标题： DNNs决策边界的某些几何和拓扑性质 显示英文标题

标题： Some Geometrical and Topological Properties of DNNs' Decision Boundaries

摘要： 决策区域的几何和拓扑结构与分类性能以及对对抗攻击的鲁棒性密切相关。 在本文中，我们使用微分几何从理论上探讨深度神经网络（DNNs）产生的决策区域的几何和拓扑特性。 目标是获得给定DNN模型的决策边界的一些几何和拓扑特性，并为DNN的设计和正则化提供一些有理论依据的指导。 首先，我们根据网络参数给出了决策边界的曲率，并给出了产生平坦或可展决策边界的网络参数的充分条件。 基于微分几何中的Gauss-Bonnet-Chern定理，我们提出了一种计算紧致决策边界欧拉特征的方法，并通过实验进行了验证。 显示英文摘要

摘要： 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.