提出多项式群卷积网络的新几何框架，揭示其参数空间结构。
The Geometry of Polynomial Group Convolutional Neural Networks
- 用分次群代数语言构建PGCNN数学框架，给出哈达玛与克罗内克两种参数化方式。
- 证明神经流形维数仅取决于层数和群大小，与具体群结构无关。
- 揭示克罗内克参数化的纤维结构，对哈达玛参数化提出合理猜想。
我们研究任意有限群 $G$ 上的多项式群卷积神经网络（PGCNN）。特别地，引入基于分次群代数的语言，建立全新的数学框架。该框架导出两种自然的参数化方式——基于哈达玛积与克罗内克积，二者通过线性映射关联。我们计算了相关神经流形的维度，验证其仅依赖于网络层数和群的大小。同时，描述了克罗内克参数化在正则群作用与缩放下的通解纤维结构，并对哈达玛参数化提出类似描述的猜想。该猜想在小群与浅层网络中经显式计算得到支持。
原文摘要 · Abstract (English)
We study polynomial group convolutional neural networks (PGCNNs) for an arbitrary finite group $G$. In particular, we introduce a new mathematical framework for PGCNNs using the language of graded group algebras. This framework yields two natural parametrizations of the architecture, based on Hadamard and Kronecker products, related by a linear map. We compute the dimension of the associated neuromanifold, verifying that it depends only on the number of layers and the size of the group. We also describe the general fiber of the Kronecker parametrization up to the regular group action and rescaling, and conjecture the analogous description for the Hadamard parametrization. Our conjecture is supported by explicit computations for small groups and shallow networks.
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