论文标题
DNNS决策边界的一些几何和拓扑特性
Some Geometrical and Topological Properties of DNNs' Decision Boundaries
论文作者
论文摘要
决策区域的几何形状和拓扑与针对对抗性攻击的分类性能和鲁棒性密切相关。在本文中，我们使用差异几何形状在理论上探讨了深神经网络（DNNS）产生的决策区域的几何和拓扑特性。目的是获得给定DNN模型的决策边界的一些几何和拓扑特性，并为DNN的设计和正规化提供一些原则指导。首先，我们在网络参数方面介绍了决策边界的曲率，并提供了足够的网络参数条件，以产生平坦或可开发的决策边界。基于差异几何形状中的高斯 - 邦网对定理，我们提出了一种计算紧凑型决策边界的Euler特征的方法，并通过实验对其进行验证。
Geometry and topology of decision regions are closely related with classification performance and robustness against adversarial attacks. In this paper, we use differential geometry to theoretically explore the geometrical and topological properties of decision regions produced by deep neural networks (DNNs). The goal is to obtain some geometrical and topological properties of decision boundaries for given DNN models, and provide some principled guidance to design and regularization of DNNs. First, we present the curvatures of decision boundaries in terms of network parameters, and give sufficient conditions on network parameters for producing flat or developable decision boundaries. Based on the Gauss-Bonnet-Chern theorem in differential geometry, we then propose a method to compute the Euler characteristics of compact decision boundaries, and verify it with experiments.
