In this paper, we introduce a novel analysis of neural networks based on
geometric (Clifford) algebra and convex optimization. We show that optimal
weights of deep ReLU neural networks are given by the wedge product of training
samples when trained with standard regularized loss. Furthermore, the training
problem reduces to convex optimization over wedge product features, which
encode the geometric structure of the training dataset. This structure is given
in terms of signed volumes of triangles and parallelotopes generated by data
vectors. The convex problem finds a small subset of samples via $\ell_1$
regularization to discover only relevant wedge product features. Our analysis
provides a novel perspective on the inner workings of deep neural networks and
sheds light on the role of the hidden layers.

基于几何 (克利福德) 代数和凸优化，我们介绍了一种新的神经网络分析方法，显示出深度 ReLU 神经网络的最优权重是通过训练样本的外积给出的，并且训练问题可以化简为对外积特征进行凸优化，这些特征编码了训练数据集的几何结构，其中几何结构通过数据向量生成的三角形和平行六面体的符号体积给出。