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2008.12.05

カーネル多変量解析

タイトルの本を買って読んでみた.

カーネル多変量解析―非線形データ解析の新しい展開 (シリーズ確率と情報の科学)
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様々な解説記事で定評のある赤穂先生によるカーネル法による解析についての本。日本語で読めるカーネルに関する本としては、導入部の丁寧さと、そのあとの展開と深さ、最新の話まで抑えている点でお勧めだと思います。カーネルの性質、汎化性能とかはもちろんのこと、例えばカーネルk-means, スぺクトラルクラスタリング、(ちょっとだけ)Gaussian Process, Leave-one-outの閉じた式、L1正則化など、他の和書ではあまり見たことない内容が多く、中身が濃いです。ただ、これらは、どれもさーっとかいて気持ちがわかって、詳しくは参考文献を見るという感じです。まぁ、それだけ詰め込んでいるから仕方ないですね。

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カーネル法とは、なんらかの対象を解析するときに、対象二つa, bを引数としてとるカーネル関数 K(a, b)を通して解析する方法です。直観的にはK(a,b)はaとbの近さのようなもので、例えばaとbが似ていれば大きな値をとるように関数を決めます(厳密にはKが半正定値性を満たすように)。

カーネル法のすごいところは、このK(a,b)さえ定義できれば対象が何であっても解析できて、たとえば対象がベクトルとかの場合は内積(とそれを凸関数にいれたもの)をKとして使うのが一般的ですが、文字列や木、グラフ、といった一見解析が難しそうな対象でさえ、カーネルさえ定義できれば同じ枠組みで解けてしまいます(人だって良い)。対象の情報はブラックボックスとしてカーネルに凝縮されて、その後はカーネルだけを通して、回帰や分類、クラスタリングなどが統一的なアルゴリズムで解くことができます。

さらに、カーネルを利用して得られる識別関数は、元々のパラメータに対しては非線形でありながら、カーネル関数に対しては線形であり、最適化が容易、過学習しにくいと、いいとこどりの形をしています。

例えば、カーネル法が有名になるきっかけとなったサポートベクターマシンでは、難しいチューニングなしに、対象に対しカーネルを定義して、サポートベクターマシンに突っこんで学習したら、手書き文字認識、文書分類、生物情報など多くの分野で従来の最高精度を大幅に超えて、90年代後半あたりに、"...using Support Vector Machine"という論文がものすごくたくさんでてました。

現在では、これもちょっと落ち着いて、ニューラルネットやパーセプトロンがまた復活してきてたり、凸でない関数でもいいじゃんとか、単純な線形モデルによる最適化、ベイジアンモデルとかが盛んに研究されていますが、(計算量を除けば)相変わらずカーネル法は安定して強く、ちゃんとおさえておく必要があるとおもいます。

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