通过决策边界体积理解深度学习的几何学
Understanding the geometry of deep learning with decision boundary volume
AI总结：
本文提出通过局部表面体积测量神经网络决策边界，揭示其与模型准确性和鲁棒性的关系，验证了决策边界体积与分类准确度的反比关系。
摘要：
对于分类任务，深度神经网络的性能由其决策边界的结构决定，其几何特性直接影响模型的准确性和鲁棒性。受Weyl的经典管公式启发，本文引入一种通过局部表面体积测量神经网络决策边界的方法，提供了一种理论上可验证且高效的度量方式，使模型效果的几何解释适用于深度学习中考虑的高维特征空间。较小的表面体积对应较低的模型复杂度和更好的泛化能力。本文在多个图像处理任务中验证了决策边界体积与分类准确度成反比。同时，局部表面体积与全连接架构泛化能力之间的关系在不同任务中表现不稳定。因此，对于适合特定数据结构的网络架构，本文展示了更平滑的决策边界导致更好性能，这与我们的直觉一致。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
