# Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks

arXiv: https://arxiv.org/abs/2409.11772 · HTML: https://mlsanity.com/paper/2409.11772/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 80 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 30 of 47)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Monarch: Expressive Structured Matrices for Efficient and Accurate Training](https://mlsanity.com/paper/2204.00595/) (2022)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Learning Fast Algorithms for Linear Transforms Using Butterfly Factorizations](https://mlsanity.com/paper/1903.05895/) (2019)
- [Residual Pathway Priors for Soft Equivariance Constraints](https://mlsanity.com/paper/2112.01388/) (2021)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Attentive Group Equivariant Convolutional Networks](https://mlsanity.com/paper/2002.03830/) (2020)
- [Equivariant Q Learning in Spatial Action Spaces](https://mlsanity.com/paper/2110.15443/) (2021)
- [Covariant Compositional Networks For Learning Graphs](https://mlsanity.com/paper/1801.02144/) (2018)
- [Sample Efficient Grasp Learning Using Equivariant Models](https://mlsanity.com/paper/2202.09468/) (2022)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Structured Transforms for Small-Footprint Deep Learning](https://mlsanity.com/paper/1510.01722/) (2015)
- [Relaxing Equivariance Constraints with Non-stationary Continuous Filters](https://mlsanity.com/paper/2204.07178/) (2022)
- [Kaleidoscope: An Efficient, Learnable Representation For All Structured Linear Maps](https://mlsanity.com/paper/2012.14966/) (2020)
- [Scale Equivariance Improves Siamese Tracking](https://mlsanity.com/paper/2007.09115/) (2020)

## Built on by (most cited first, 7 of 7)

- [Approximate Equivariance in Reinforcement Learning](https://mlsanity.com/paper/2411.04225/) (2024)
- [Flopping for FLOPs: Leveraging equivariance for computational efficiency](https://mlsanity.com/paper/2502.05169/) (2025)
- [Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetry](https://mlsanity.com/paper/2512.11855/) (2025)
- [PEnGUiN: Partially Equivariant Graph NeUral Networks for Sample Efficient MARL](https://mlsanity.com/paper/2503.15615/) (2025)
- [Approximate Equivariance via Projection-based Regularisation](https://mlsanity.com/paper/2601.05028/) (2026)
- [EdgeLDR: Quaternion Low-Displacement Rank Neural Networks for Edge-Efficient Deep Learning](https://mlsanity.com/paper/2601.05379/) (2026)
- [Tunable Soft Equivariance with Guarantees](https://mlsanity.com/paper/2603.26657/) (2026)

## Abstract (arXiv)

There has been much recent interest in designing neural networks (NNs) with relaxed equivariance, which interpolate between exact equivariance and full flexibility for consistent performance gains. In a separate line of work, structured parameter matrices with low displacement rank (LDR) -- which permit fast function and gradient evaluation -- have been used to create compact NNs, though primarily benefiting classical convolutional neural networks (CNNs). In this work, we propose a framework based on symmetry-based structured matrices to build approximately equivariant NNs with fewer parameters. Our approach unifies the aforementioned areas using Group Matrices (GMs), a forgotten precursor to the modern notion of regular representations of finite groups. GMs allow the design of structured matrices similar to LDR matrices, which can generalize all the elementary operations of a CNN from cyclic groups to arbitrary finite groups. We show GMs can also generalize classical LDR theory to general discrete groups, enabling a natural formalism for approximate equivariance. We test GM-based architectures on various tasks with relaxed symmetry and find that our framework performs competitively with approximately equivariant NNs and other structured matrix-based methods, often with one to two orders of magnitude fewer parameters.