Artificial neural networks grow on the number of applications and complexity, which require a minimization on the number of units for some practical implementations. A particular problem is the minimum number of units that a feed forward neural network needs on its first layer. In order to study this problem, it is defined a family of classification problems following a continuity hypothesis, where inputs that are close to some set of points may share the same category. Given a set S of k −dimensional inputs and let N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {N}$\end{document} be a feed forward neural network that classifies any input in S within a fixed error, there is proved that N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {N}$\end{document} requires Θk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\Theta } \left (k \right )$\end{document} units in the first layer, if N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {N}$\end{document} can solve any instance from the given family of classification problems. Furthermore, this asymptotic result is optimal.

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神经网络中输入和第一层之间的线性关系

人工神经网络随着应用程序的数量和复杂性的增加而增长，这需要最小化一些实际实现的单元数量。一个特殊的问题是前馈神经网络在其第一层所需的最小单元数。为了研究这个问题，它定义了一系列遵循连续性假设的分类问题，其中靠近某些点集的输入可能共享相同的类别。if N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin }{-69pt} \begin{document}$\mathcal {N}$\end{document} 可以解决给定分类问题系列中的任何实例。此外，这种渐近结果是最优的。