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.

翻译：本文引入并研究了非交换卷积神经网络的代数泛化方法。我们利用代数信号处理理论对卷积非交换架构进行建模，推导出具体的稳定性界限，这些界限扩展了文献中针对交换卷积神经网络所获得的结果。研究表明，非交换卷积架构能够在算子空间上对形变保持稳定。我们发展非交换信号模型的谱表示，以揭示非交换滤波器独立处理傅里叶分量的特性。特别地，我们证明：尽管非交换模型中信号的谱分解对应维数大于1的特征空间，但稳定性与选择性之间存在由低维矩阵空间中的矩阵多项式函数控制的权衡关系。这种权衡表明，当代数中的滤波器被限制为稳定时，网络判别性能的损失可通过逐点非线性激活得到补偿。本文所得结果对群神经网络、多图神经网络和四元数神经网络等非交换卷积架构具有直接应用价值，我们通过一系列数值实验展示了这些架构在存在扰动时的行为特性。