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.

中文翻译：

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Simplicial-Map神经网络的鲁棒对抗示例

广义上讲，当输入数据点上的小扰动导致模型分配的输出标签发生变化时，就会发生针对分类模型的对抗性示例。这样的对抗性示例代表了神经网络应用程序安全性的弱点，并且已经提出了许多不同的解决方案来最小化它们的影响。在本文中，我们通过称为简单映射神经网络的一系列神经网络提出了一种新方法。从代数拓扑的角度构造。我们的建议基于三个主要思想。首先，给定一个分类问题，输入数据集及其单热点标签集都将具有简单复杂结构，并且将定义这些复杂之间的简单映射。其次，将从这种简单映射图构建表征分类问题的神经网络。最后，通过考虑简单复合体的重心细分，将计算决策边界，以使神经网络对给定大小的对抗攻击具有鲁棒性。