On the finite representation of linear group equivariant operators via permutant measures

Recent advances in machine learning have highlighted the importance of using group equivariant non-expansive operators for building neural networks in a more transparent and interpretable way. An operator is called equivariant with respect to a group if the action of the group commutes with the operator. Group equivariant non-expansive operators can be seen as multi-level components that can be joined and connected in order to form neural networks by applying the operations of chaining, convex combination and direct product. In this paper we prove that each linear G-equivariant non-expansive operator (GENEO) can be produced by a weighted summation associated with a suitable permutant measure, provided that the group G transitively acts on a finite signal domain. This result is based on the Birkhoff–von Neumann decomposition of doubly stochastic matrices and some well known facts in group theory. Our theorem makes available a new method to build all linear GENEOs with respect to a transitively acting group in the finite setting. This work is part of the research devoted to develop a good mathematical theory of GENEOs, seen as relevant components in machine learning.