Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations
This work addresses stability issues for non-commutative architectures like group and quaternion neural networks, but it is incremental as it builds on existing algebraic signal processing tools.
The paper tackles the problem of stability to small deformations in non-commutative convolutional filters used in neural networks, showing that these filters can be stable to perturbations on operator spaces and revealing a trade-off between stability and selectivity similar to commutative models.
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.