Convolutional Filtering and Neural Networks with Non Commutative Algebras

In this paper we provide stability results for algebraic neural networks (AlgNNs) based on non commutative algebras. AlgNNs are stacked layered structures with each layer associated to an algebraic signal model (ASM) determined by an algebra, a vector space, and a homomorphism. Signals are modeled as elements of the vector space, filters are elements in the algebra, while the homomorphism provides a realization of the filters as concrete operators. We study the stability of the algebraic filters in non commutative algebras to perturbations on the homomorphisms, and we provide conditions under which stability is guaranteed. We show that the commutativity between shift operators and between shifts and perturbations does not affect the property of an architecture of being stable. This provides an answer to the question of whether shift invariance was a necessary attribute of convolutional architectures to guarantee stability. Additionally, we show that although the frequency responses of filters in non commutative algebras exhibit substantial differences with respect to filters in commutative algebras, their derivatives for stable filters have a similar behavior.