# Learning Equivariant Functions via Quadratic Forms

arXiv: https://arxiv.org/abs/2509.22184 · HTML: https://mlsanity.com/paper/2509.22184/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 39 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [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)
- [B-Spline CNNs on Lie Groups](https://mlsanity.com/paper/1909.12057/) (2019)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Learning Invariances in Neural Networks](https://mlsanity.com/paper/2010.11882/) (2020)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Physics-Embedded Neural Networks: Graph Neural PDE Solvers with Mixed Boundary Conditions](https://mlsanity.com/paper/2205.11912/) (2022)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Equivariant Graph Hierarchy-Based Neural Networks](https://mlsanity.com/paper/2202.10643/) (2022)
- [A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning](https://mlsanity.com/paper/2311.00212/) (2023)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [Learning Infinitesimal Generators of Continuous Symmetries from Data](https://mlsanity.com/paper/2410.21853/) (2024)
- [Symmetry Discovery Beyond Affine Transformations](https://mlsanity.com/paper/2406.03619/) (2024)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)

## Built on by (most cited first, 2 of 2)

- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [Efficient Learning and Symmetry Discovery under Exact Invariances](https://mlsanity.com/paper/2609.07031/) (2026)

## Abstract (arXiv)

In this study, we introduce a method for learning group (known or unknown) equivariant functions by learning the associated quadratic form xT A x corresponding to the group from the data. Certain groups, known as orthogonal groups, preserve a specific quadratic form, and we leverage this property to uncover the underlying symmetry group under the assumption that it is orthogonal. By utilizing the corresponding unique symmetric matrix and its inherent diagonal form, we incorporate suitable inductive biases into the neural network architecture, leading to models that are both simplified and efficient. Our approach results in an invariant model that preserves norms, while the equivariant model is represented as a product of a norm-invariant model and a scale-invariant model, where the ``product'' refers to the group action. Moreover, we extend our framework to a more general setting where the function acts on tuples of input vectors via a diagonal (or product) group action. In this extension, the equivariant function is decomposed into an angular component extracted solely from the normalized first vector and a scale-invariant component that depends on the full Gram matrix of the tuple. This decomposition captures the inter-dependencies between multiple inputs while preserving the underlying group symmetry. We assess the effectiveness of our framework across multiple tasks, including polynomial regression, top quark tagging, and moment of inertia matrix prediction. Comparative analysis with baseline methods demonstrates that our model consistently excels in both discovering the underlying symmetry and efficiently learning the corresponding equivariant function.