# Equivariance Through Parameter-Sharing

arXiv: https://arxiv.org/abs/1702.08389 · HTML: https://mlsanity.com/paper/1702.08389/

## Key facts

- Cited by 141 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2017.

## Built on by (most cited first, 30 of 141)

- [Pure Transformers are Powerful Graph Learners](https://mlsanity.com/paper/2207.02505/) (2022)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Equivariant Architectures for Learning in Deep Weight Spaces](https://mlsanity.com/paper/2301.12780/) (2023)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Provably Strict Generalisation Benefit for Equivariant Models](https://mlsanity.com/paper/2102.10333/) (2021)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Generalization on the Unseen, Logic Reasoning and Degree Curriculum](https://mlsanity.com/paper/2301.13105/) (2023)
- [Permutation Equivariant Neural Functionals](https://mlsanity.com/paper/2302.14040/) (2023)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [Graph Metanetworks for Processing Diverse Neural Architectures](https://mlsanity.com/paper/2312.04501/) (2023)
- [Neural Functional Transformers](https://mlsanity.com/paper/2305.13546/) (2023)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Symmetry in Neural Network Parameter Spaces](https://mlsanity.com/paper/2506.13018/) (2025)
- [Equivariant Deep Weight Space Alignment](https://mlsanity.com/paper/2310.13397/) (2023)
- [Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions](https://mlsanity.com/paper/2205.11912/) (2022)
- [Robust (Controlled) Table-to-Text Generation with Structure-Aware Equivariance Learning](https://mlsanity.com/paper/2205.03972/) (2022)
- [Provably Strict Generalisation Benefit for Invariance in Kernel Methods](https://mlsanity.com/paper/2106.02346/) (2021)
- [Symmetry-via-Duality: Invariant Neural Network Densities from Parameter-Space Correlators](https://mlsanity.com/paper/2106.00694/) (2021)
- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Universal Neural Functionals](https://mlsanity.com/paper/2402.05232/) (2024)
- [Ultra-High-Resolution Detector Simulation with Intra-Event Aware GAN and Self-Supervised Relational Reasoning](https://mlsanity.com/paper/2303.08046/) (2023)
- [Symmetry Breaking and Equivariant Neural Networks](https://mlsanity.com/paper/2312.09016/) (2023)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Equi-Tuning: Group Equivariant Fine-Tuning of Pretrained Models](https://mlsanity.com/paper/2210.06475/) (2022)

## Abstract (arXiv)

We propose to study equivariance in deep neural networks through parameter symmetries. In particular, given a group G that acts discretely on the input and output of a standard neural network layer φW: ReM → ReN, we show that φW is equivariant with respect to G-action iff G explains the symmetries of the network parameters W. Inspired by this observation, we then propose two parameter-sharing schemes to induce the desirable symmetry on W. Our procedures for tying the parameters achieve G-equivariance and, under some conditions on the action of G, they guarantee sensitivity to all other permutation groups outside G.