论文标题
Barron函数的表示公式和点的特性
Representation formulas and pointwise properties for Barron functions
论文作者
论文摘要
我们研究具有Relu激活（Barron空间）的无限宽两层神经网络的自然功能空间，并建立了不同的表示公式。在两种情况下，我们明确描述了同构的空间。 使用方便的表示，我们研究了两层网络的点特性，并表明其奇异集是分形或弯曲的函数（例如，来自平滑亚元号的距离函数）不能由具有有限路径 - 摩尔的无限宽的两层网络表示。我们使用此结构定理来表明Barron空间唯一的$ C^1 $ -Diffeomormormorphisms。 此外，我们表明，每个Barron函数都可以分解为有界和一个积极的一系列函数的总和，并且存在Barron函数在无穷大迅速衰减，并且在全球范围内均可腐烂。该结果表明，两层神经网络可能能够近似比通常认为的更多功能。
We study the natural function space for infinitely wide two-layer neural networks with ReLU activation (Barron space) and establish different representation formulae. In two cases, we describe the space explicitly up to isomorphism. Using a convenient representation, we study the pointwise properties of two-layer networks and show that functions whose singular set is fractal or curved (for example distance functions from smooth submanifolds) cannot be represented by infinitely wide two-layer networks with finite path-norm. We use this structure theorem to show that the only $C^1$-diffeomorphisms which Barron space are affine. Furthermore, we show that every Barron function can be decomposed as the sum of a bounded and a positively one-homogeneous function and that there exist Barron functions which decay rapidly at infinity and are globally Lebesgue-integrable. This result suggests that two-layer neural networks may be able to approximate a greater variety of functions than commonly believed.
