Graded Representation Theory of Equivariant Neural Networks
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Abstract (arXiv)
Nonlinear activations can create equivariant interactions between irreducible representations that linear maps cannot. We use the Gaussian degree decomposition to extend ordinary polynomial degree to such nonlinear maps, and prove that for a fixed coordinatewise equivariant layer each degree factors into a polynomial determined by the linear maps and a scalar determined by the activation. This separates three distinct obstructions, coming from symmetry, coordinates, and activation.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 10 of its 18 references that are in the map. First posted on arXiv 22 Sep 2026.
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References · 10
- 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
- 2018Invariant and Equivariant Graph Networkscited by 346
- 2016Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivitycited by 138
- 2018Universal approximations of invariant maps by neural networkscited by 127
- 2020Universal Equivariant Multilayer Perceptronscited by 49
- 2023Equivariant Polynomials for Graph Neural Networkscited by 38
- 2023Any-dimensional equivariant neural networkscited by 9
- 2024A Characterization Theorem for Equivariant Networks with Point-wise Activationscited by 9
- 2024Equivariant neural networks and piecewise linear representation theorycited by 3
- 2026Piecewise Linear Equivariant Maps for Compact Groupscited by 1
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