揭示等变与不变映射的对应关系，助力设计更优对称神经网络。
Decomposition of Equivariant Maps via Invariant Maps: Application to Universal Approximation under Symmetry
- 发现等变映射与特定不变映射间的一一对应关系
- 构建基于不变网络的通用等变架构，参数量更少
- 给出有限群对称下ReLU网络的逼近速率
本文研究了群 $G$ 作用下不变映射与等变映射之间的关系，建立两者间的一一对应。这一理论使等变映射的分析可转化为不变映射问题，反之亦然。基于此，我们提出一种由通用不变网络构建的通用等变架构。该构造揭示了其与已有通用等变架构的本质差异。同时分析模型复杂度，探讨不变与等变网络在自由参数数量上的关系。最后，针对有限群 $G$ 的情形，给出使用 ReLU 激活函数的 $G$-等变深度神经网络的逼近速率。
原文摘要 · Abstract (English)
In this paper, we develop a theory about the relationship between invariant and equivariant maps with regard to a group $G$. We then leverage this theory in the context of deep neural networks with group symmetries in order to obtain novel insight into their mechanisms. More precisely, we establish a one-to-one relationship between equivariant maps and certain invariant maps. This allows us to reduce arguments for equivariant maps to those for invariant maps and vice versa. As an application, we propose a construction of universal equivariant architectures built from universal invariant networks. We, in turn, explain how the universal architectures arising from our construction differ from standard equivariant architectures known to be universal. Furthermore, we explore the complexity, in terms of the number of free parameters, of our models, and discuss the relation between invariant and equivariant networks' complexity. Finally, we also give an approximation rate for G-equivariant deep neural networks with ReLU activation functions for finite group G.
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