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Graded Representation Theory of Equivariant Neural Networks
Abstract
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.
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BibTeX
@misc{shayestehfar2026graded,
title = {{Graded Representation Theory of Equivariant Neural Networks}},
author = {Mani Shayestehfar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25776},
eprint = {2609.25776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25776},
}
RIS
TY - UNPB
TI - Graded Representation Theory of Equivariant Neural Networks
AU - Mani Shayestehfar
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25776
UR - https://arxiv.org/abs/2609.25776
AB - 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.
ER -