On the finite representation of group equivariant operators via permutant measures
This work addresses the challenge of designing group-equivariant neural network architectures, offering a theoretical foundation for constructing linear operators, but it is incremental as it builds on existing equivariance concepts.
The paper tackles the problem of constructing linear G-equivariant operators for neural networks by proving that each such operator can be generated using a permutant measure when the group G acts transitively on a finite domain X, providing a new method for building these operators in finite settings.
The study of $G$-equivariant operators is of great interest to explain and understand the architecture of neural networks. In this paper we show that each linear $G$-equivariant operator can be produced by a suitable permutant measure, provided that the group $G$ transitively acts on a finite signal domain $X$. This result makes available a new method to build linear $G$-equivariant operators in the finite setting.