Non Commutative Convolutional Signal Models in Neural Networks:
  Stability to Small Deformations
Alejandro Parada-Mayorga Landon Butler Alejandro Ribeiro
In this paper we discuss the results recently published in~[1] about
algebraic signal models (ASMs) based on non commutative algebras and their use
in convolutional neural networks. Relying on the general tools from algebraic
signal processing (ASP), we study the filtering and stability properties of non
commutative convolutional filters. We show how non commutative filters can be
stable to small perturbations on the space of operators. We also show that
although the spectral components of the Fourier representation in a non
commutative signal model are associated to spaces of dimension larger than one,
there is a trade-off between stability and selectivity similar to that observed
for commutative models. Our results have direct implications for group neural
networks, multigraph neural networks and quaternion neural networks, among
other non commutative architectures. We conclude by corroborating these results
through numerical experiments.