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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 Tensor rank decomposition ： ウィキペディア英語版 Tensor rank decomposition In multilinear algebra, the tensor rank decomposition or canonical polyadic decomposition (CPD) may be regarded as a generalization of the matrix singular value decomposition (SVD) to tensors, which has found application in statistics, signal processing, psychometrics, linguistics and chemometrics. It was introduced by Hitchcock in 1927 and later rediscovered several times, notably in psychometrics. For this reason, the tensor rank decomposition is sometimes historically referred to as PARAFAC〔 or CANDECOMP.〔 == Definition == Consider a tensor space $\backslash mathbb^\; \backslash cong\; \backslash mathbb^\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbb^$, where $\backslash mathbb$ is either the real field $\backslash mathbb$ or the complex field $\backslash mathbb$. Every (order-$d$) tensor in this space may then be represented with a suitably large $r$ as a linear combination of $r$ rank-1 tensors: : $\backslash mathcal\; =\; \backslash sum\_^\; \backslash lambda\_i\; \backslash mathbf\_i^1\; \backslash otimes\; \backslash mathbf\_i^2\; \backslash otimes\; \backslash cdots\; \backslash otimes\; \backslash mathbf\_i^d,$ where $\backslash lambda\_i\; \backslash in\; \backslash mathbb$ and $\backslash mathbf\_i^k\; \backslash in\; \backslash mathbb^$; note that the superscript $k$ should not be interpreted as an exponent, it is merely another index. When the number of terms $r$ is minimal in the above expression, then $r$ is called the rank of the tensor, and the decomposition is often referred to as a ''(tensor) rank decomposition'', ''minimal CP decomposition'', or ''Canonical Polyadic Decomposition (CPD)''. Contrariwise, if the number of terms is not minimal, then the above decomposition is often referred to as ''$r$-term decomposition'', ''CANDECOMP/PARAFAC'' or ''Polyadic decomposition''. 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「Tensor rank decomposition」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.