# Convolutional Filtering and Neural Networks with Non Commutative Algebras

arXiv: https://arxiv.org/abs/2108.09923 · HTML: https://mlsanity.com/paper/2108.09923/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 65 references that are in the map.
- First posted on arXiv 23 Aug 2021.

## Builds on (most cited first, 22 of 22)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Deep Quaternion Networks](https://mlsanity.com/paper/1712.04604/) (2017)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [Diffusion Scattering Transforms on Graphs](https://mlsanity.com/paper/1806.08829/) (2018)
- [Quaternion Recurrent Neural Networks](https://mlsanity.com/paper/1806.04418/) (2018)
- [Quaternion Convolutional Neural Networks for Heterogeneous Image Processing](https://mlsanity.com/paper/1811.02656/) (2018)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [Quaternion Generative Adversarial Networks](https://mlsanity.com/paper/2104.09630/) (2021)
- [Quaternion Graph Neural Networks](https://mlsanity.com/paper/2008.05089/) (2020)
- [Spectral Multigraph Networks for Discovering and Fusing Relationships in Molecules](https://mlsanity.com/paper/1811.09595/) (2018)
- [Stability of Algebraic Neural Networks to Small Perturbations](https://mlsanity.com/paper/2010.11544/) (2020)
- [AlgebraNets](https://mlsanity.com/paper/2006.07360/) (2020)
- [Quaternion Neural Networks for Multi-channel Distant Speech Recognition](https://mlsanity.com/paper/2005.08566/) (2020)
- [Quiver Signal Processing (QSP)](https://mlsanity.com/paper/2010.11525/) (2020)

## Built on by (most cited first, 8 of 8)

- [Convolutional Learning on Multigraphs](https://mlsanity.com/paper/2209.11354/) (2022)
- [Sampling and Uniqueness Sets in Graphon Signal Processing](https://mlsanity.com/paper/2401.06279/) (2024)
- [Graph neural networks and non-commuting operators](https://mlsanity.com/paper/2411.04265/) (2024)
- [Towards Explaining Hypercomplex Neural Networks](https://mlsanity.com/paper/2403.17929/) (2024)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [RKHS Representation of Algebraic Convolutional Filters with Integral Operators](https://mlsanity.com/paper/2602.19094/) (2026)
- [Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations](https://mlsanity.com/paper/2310.03879/) (2023)
- [Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions](https://mlsanity.com/paper/2609.36106/) (2026)

## Abstract (arXiv)

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.