# A Generative Model of Symmetry Transformations

arXiv: https://arxiv.org/abs/2403.01946 · HTML: https://mlsanity.com/paper/2403.01946/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 57 references that are in the map.
- First posted on arXiv 4 Mar 2024.

## Builds on (most cited first, 30 of 41)

- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [The Variational Fair Autoencoder](https://mlsanity.com/paper/1511.00830/) (2015)
- [DIVA: Domain Invariant Variational Autoencoders](https://mlsanity.com/paper/1905.10427/) (2019)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Multi-Level Variational Autoencoder: Learning Disentangled Representations from Grouped Observations](https://mlsanity.com/paper/1705.08841/) (2017)
- [Automatic Symmetry Discovery with Lie Algebra Convolutional Network](https://mlsanity.com/paper/2109.07103/) (2021)
- [The Effects of Regularization and Data Augmentation are Class Dependent](https://mlsanity.com/paper/2204.03632/) (2022)
- [Deforming Autoencoders: Unsupervised Disentangling of Shape and Appearance](https://mlsanity.com/paper/1806.06503/) (2018)
- [Importance Weighting and Variational Inference](https://mlsanity.com/paper/1808.09034/) (2018)
- [Lossy Compression for Lossless Prediction](https://mlsanity.com/paper/2106.10800/) (2021)
- [Learning Invariances using the Marginal Likelihood](https://mlsanity.com/paper/1808.05563/) (2018)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Invariance Learning in Deep Neural Networks with Differentiable Laplace Approximations](https://mlsanity.com/paper/2202.10638/) (2022)
- [Generative Adversarial Symmetry Discovery](https://mlsanity.com/paper/2302.00236/) (2023)
- [Galaxy Zoo DECaLS: Detailed Visual Morphology Measurements from Volunteers and Deep Learning for 314,000 Galaxies](https://mlsanity.com/paper/2102.08414/) (2021)
- [Equivariant Adaptation of Large Pretrained Models](https://mlsanity.com/paper/2310.01647/) (2023)
- [Unsupervised Learning of Group Invariant and Equivariant Representations](https://mlsanity.com/paper/2202.07559/) (2022)
- [Grounding inductive biases in natural images:invariance stems from variations in data](https://mlsanity.com/paper/2106.05121/) (2021)
- [Learning Probabilistic Symmetrization for Architecture Agnostic Equivariance](https://mlsanity.com/paper/2306.02866/) (2023)
- [Last Layer Marginal Likelihood for Invariance Learning](https://mlsanity.com/paper/2106.07512/) (2021)
- [Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitions](https://mlsanity.com/paper/2207.12067/) (2022)
- [Self-Supervised Disentanglement by Leveraging Structure in Data Augmentations](https://mlsanity.com/paper/2311.08815/) (2023)
- [Learning Invariant Weights in Neural Networks](https://mlsanity.com/paper/2202.12439/) (2022)
- [Group Equivariant Subsampling](https://mlsanity.com/paper/2106.05886/) (2021)
- [Learning Instance-Specific Augmentations by Capturing Local Invariances](https://mlsanity.com/paper/2206.00051/) (2022)

## Built on by (most cited first, 10 of 10)

- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [Test-Time Canonicalization by Foundation Models for Robust Perception](https://mlsanity.com/paper/2507.10375/) (2025)
- [Symmetry-Aware Generative Modeling through Learned Canonicalization](https://mlsanity.com/paper/2501.07773/) (2025)
- [Symmetry Understanding of 3D Shapes via Chirality Disentanglement](https://mlsanity.com/paper/2508.05505/) (2025)
- [Discovering Symmetry Groups with Flow Matching](https://mlsanity.com/paper/2512.20043/) (2025)
- [CP2: Leveraging Geometry for Conformal Prediction via Canonicalization](https://mlsanity.com/paper/2506.16189/) (2025)
- [Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions](https://mlsanity.com/paper/2510.21706/) (2025)
- [Inverting Data Transformations via Diffusion Sampling](https://mlsanity.com/paper/2602.08267/) (2026)
- [RECON: Robust symmetry discovery via Explicit Canonical Orientation Normalization](https://mlsanity.com/paper/2505.13289/) (2025)
- [Symmetry Matters: Auditing and Symmetrizing 3D Generative Models](https://mlsanity.com/paper/2512.18953/) (2025)

## Abstract (arXiv)

Correctly capturing the symmetry transformations of data can lead to efficient models with strong generalization capabilities, though methods incorporating symmetries often require prior knowledge. While recent advancements have been made in learning those symmetries directly from the dataset, most of this work has focused on the discriminative setting. In this paper, we take inspiration from group theoretic ideas to construct a generative model that explicitly aims to capture the data's approximate symmetries. This results in a model that, given a prespecified but broad set of possible symmetries, learns to what extent, if at all, those symmetries are actually present. Our model can be seen as a generative process for data augmentation. We provide a simple algorithm for learning our generative model and empirically demonstrate its ability to capture symmetries under affine and color transformations, in an interpretable way. Combining our symmetry model with standard generative models results in higher marginal test-log-likelihoods and improved data efficiency.