Classification regions of deep neural networks

While the geometry of classification regions and decision functions induced by traditional classifiers (such as linear and kernel SVM) is fairly well understood, these fundamental geometric properties are to a large extent unknown for state-of-the-art deep neural networks. Yet, to understand the recent success of deep neural networks and potentially address their weaknesses (such as their instability to perturbations [1]), an understanding of these geometric properties remains primordial. While many fundamental properties of deep networks have recently been studied, such as their optimization landscape in [2], [3], their generalization in [4], [5], and their expressivity in [6], [7], the geometric properties of the decision boundary and classification regions of deep networks has comparatively received little attention. The goal of this paper is to analyze these properties, and leverage them to improve the robustness of such classifiers to perturbations. In this paper, we specifically view classification regions as topological spaces, and decision boundaries as hypersurfaces and examine their geometric properties. We first study the classification regions induced by state-of-the-art deep networks, and provide empirical evidence suggesting that these classification regions are connected; that is, there exists a continuous path that remains in the region between any two points of the same label. Up to our knowledge, this represents the first instance where the connectivity of classification regions is empirically shown. Then, to study the complexity of the functions learned by the deep network, we analyze the curvature of their decision boundary.