Geometry and Topology of Deep Neural Networks' Decision Boundaries
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Geometry and topology of decision regions are closely related with classification performance and robustness against adversarial attacks. In this paper, we use differential geometry and topology to explore theoretically the geometrical and topological properties of decision regions produced by deep neural networks (DNNs). The goals are to obtain some geometrical and topological properties of decision regions for given DNN models, and provide some principled guidances to designing and regularizing DNNs. At first, we give the curvatures of decision boundaries in terms of network weights. Based on the rotation index theorem and Gauss-Bonnet-Chern theorem, we then propose methods to identify the closeness and connectivity of given decision boundaries, and obtain the Euler characteristics of closed ones, all without the need to solve decision boundaries explicitly. Finally, we give necessary conditions on network architectures in order to produce closed decision boundaries, and sufficient conditions on network weights for producing zero curvature (flat or developable) decision boundaries.
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