Simplicial-Map Neural Networks Robust to Adversarial Examples

Eduardo Paluzo-Hidalgo,Miguel A Gutiérrez-Naranjo,Jónathan Heras,Rocio Gonzalez-Diaz

doi:10.3390/math9020169

Eduardo Paluzo-Hidalgo, Miguel A Gutiérrez-Naranjo + Show 2 more

Open Access

https://doi.org/10.3390/math9020169

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Journal: Mathematics	Publication Date: Jan 15, 2021
Citations: 3	License type: CC BY 4.0

Affiliation: Universidad de Sevilla, University of La Rioja

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.

Highlights

Adversarial examples are currently one of the main problems for the robustness of neural networks applications [1]
From a mathematical point of view, neural network can be seen as the composition of a big amount of simple functions, mainly from linear algebra, and the so-called activation functions
Since the efficiency of such neural networks depends of the choice of an appropriate set of parameters, most of the efforts in the study of such networks has been focused on optimization techniques

Summary

Introduction

Adversarial examples are currently one of the main problems for the robustness of neural networks applications [1]. Regarding other approaches to these ideas found in the literature, in [13], the authors proved the existence of a two-hidden-layer neural network which can approximate any continuous multivariable function with arbitrary precision, and, in [14], they provided a constructive method through a numerical analysis approach Such papers can be seen as alternative constructive proofs to the Universal Approximation Theorem where no adversarial examples on classification problems were considered.

Background

Simplicial-Map Neural Networks

Classification with Simplicial-Map Neural Networks

Findings

Conclusions and Future Work