  In this paper we introduce and study the algebraic generalization of non
commutative convolutional neural networks. We leverage the theory of algebraic
signal processing to model convolutional non commutative architectures, and we
derive concrete stability bounds that extend those obtained in the literature
for commutative convolutional neural networks. We show that non commutative
convolutional architectures can be stable to deformations on the space of
operators. We develop the spectral representation of non commutative signal
models to show that non commutative filters process Fourier components
independently of each other. In particular we prove that although the spectral
decompositions of signals in non commutative models are associated to
eigenspaces of dimension larger than one, there exists a trade-off between
stability and selectivity, which is controlled by matrix polynomial functions
in spaces of matrices of low dimension. This tradeoff shows how when the
filters in the algebra are restricted to be stable, there is a loss in
discriminability that is compensated in the network by the pointwise
nonlinearities. The results derived in this paper have direct applications and
implications in non commutative convolutional architectures such as group
neural networks, multigraph neural networks, and quaternion neural networks,
for which we provide a set of numerical experiments showing their behavior when
perturbations are present.
