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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 LogSumExp ： ウィキペディア英語版 LogSumExp The LogSumExp (LSE) function is a smooth approximation to the maximum function, mainly used by machine learning algorithms. It's defined as the logarithm of the sum of the exponentials of the arguments: :$LSE(x\_1,\; \backslash dots,\; x\_n)\; =\; \backslash log\backslash left(\; \backslash exp(x\_1)+\; \backslash cdots\; +\; \backslash exp(x\_n)\; \backslash right)$ The LogSumExp function domain is $\backslash R^n$, the real coordinate space, and its range is $\backslash R$, the real line. The larger the values of $x\_k$ or their deviation, the better the approximation becomes. The LogSumExp function is convex, and is strictly monotonically increasing everywhere in its domain (but not strictly convex everywhere 〔(【引用サイトリンク】title=convex analysis - About the strictly convexity of log-sum-exp function - Mathematics Stack Exchange )〕). On the otherhand, when directly encountered, LSE can be well-approximated by $\backslash max\}\backslash leq\; LSE(x\_1,\; \backslash dots,\; x\_n)\backslash leq\; \backslash max\{\backslash \{x\_1,\; \backslash dots,\; x\_n\backslash \}\}\; +\; \backslash log(n)$ The lower bound is met when only one of the argument is non-zero, while the upper bound is met when all the arguments are equal. ==log-sum-exp trick for log-domain calculations== The LSE function is often encountered when the usual arithmetic computations are performed in log-domain or log-scale. Like multiplication operation in linear-scale becoming simple addition in log-scale; an addition operation in linear-scale becomes the LSE in the log-domain. A common purpose of using log-domain computations is to increase accuracy and avoid underflow and overflow problems when very small or very large numbers are represented directly (i.e. in a linear domain) using a limited-precision, floating point numbers. Unfortunately, the use of LSE directly in this case can again cause overflow/underflow problems. Therefore, the following equivalent must be used instead (especially when the accuracy of the above 'max' approx. is not sufficient). Therefore, many math libraries such as boost,IT++ etc. provide a default routine of LSE and use this formula internally. :$LSE(x\_1,\; \backslash dots,\; x\_n)\; =\; x^$ * + \log\left( \exp(x_1-x^ *)+ \cdots + \exp(x_n-x^ *) \right) where $x^$ * = \max} 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「LogSumExp」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.