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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 Overcompleteness ： ウィキペディア英語版 Overcompleteness A subset $\backslash \_$ of a Banach space $X$, sometimes called a "system", is complete if every element in $X$ can be approximated arbitrarily well in norm by finite linear combinations of elements in $\backslash \_$.〔C. Heil, A Basis Theory Primer: Expanded Edition. Boston, MA: Birkhauser, 2010.〕 Such a complete system is overcomplete if removal of a $\backslash phi\_j$ from the system results in a complete system (i.e., $\backslash \_\{i\backslash in\; J\backslash backslash\backslash \{j\backslash \}\}$ is still complete). In different research, such as signal processing and function approximation, overcompleteness can help researchers to achieve a more stable, more robust, or more compact decomposition than using a basis.〔R. Balan, P. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames. I. theory, The Journal of Fourier Analysis and Applications, vol. 12, no. 2, 2006.〕 Overcomplete frames are widely used in mathematics, computer science, engineering, and statistics. ==Relation between overcompleteness and frames== Overcompleteness is usually discussed as a property of overcomplete frames. The theory of frame originates in a paper by Duffin and Schaeffer on non-harmonic Fourier series.〔R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, vol. 72, no. 2, pp. 341 such that for an arbitrary $f\backslash in\backslash mathcal$, : $A\backslash |f\backslash |^2\backslash leq\backslash sum\_|\backslash langle\; f,\; \backslash phi\_i\; \backslash rangle|^2\backslash leq\; B\backslash |f\backslash |^2$ where $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ denotes the inner product, $A$ and $B$ are positive constants called bounds of the frame. When $A$ and $B$ can be chosen such that $A=B$, the frame is called a tight frame.〔K. Grochenig, ''Foundations of time-frequency analysis''. Boston, MA: Birkhauser, 2000.〕 It can be seen that $\backslash mathcal=\backslash operatorname\backslash$. An example of frame can be given as follows. Let each of $\backslash \_^$ and $\backslash \_^$ be an orthonormal basis of $\backslash mathcal$, then : $\backslash \_^=\backslash \_^\backslash cup\backslash \_^$ is a frame of $\backslash mathcal$ with bounds $A=B=2$. Let $S$ be the frame operator, : $Sf=\backslash sum\_\backslash langle\; f,\; \backslash phi\_i\; \backslash rangle\backslash phi\_i$ A frame that is not a Riesz basis, in which case it consists of a set of functions more than a basis, is said to be overcomplete. In this case, given $f\backslash in\backslash mathcal$, it can have different decompositions based on the frame. The frame given in the example above is an overcomplete frame. When frames are used for function estimation, one may want to compare the performance of different frames. The parsimony of the approximating functions by different frames may be considered as one way to compare their performances.〔(), STA218, Data Mining Class Note at Duke University〕 Given a tolerance $\backslash epsilon$ and a frame $F=\backslash \_$ in $L^2(\backslash mathbb)$, for any function $f\backslash in\; L^2(\backslash mathbb)$, define the set of all approximating functions that satisfy : Then let : $k\_(f,\backslash epsilon)=\backslash inf\backslash$ $k(f,\backslash epsilon)$ indicates the parsimony of utilizing frame $F$ to approximate $f$. Different $f$ may have different $k$ based on the hardness to be approximated with elements in the frame. The worst case to estimate a function in $L^2(\backslash mathbb)$ is defined as : $k\_F\; (\backslash epsilon)=\backslash sup\_\backslash$ For another frame $G$, if Overcomplete frames are usually constructed in three ways. # Combine a set of bases, such as wavelet basis and Fourier basis, to obtain an overcomplete frame. # Enlarge the range of parameters in some frame, such as in Gabor frame and wavelet frame, to have an overcomplete frame. # Add some other functions to an existing complete basis to achieve an overcomplete frame. An example of an overcomplete frame is shown below. The collected data is in a two-dimensional space, and in this case a basis with two elements should be able to explain all the data. However, when noise is included in the data, a basis may not be able to express the properties of the data. If an overcomplete frame with four elements corresponding to the four axes in the figure is used to express the data, each point would be able to have a good expression by the overcomplete frame. Image:OvercompleteframeGuoxian.jpg|An example of an overcomplete frame The flexibility of the overcomplete frame is one of its key advantages when used in expressing a signal or approximating a function. However, because of this redundancy, a function can have multiple expressions under an overcomplete frame.〔M. S. Lewicki and T. J. Sejnowski, Learning overcomplete representations, Neural Computation, vol. 12, no. 2, pp. 337) may be used in solving this equation. This should be equivalent to the ] regression in statistics community. Bayesian approach is also used to eliminate the redundancy in an overcompete frame. Lweicki and Sejnowski proposed an algorithm for overcomplete frame by viewing it as a probabilistic model of the observed data.〔 Recently, the overcomplete Gabor frame has been combined with bayesian variable selection method to achieve both small norm expansion coefficients in $L^2(\backslash mathbb)$ and sparsity in elements.〔P. Wolfe, S. Godsill, and W. Ng, Bayesian variable selection and regularization for time-frequency surface estimation, J. R. Statist. Soc. B, vol. 66, no. 3, 2004.〕 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「Overcompleteness」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.