# Algebraic Neural Networks: Stability to Deformations

arXiv: https://arxiv.org/abs/2009.01433 · HTML: https://mlsanity.com/paper/2009.01433/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Sep 2020.

## Built on by (most cited first, 22 of 22)

- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
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- [Tangent Bundle Filters and Neural Networks: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2210.15058/) (2022)
- [Convolutional Filtering and Neural Networks with Non Commutative Algebras](https://mlsanity.com/paper/2108.09923/) (2021)
- [Graphon Pooling for Reducing Dimensionality of Signals and Convolutional Operators on Graphs](https://mlsanity.com/paper/2212.08171/) (2022)
- [Stability of Aggregation Graph Neural Networks](https://mlsanity.com/paper/2207.03678/) (2022)
- [Sparse Covariance Neural Networks](https://mlsanity.com/paper/2410.01669/) (2024)
- [Learning Stochastic Graph Neural Networks with Constrained Variance](https://mlsanity.com/paper/2201.12611/) (2022)
- [Learning with Multigraph Convolutional Filters](https://mlsanity.com/paper/2210.16272/) (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)
- [Quiver neural networks](https://mlsanity.com/paper/2207.12773/) (2022)
- [Convolutional Filtering with RKHS Algebras](https://mlsanity.com/paper/2411.01341/) (2024)
- [Fully tensorial approach to hypercomplex-valued neural networks](https://mlsanity.com/paper/2407.00449/) (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)
- [On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective](https://mlsanity.com/paper/2506.01213/) (2025)
- [Algebraic Approach to Ridge-Regularized Mean Squared Error Minimization in Minimal ReLU Neural Network](https://mlsanity.com/paper/2508.17783/) (2025)
- [Neural Isomorphic Fields: A Transformer-based Algebraic Numerical Embedding](https://mlsanity.com/paper/2601.12095/) (2026)
- [Change the Product, Keep the Parameters: Associative Algebra Layers for Transformers](https://mlsanity.com/paper/2609.32814/) (2026)
- [Fundamental Limits of Transferability and Equivariance in Algebraic Signal Models I: Finite Dimensions](https://mlsanity.com/paper/2609.36106/) (2026)

## Abstract (arXiv)

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