发表机构
London School of Geometry and Number Theory, Imperial College London, UK; Warwick Mathematical Institute, University of Warwick, UK（几何与数论学校，帝国学院伦敦，英国; 沃里克数学研究所，沃里克大学，英国）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究普法夫超曲面管状邻域体积界，推广代数簇相关结果。通过普法夫格式给出界，并应用于神经网络分类器，在均匀和高斯设置下得到条件数概率分布尾部界，特殊单隐藏层 sigmoid 网络有宽度多项式界。
AI 中文摘要
我们推导了光滑普法夫超曲面管状邻域体积的界，推广了代数簇的已知结果。这些界是根据定义函数的普法夫格式给出的。作为应用，我们在均匀和高斯设置下，得到了衡量具有普法夫激活函数的神经网络分类器鲁棒性的条件数概率分布的尾部界。在具有有理权重的单隐藏层 sigmoid 网络的特殊情况下，我们推导了决策边界管状邻域的宽度多项式界。
英文摘要
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.
Comments32 pages, 1 figure
