Simplicial-Map Neural Networks Robust to Adversarial Examples

Eduardo Paluzo-Hidalgo, Rocio Gonzalez-Diaz, Miguel A. Gutiérrez-Naranjo and Jónathan Heras
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Eduardo Paluzo-Hidalgo: Department of Applied Mathematics I, University of Seville, 41012 Seville, Spain
Rocio Gonzalez-Diaz: Department of Applied Mathematics I, University of Seville, 41012 Seville, Spain
Miguel A. Gutiérrez-Naranjo: Department of Computer Sciences and Artificial Intelligence, University of Seville, 41012 Seville, Spain
Jónathan Heras: Department of Mathematics and Computer Sciences, University of La Rioja, 26006 Logroño, Spain

Mathematics, 2021, vol. 9, issue 2, 1-16

Abstract: Broadly speaking, an adversarial example against a classification model occurs when a small perturbation on an input data point produces a change on the output label assigned by the model. Such adversarial examples represent a weakness for the safety of neural network applications, and many different solutions have been proposed for minimizing their effects. In this paper, we propose a new approach by means of a family of neural networks called simplicial-map neural networks constructed from an Algebraic Topology perspective. Our proposal is based on three main ideas. Firstly, given a classification problem, both the input dataset and its set of one-hot labels will be endowed with simplicial complex structures, and a simplicial map between such complexes will be defined. Secondly, a neural network characterizing the classification problem will be built from such a simplicial map. Finally, by considering barycentric subdivisions of the simplicial complexes, a decision boundary will be computed to make the neural network robust to adversarial attacks of a given size.

Keywords: algebraic topology; neural network; adversarial examples (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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