Understanding the geometry of deep learning with decision boundary volume
This work provides a geometric interpretation for model effectiveness, potentially aiding in network design and analysis for image processing, though it appears incremental in linking known concepts to deep learning.
The authors tackled the problem of understanding deep learning performance by measuring decision boundary geometry, finding that smaller decision boundary volume correlates with higher classification accuracy in convolutional architectures for image tasks.
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.