Algebraic Neural Networks: Stability to Deformations

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