Exact Visualization of Deep Neural Network Geometry and Decision Boundary

Abstract

Visualizing Deep Network (DN) geometry and decision boundaries remains a key challenge even today. In fact, despite the dire need for such methods e.g. to assess the quality of a trained model, to compare models, to interpret decisions, the community at large still relies on crude approximations. For example, computing the decision boundary of a model, say on a 2d slice of their input space, is done through gradient descent and sampling with dichotomy search. In this paper, we lean on the rich theory of Continuous Piece-Wise Linear (CPWL) DNs to provide, for the first time, a method that provably produces the exact geometry (CPWL partition) and decision boundary of any DN employing nonlinearities such as ReLU, Leaky-ReLU, and max-pooling. Using the proposed method, we are able to not only visualize the decision boundary but also obtain its spanning space, i.e., we can sample arbitrarily many inputs that provably lie on the model's decision boundary, up to numerical precision. We explore how such methods can be used to interpret architectural choices, e.g., using convolutional architectures instead of fully-connected neural networks.