研究了神经网络决策边界的几何体积，给出鲁棒性概率的严格上限。
Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks
- 基于普拉菲安函数定义边界，推导体积上界
- 在单隐层有理权Sigmoid网络中得到宽度的多项式界
- 适用于分析神经网络分类器对扰动的敏感性
我们推导了光滑普拉菲安超曲面管状邻域体积的上界，推广了已有代数簇结果。上界以定义函数的普拉菲安格式表示。作为应用，我们得到了具有普拉菲安激活函数的神经网络分类器条件数的概率分布尾部界限，在均匀与高斯设定下均成立。在单隐层有理权重Sigmoid网络的特例中，我们导出了决策边界管状邻域的宽度多项式上界。
原文摘要 · Abstract (English)
We derive bounds for the volume of tubular neighbourhoods of smooth Pfaffian hypersurfaces, generalising known results for algebraic varieties. The bounds are given in terms of the Pfaffian format of the defining functions. As an application, we obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions, in both the uniform and Gaussian settings. In the special case of single-hidden-layer sigmoid networks with rational weights, we derive polynomial-in-width bounds for tubular neighbourhoods of the decision boundary.
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