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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 Regularization by spectral filtering ： ウィキペディア英語版 Regularization by spectral filtering Spectral regularization is any of a class of regularization techniques used in machine learning to control the impact of noise and prevent overfitting. Spectral regularization can be used in a broad range of applications, from deblurring images to classifying emails into a spam folder and a non-spam folder. For instance, in the email classification example, spectral regularization can be used to reduce the impact of noise and prevent overfitting when a machine learning system is being trained on a labeled set of emails to learn how to tell a spam and a non-spam email apart. Spectral regularization algorithms rely on methods that were originally defined and studied in the theory of ill-posed inverse problems (for instance, see〔H. W. Engl, M. Hanke, and A. Neubauer. ''Regularization of inverse problems''. Kluwer, 1996.〕) focusing on the inversion of a linear operator (or a matrix) that possibly has a bad condition number or an unbounded inverse. In this context, regularization amounts to substituting the original operator by a bounded operator called the "regularization operator" that has a condition number controlled by a regularization parameter,〔L. Lo Gerfo, L. Rosasco, F. Odone, E. De Vito, and A. Verri. Spectral Algorithms for Supervised Learning, ''Neural Computation'', 20(7), 2008.〕 a classical example being Tikhonov regularization. To ensure stability, this regularization parameter is tuned based on the level of noise.〔 The main idea behind spectral regularization is that each regularization operator can be described using spectral calculus as an appropriate filter on the eigenvalues of the operator that defines the problem, and the role of the filter is to "suppress the oscillatory behavior corresponding to small eigenvalues".〔 Therefore, each algorithm in the class of spectral regularization algorithms is defined by a suitable filter function (which needs to be derived for that particular algorithm). Three of the most commonly used regularization algorithms for which spectral filtering is well-studied are Tikhonov regularization, Landweber iteration, and truncated singular value decomposition (TSVD). As for choosing the regularization parameter, examples of candidate methods to compute this parameter include the discrepancy principle, generalized cross validation, and the L-curve criterion.〔P. C. Hansen, J. G. Nagy, and D. P. O'Leary. ''Deblurring Images: Matrices, Spectra, and Filtering'', Fundamentals of Algorithms 3, SIAM, Philadelphia, 2006.〕 It is of note that the notion of spectral filtering studied in the context of machine learning is closely connected to the literature on function approximation (in signal processing). == Notation == The training set is defined as $S\; =\; \backslash$, where $X$ is the $n\; \backslash times\; d$ input matrix and $Y\; =\; (y\_1,\backslash dots,y\_n)$ is the output vector. Where applicable, the kernel function is denoted by $k$, and the $n\; \backslash times\; n$ kernel matrix is denoted by $K$ which has entries $K\_\; =\; k(x\_i,x\_j)$ and $\backslash mathcal$ denotes the Reproducing Kernel Hilbert Space (RKHS) with kernel $k$. The regularization parameter is denoted by $\backslash lambda$. ''(Note: For $g\; \backslash in\; G$ and $f\; \backslash in\; F$, with $G$ and $F$ being Hilbert spaces, given a linear, continuous operator $L$, assume that $g\; =\; Lf$ holds. In this setting, the direct problem would be to solve for $g$ given $f$ and the inverse problem would be to solve for $f$ given $g$. If the solution exists, is unique and stable, the inverse problem (i.e. the problem of solving for $f$) is well-posed; otherwise, it is ill-posed.) '' 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「Regularization by spectral filtering」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.