2021/08/23 by Alejandro Parada-Mayorga, Parada-Mayorga, Alejandro, Butler, Landon +3 · 1 citation
Computer Science · Mathematics · #Advanced Graph Neural Networks #FOS: Computer and information sciences #Machine Learning (cs.LG) #Mathematical Analysis and Transform Methods #Matrix Theory and Algorithms #Neural Networks Stability and Synchronization #Neural Networks and Applications
paper · pdf · doi:10.48550/arxiv.2108.09923
openalex publication_date 2021/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce and study the algebraic generalization of non\ncommutative convolutional neural networks. We leverage the theory of algebraic\nsignal processing to model convolutional non commutative architectures, and we\nderive concrete stability bounds that extend those obtained in the literature\nfor commutative convolutional neural networks. We show that non commutative\nconvolutional architectures can be stable to deformations on the space of\noperators. We develop the spectral representation of non commutative signal\nmodels to show that non commutative filters process Fourier components\nindependently of each other. In particular we prove that although the spectral\ndecompositions of signals in non commutative models are associated to\neigenspaces of dimension larger than one, there exists a trade-off between\nstability and selectivity, which is controlled by matrix polynomial functions\nin spaces of matrices of low dimension. This tradeoff shows how when the\nfilters in the algebra are restricted to be stable, there is a loss in\ndiscriminability that is compensated in the network by the pointwise\nnonlinearities. The results derived in this paper have direct applications and\nimplications in non commutative convolutional architectures such as group\nneural networks, multigraph neural networks, and quaternion neural networks,\nfor which we provide a set of numerical experiments showing their behavior when\nperturbations are present.\n