On the Decision Boundaries of Deep Neural Networks: A Tropical Geometry Perspective

Motasem Alfarra, Adel Bibi, Hasan Hammoud, Mohamed Gaafar, Bernard Ghanem

TL;DR: Tropical geometry can be leveraged to represent the decision boundaries of neural networks and bring to light interesting insights.

Abstract: This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piece-wise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to provide a characterization of the decision boundaries of a simple neural network of the form (Affine, ReLU, Affine). Specifically, we show that the decision boundaries are a subset of a tropical hypersurface, which is intimately related to a polytope formed by the convex hull of two zonotopes. The generators of the zonotopes are precise functions of the neural network parameters. We utilize this geometric characterization to shed light and new perspective on three tasks. In doing so, we propose a new tropical perspective for the lottery ticket hypothesis, where we see the effect of different initializations on the tropical geometric representation of the decision boundaries. Also, we leverage this characterization as a new set of tropical regularizers, which deal directly with the decision boundaries of a network. We investigate the use of these regularizers in neural network pruning (removing network parameters that do not contribute to the tropical geometric representation of the decision boundaries) and in generating adversarial input attacks (with input perturbations explicitly perturbing the decision boundaries geometry to change the network prediction of the input).

Code: https://drive.google.com/file/d/1igQUriSYnUMKo6GafsRz5P_J45YYiImh/view?usp=sharing

Keywords: Decision boundaries, Neural Network, Tropical Geometry, Network Pruning, Adversarial Attacks, Lottery Ticket Hypothesis