0zxzhhA9QPz7df8E2fnYUiyeTkSSq/5LBDGOKczQdS8giXWkkKVtj9HEDW6FbmFOmBcS7VIh4GtG8SWElQ/3NJY4</latexit> <latexit sha1_base64="b5BVzJ9XHllA/GejBK90yf4lZBE=">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</latexit> <latexit sha1_base64="JpmtjMERbhxwkjeIlagqTcb58Jg=">AAACd3ichVHLSsNAFD2Nr1pfVTeCm2BRXJUbEXysim5cqrUqWClJHHU0TWKSFmrxB/wBFy5EQbT4GW78ARf9BHFZQRAX3qYBUVHvMDNnztxz58yM4VrSD4jqMaWtvaOzK96d6Ont6x9IDg6t+07JM0XOdCzH2zR0X1jSFrlABpbYdD2hFw1LbBiHi839jbLwfOnYa0HFFdtFfc+Wu9LUA6YKyWFS85Y4UvOuLBy0oFZIpihNYag/gRaBFKJYdpI3yGMHDkyUUISAjYCxBR0+ty1oILjMbaPKnMdIhvsCJ0iwtsRZgjN0Zg953OPVVsTavG7W9EO1yadY3D1WqhinR6pRgx7ojp7o/dda1bBG00uFZ6OlFW5h4HQk+/qvqshzgP1P1Z+eA+xiNvQq2bsbMs1bmC19+fiskZ1fHa9O0BU9s/9LqtM938Auv5jXK2L1HAn+AO37c/8Euan0XFpbmU5lFqKfiGMUY5jk555BBktYRo6PreACt6jF3hRVmVAmW6lKLNIM40so2gcHAZBU</latexit>