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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 Manifold alignment ： ウィキペディア英語版 Manifold alignment Manifold alignment is a class of machine learning algorithms that produce projections between sets of data, given that the original data sets lie on a common manifold. The concept was first introduced as such by Ham, Lee, and Saul in 2003, adding a manifold constraint to the general problem of correlating sets of high-dimensional vectors. ==Overview== Manifold alignment assumes that disparate data sets produced by similar generating processes will share a similar underlying manifold representation. By learning projections from each original space to the shared manifold, correspondences are recovered and knowledge from one domain can be transferred to another. Most manifold alignment techniques consider only two data sets, but the concept extends to arbitrarily many initial data sets. Consider the case of aligning two data sets, $X$ and $Y$, with $X\_i\; \backslash in\; \backslash mathbb^m$ and $Y\_i\; \backslash in\; \backslash mathbb^n$. Manifold alignment algorithms attempt to project both $X$ and $Y$ into a new ''d''-dimensional space such that the projections both minimize distance between corresponding points and preserve the local manifold structure of the original data. The projection functions are denoted: $\backslash phi\_:\backslash ,\backslash mathbb^\backslash rightarrow\backslash mathbb^$ $\backslash phi\_:\backslash ,\backslash mathbb^\backslash rightarrow\backslash mathbb^$ Let $W$ represent the binary correspondence matrix between points in $X$ and $Y$: $W\_=\backslash begin$ 1 & if\, X_\leftrightarrow Y_\\ 0 & otherwise \end Let $S\_X$ and $S\_Y$ represent pointwise similarities within data sets. This is usually encoded as the heat kernel of the adjacency matrix of a ''k''-nearest neighbor graph. Finally, introduce a coefficient $0\; \backslash le\; \backslash mu\; \backslash le\; 1$, which can be tuned to adjust the weight of the 'preserve manifold structure' goal, versus the 'minimize corresponding point distances' goal. With these definitions in place, the loss function for manifold alignment can be written: $\backslash arg\backslash min\_\}\backslash mu\backslash sum\_\backslash left\backslash Vert\; \backslash phi\_\backslash left(X\_\backslash right)-\backslash phi\_\backslash left(X\_\backslash right)\backslash right\backslash Vert\; ^S\_+\backslash mu\backslash sum\_\backslash left\backslash Vert\; \backslash phi\_\backslash left(Y\_\backslash right)-\backslash phi\_\backslash left(Y\_\backslash right)\backslash right\backslash Vert\; ^S\_+\backslash left(1-\backslash mu\backslash right)\backslash sum\_\backslash Vert\backslash phi\_\backslash left(X\_\backslash right)-\backslash phi\_\backslash left(Y\_\backslash right)\backslash Vert^W\_$ Solving this optimization problem is equivalent to solving a generalized eigenvalue problem using the graph laplacian of the joint matrix, ''G'': \left(1-\mu\right)W^T & \mu S_Y \end\right ) 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「Manifold alignment」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.