Adversarial examples are a pervasive phenomenon of machine learning models
where seemingly imperceptible perturbations to the input lead to
misclassifications for otherwise statistically accurate models. We propose a
geometric framework, drawing on tools from the manifold reconstruction
literature, to analyze the high-dimensional geometry of adversarial examples.
In particular, we highlight the importance of codimension: for low-dimensional
data manifolds embedded in high-dimensional space there are many directions off
the manifold in which an adversary could construct adversarial examples.
Adversarial examples are a natural consequence of learning a decision boundary
that classifies the low-dimensional data manifold well, but classifies points
near the manifold incorrectly. Using our geometric framework we prove that
adversarial training is sample inefficient, and show sufficient sampling
conditions under which nearest neighbor classifiers and ball-based adversarial
training are robust. Finally we introduce adversarial training with Voronoi
constraints, which replaces the norm ball constraint with the Voronoi cell for
each point in the training set. We show that adversarial training with Voronoi
constraints produces robust models which significantly improve over the
state-of-the-art on MNIST and are competitive on CIFAR-10.

该论文提出了一个几何框架以分析对抗性样本的高维几何，强调了余维数的重要性，并介绍了带有 Voronoi 约束的对抗性训练，证明它比基于球形的对抗性训练在 MNIST 数据上更具鲁棒性。

使用 Voronoi 约束的对抗训练

Adversarial Training with Voronoi Constraints

Adversarial examples are a pervasive phenomenon of machine learning models
where seemingly imperceptible perturbations to the input lead to
misclassifications for otherwise statistically accurate models. We propose a
geometric framework, drawing on tools from the manifold reconstruction
literature, to analyze the high-dimensional geometry of adversarial examples.
In particular, we highlight the importance of codimension: for low-dimensional
data manifolds embedded in high-dimensional space there are many directions off
the manifold in which to construct adversarial examples. Adversarial examples
are a natural consequence of learning a decision boundary that classifies the
low-dimensional data manifold well, but classifies points near the manifold
incorrectly. Using our geometric framework we prove (1) a tradeoff between
robustness under different norms, (2) that adversarial training in balls around
the data is sample inefficient, and (3) sufficient sampling conditions under
which nearest neighbor classifiers and ball-based adversarial training are
robust.

该研究提出了一种基于几何框架和流形重建方法的方法，以分析对抗样本的高维几何形状，并证明了不同规范的鲁棒性、球形对抗性训练的样本编号和最近邻分类器与基于球面的对抗训练的充分采样条件。