Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations

TL;DR

This paper explores algebraic signal models based on non commutative algebras in neural networks, analyzing their stability to small deformations and implications for various non commutative architectures.

Contribution

It provides a theoretical analysis of stability and filtering properties of non commutative convolutional filters within algebraic signal processing framework.

Findings

Non commutative filters can be stable to small operator perturbations.

Spectral components in non commutative models relate to higher-dimensional spaces.

Trade-off exists between stability and selectivity in non commutative signal models.

Abstract

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.