Barron函数的表示公式和点态性质
Representation formulas and pointwise properties for Barron functions
摘要：
我们研究了具有ReLU激活的无限宽两层神经网络的自然函数空间（Barron空间），并建立了不同的表示公式。在两种情况下，我们明确地描述空间同构。使用一个方便的表示，我们研究了两层网络的逐点性质，并表明，其奇异集是分形或弯曲的功能（例如距离函数光滑子流形）不能表示为无限宽的两层网络与有限的路径范数。利用这一结构定理，证明了Barron空间中唯一的$C^1$ -仿射射同态是仿射的.此外，我们表明，每一个Barron函数可以分解为一个有界的和一个积极的单齐次函数的总和，并存在Barron功能迅速衰减到无穷大，是全球勒贝格可积。这一结果表明，双层神经网络可能能够近似比通常认为的更多种类的功能。
英文摘要：
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.
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