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

Motasem Alfarra, Adel Bibi, Hasan Abed Al Kader Hammoud, Mohamed Gaafar, Bernard Ghanem

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


This work tackles the problem of characterizing and understanding the decision boundaries of neural networks with piecewise linear non-linearity activations. We use tropical geometry, a new development in the area of algebraic geometry, to characterize the decision boundaries of a simple network of the form (Affine, ReLU, Affine). Our main finding is 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 these zonotopes are functions of the network parameters. This geometric characterization provides new perspectives to three tasks. (i) We propose a new tropical perspective to the lottery ticket hypothesis, where we view the effect of different initializations on the tropical geometric representation of a network's decision boundaries. (ii) Moreover, we propose new tropical based optimization reformulations that directly influence the decision boundaries of the network for the task of network pruning. (iii) At last, we discuss the reformulation of the generation of adversarial attacks in a tropical sense. We demonstrate that one can construct adversaries in a new tropical setting by perturbing a specific set of decision boundaries by perturbing a set of parameters in the network.
Original languageEnglish (US)
Pages (from-to)1-12
Number of pages12
JournalIEEE Transactions on Pattern Analysis and Machine Intelligence
StatePublished - Aug 24 2022

Bibliographical note

KAUST Repository Item: Exported on 2022-09-14
Acknowledgements: This work was supported by King Abdullah University of Science and Technology Office of Sponsored Research. Part of the work was done while MG was at itemis AG.


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