  We study algebraic neural networks (AlgNNs) with commutative algebras which
unify diverse architectures such as Euclidean convolutional neural networks,
graph neural networks, and group neural networks under the umbrella of
algebraic signal processing. An AlgNN is a stacked layered information
processing structure where each layer is conformed by an algebra, a vector
space and a homomorphism between the algebra and the space of endomorphisms of
the vector space. Signals are modeled as elements of the vector space and are
processed by convolutional filters that are defined as the images of the
elements of the algebra under the action of the homomorphism. We analyze
stability of algebraic filters and AlgNNs to deformations of the homomorphism
and derive conditions on filters that lead to Lipschitz stable operators. We
conclude that stable algebraic filters have frequency responses -- defined as
eigenvalue domain representations -- whose derivative is inversely proportional
to the frequency -- defined as eigenvalue magnitudes. It follows that for a
given level of discriminability, AlgNNs are more stable than algebraic filters,
thereby explaining their better empirical performance. This same phenomenon has
been proven for Euclidean convolutional neural networks and graph neural
networks. Our analysis shows that this is a deep algebraic property shared by a
number of architectures.

Machine Learning;Machine Learning