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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Cheeger bound ： ウィキペディア英語版 Cheeger bound In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs. Let $X$ be a finite set and let $K(x,y)$ be the transition probability for a reversible Markov chain on $X$. Assume this chain has stationary distribution $\backslash pi$. Define :$Q(x,y)\; =\; \backslash pi(x)\; K(x,y)$ and for $A,B\; \backslash subset\; X$ define : $Q(A\; \backslash times\; B)\; =\; \backslash sum\_\; Q(x,y).$ Define the constant $\backslash Phi$ as : $\backslash Phi\; =\; \backslash min\_\}\; \backslash frac.$ The operator $K,$ acting on the space of functions from $|X|$ to $|X|$, defined by : $(K\; \backslash phi)(x)\; =\; \backslash sum\_y\; K(x,y)\; \backslash phi(y)\; \backslash ,$ has eigenvalues $\backslash lambda\_1\; \backslash geq\; \backslash lambda\_2\; \backslash geq\; \backslash cdots\; \backslash geq\; \backslash lambda\_n$. It is known that $\backslash lambda\_1\; =\; 1$. The Cheeger bound is a bound on the second largest eigenvalue $\backslash lambda\_2$. Theorem (Cheeger bound): :$1\; -\; 2\; \backslash Phi\; \backslash leq\; \backslash lambda\_2\; \backslash leq\; 1\; -\; \backslash frac.$ == See also == * Poincaré bound * Stochastic matrix * Cheeger constant 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cheeger bound」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.