We study robustness verification of neural networks via metric algebraic geometry. For polynomial neural networks, certifying a robustness radius amounts to computing the distance to the algebraic decision boundary. We use the Euclidean distance (ED) degree as an intrinsic measure of the complexity of this problem, analyze the associated ED discriminant, and introduce a parameter discriminant that detects parameter values at which the ED degree drops. We derive formulas for the ED degree for several network architectures and characterize the expected number of real critical points in the infinite-width limit. We develop symbolic elimination methods to compute these quantities and homotopy-continuation methods for exact robustness certification. Finally, experiments on lightning self-attention modules reveal decision boundaries with strictly smaller ED degree than generic cubic hypersurfaces of the same ambient dimension.

翻译：我们通过度量代数几何研究神经网络的鲁棒性验证。对于多项式神经网络，验证鲁棒半径等价于计算到代数决策边界的欧氏距离。我们将欧氏距离（ED）度作为该问题复杂度的内蕴度量，分析相关的ED判别式，并引入一种参数判别式来检测ED度下降的参数值。我们推导了若干网络架构的ED度公式，刻画了无限宽度极限下实临界点的期望数量。发展了用于计算这些量的符号消去方法以及用于精确鲁棒性认证的同伦延拓方法。最后，在闪电自注意力模块上的实验揭示了决策边界的ED度严格小于相同环境维度的通用三次超曲面。