For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a classical tube formula due to Weyl, we introduce a method to measure the decision boundary of a neural network through local surface volumes, providing a theoretically justifiable and efficient measure enabling a geometric interpretation of the effectiveness of the model applicable to the high dimensional feature spaces considered in deep learning. A smaller surface volume is expected to correspond to lower model complexity and better generalisation. We verify, on a number of image processing tasks with convolutional architectures that decision boundary volume is inversely proportional to classification accuracy. Meanwhile, the relationship between local surface volume and generalisation for fully connected architecture is observed to be less stable between tasks. Therefore, for network architectures suited to a particular data structure, we demonstrate that smoother decision boundaries lead to better performance, as our intuition would suggest.

翻译：对于分类任务而言，深度神经网络的性能由其决策边界结构决定，该边界的几何特性直接影响模型的关键性质，包括准确性与鲁棒性。受惠于Weyl的经典管状体积公式启发，我们提出一种通过局部曲面体积度量神经网络决策边界的方法，该度量具备理论依据且计算高效，能够为深度学习所涉及的高维特征空间中模型的有效性提供几何解释。更小的曲面体积预期对应更低的模型复杂度和更好的泛化能力。我们在多项采用卷积架构的图像处理任务中验证了决策边界体积与分类准确率呈反比关系。同时观察到，全连接架构的局部曲面体积与泛化能力之间的关系在不同任务间稳定性较弱。因此，对于适配特定数据结构的网络架构，我们证明了更平滑的决策边界会带来更优的性能，这与直观认知相符。