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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 Basis pursuit denoising ： ウィキペディア英語版 Basis pursuit denoising In mathematics and machine learning, basis pursuit denoising (BPDN) is one approach to solving a mathematical optimization problem of the form: :$\backslash min\_x\; \backslash frac\backslash |y-Ax\backslash |^2\_2+\backslash lambda\backslash |x\backslash |\_1.$ where $\backslash lambda$ is a parameter that controls the trade-off between sparsity and reconstruction fidelity, $x$ is an $N\; \backslash times\; 1$ solution vector, $y$ is an $M\; \backslash times\; 1$ vector of observations, $A$ is an $M\; \backslash times\; N$ transform matrix and . This is an instance of convex optimization and also of quadratic programming. Some authors refer to basis pursuit denoising as the following closely related problem: :$\backslash min\_x\; \backslash |x\backslash |\_1\; \backslash ;\backslash textrm\; \backslash \; \backslash textrm\backslash ;\backslash ;\backslash |y-Ax\backslash |^2\_2\; \backslash le\; \backslash delta$ which, for any given $\backslash lambda$, is equivalent to the unconstrained formulation for some (usually unknown ''a priori'') value of $\backslash delta$. The two problems are quite similar, but most specialized numerical algorithms can only solve the unconstrained formulation. Either type of basis pursuit denoising solves a regularization problem with a trade-off between having a small residual (making $y$ close to $Ax$ in terms of an $L\_2$ distance) and making $x$ simple in the $L\_1$ sense. It can be thought of as a mathematical statement of Occam's razor, finding the simplest possible explanation ($\backslash min\_x\; \backslash |x\backslash |\_1$) capable of accounting for the observations ($\backslash min\_x\; \backslash |y-Ax\backslash |^2\_2$). Exact solutions to basis pursuit denoising are often the best computationally tractable approximation of an underdetermined system of equations. Basis pursuit denoising has potential applications in statistics (c.f. the LASSO method of regularization), image compression and compressed sensing. As $\backslash lambda\; \backslash rightarrow\; \backslash infty$ (or when $\backslash delta\; =\; 0$), this problem becomes basis pursuit. ==Solving basis pursuit denoising== The problem is a convex quadratic problem, so it can be solved by many general solvers, such as interior point methods. For very large problems, many specialized methods that are faster than interior point methods have been proposed. Several popular methods for solving basis pursuit denoising include the in-crowd algorithm (a fast solver for large, sparse problems〔See ''The In-Crowd Algorithm for Fast Basis Pursuit Denoising'', IEEE Trans Sig Proc 59 (10), Oct 1 2011, pp. 4595 - 4605, (), demo MATLAB code available ()〕), homotopy continuation, fixed-point continuation (a special case of the (forward backward algorithm )) and spectral projected gradient for L1 minimization (which actually solves LASSO, a related problem). 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「Basis pursuit denoising」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.