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A Generative Model of Symmetry Transformations
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Proposes a Symmetry-aware Generative Model (SGM) that factors data into an invariant prototype and a learned distribution over group transformation parameters, learned via self-supervised invariance plus maximum likelihood, and uses it for data augmentation in VAEs.
Problem
Generative models ignore the approximate symmetries present in data, and symmetry learning has mostly targeted discriminative models where labels are assumed invariant; generative models must capture symmetry factors of variation, yet prior symmetry knowledge is often required.
Approach
Decompose each observation as a transformation of an invariant prototype, x = T_eta(prototype), where a self-supervised transformation inference network maps observations to prototypes and a normalizing flow models p(eta | prototype); the prototype density is never modeled, and the flow learns which transformations from a prespecified broad group are actually present.
Key results
- On rotated MNIST, InvVAE obtains roughly 500 nats higher test IWLB than the standard VAE and is nearly robust to added rotation.
- AugVAE performance is unaffected when training observations are reduced by three quarters, whereas the standard VAE degrades.
- On GalaxyMNIST with affine and color transformations, AugVAE outperforms the standard VAE and stays robust when trained on half the dataset.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| incorporate learned symmetries with AugVAE data augmentation | VAE · Rotated MNIST | importance-weighted lower bound | Helped |
| use InvVAE to represent examples by their inferred prototypes | VAE · Rotated MNIST | likelihood: 500 nat larger | Helped |
| use InvVAE prototype representations | VAE · Rotated MNIST | performance under rotation | Helped |
| train InvVAE on half the training data | InvVAE · Rotated MNIST | metrics: <10% | No clear effect |
| use standard data augmentation instead of learned, prototype-conditioned augmentation | VAE · Rotated MNIST | importance-weighted lower bound | Mixed |
| incorporate learned symmetries with AugVAE data augmentation | VAE · GalaxyMNIST | importance-weighted lower bound | Helped |
| train AugVAE on half the dataset | AugVAE · GalaxyMNIST | importance-weighted lower bound | No clear effect |
| use the x-space SSL objective instead of the h-space objective (ablation) | Rotation inference net · Fully rotated MNIST | x-mse: 0.3567 -> 0.2387 | Helped |
| use the x-space SSL objective instead of the h-space objective (ablation) | Rotation inference net · Fully rotated MNIST | eta-mse: 0.4736 -> 0.9715 | Hurt |
| average the x-space and h-space SSL objectives instead of using h-space alone (ablation) | Rotation inference net · Fully rotated MNIST | x-mse: 0.3567 -> 0.3129 | Hurt |
| average the SSL loss over 5 samples instead of 1 (ablation) | Rotation inference net · Fully rotated MNIST | x-mse: 0.0981 -> 0.0840 | Helped |
| use the symmetric SSL loss (ablation) | SGM inference network · MNIST | training performance | Helped |
Surprising findings
- Standard VAE with fixed random data augmentation becomes substantially worse than no augmentation when more data is available, because augmentations are independent of the sample.
- An invariant VAE encoder built by averaging over transformations collapsed to circles/rings prototypes and failed, motivating the SSL approach.
Limitations
- Requires specifying a superset of possible symmetries in advance.
- Approximate invertibility and information loss from discrete image grids and out-of-bounds transformations.
- Inference network is only approximately invariant; prototypes drift by about 5% of max transformation magnitude across iterations.
Experimental setup
Datasets: MNIST dSprites GalaxyMNIST PatchCamelyon
Benchmarks: MNIST under affine transformations MNIST under color transformations dSprites GalaxyMNIST
Metrics: importance-weighted lower bound (test log-likelihood) reconstruction mse prototype mse
Baselines: standard VAE VAE with standard data augmentation InvVAE (invariant-encoder VAE variant)
Builds on: BYOL BINCE Neural Spline Flow VAE importance-weighted bound
Closest papers by meaning
- Learning Infinitesimal Generators of Continuous Symmetries from Data (NeurIPS 2024 · similarity 0.92)
- Any-Subgroup Equivariant Networks via Symmetry Breaking (ICLR 2026 · similarity 0.88)
- Symmetry Discovery Beyond Affine Transformations (NeurIPS 2024 · similarity 0.88)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.88)
- SymDrift: One-Shot Generative Modeling under Symmetries (NeurIPS 2026 · similarity 0.86)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.85)
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