Research map / Probabilistic and scientific modeling / Molecules and proteins
Symmetry Discovery Beyond Affine Transformations
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes a vector-field-based framework that estimates continuous (including non-affine) symmetries of data via polynomial level-set and vector-field estimation, plus a discrete-symmetry variant, and compares against LieGAN.
Problem
Continuous symmetry detection in data has been limited to affine transformation groups (e.g., LieGAN), and existing methods are computationally expensive and struggle with small samples.
Approach
Estimate vector fields (infinitesimal generators) that annihilate learned machine learning functions (probability densities, regression targets, or level-set functions) via constrained polynomial regression with an extended feature matrix, then construct invariant features from those vector fields; this avoids the discretization of group elements used by LieGAN and extends to non-affine symmetries
Key results
- On Gaussian data with N=200, the method reaches similarity 1.0000 vs LieGAN 0.7636 ± 0.2862; at N=20000 it takes 5.6 s vs LieGAN 195.6 s.
- On non-affine data f=x^3-y^2, the method achieves similarity 0.9983 ± 0.0006 while LieGAN achieves 0.0340 ± 0.0142.
- On discrete rotation with 20,000 points, the method recovers theta ≈ 0.8972 (≈2π/7) with similarity 1.0000 in about 48 s.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed vector-field method instead of LieGAN | Gaussian affine symmetry detection with 200 samples · Gaussian affine symmetry, N=200 | similarity: $0.7636 \pm 0.2862$ -> $1.0000 \pm 5.6 \cdot 10^{-7}$ | Helped |
| Use the proposed vector-field method instead of LieGAN | Gaussian affine symmetry detection · Gaussian affine symmetry, N=200 | time (s): $2.1717 \pm 0.0481$ vs $2.1778 \pm 0.0125$ | No clear effect |
| Use the proposed vector-field method instead of LieGAN | Gaussian affine symmetry detection · Gaussian affine symmetry, N=2000 | time (s): $18.1289 \pm 0.2528$ -> $2.6412 \pm 0.0208$ | Helped |
| Use the proposed vector-field method instead of LieGAN | Gaussian affine symmetry detection · Gaussian affine symmetry, N=20000 | similarity: $1.0000 \pm 1.5\cdot 10^{-5}$ vs $1.0000 \pm 3.2\cdot 10^{-8}$ | No clear effect |
| Use the proposed vector-field method instead of LieGAN | Gaussian affine symmetry detection · Gaussian affine symmetry, N=20000 | time (s): $195.5748 \pm 2.6080$ -> $5.6177 \pm 0.0585$ | Helped |
| Use the proposed vector-field method instead of LieGAN | Non-affine symmetry detection · Non-affine symmetry, f=x^3-y^2 | similarity: $0.0340 \pm 0.0142$ -> $0.9983 \pm 0.0006$ | Helped |
| Reduce the sample count from 20,000 to 1000 | Our method for discrete rotation detection · Discrete rotation, k=7 | similarity: 1.0000 -> 0.9979 | No clear effect |
| Use the proposed method instead of LieGAN on reduced data | Discrete rotation detection · Discrete rotation, reduced data | similarity: 0.1956 vs 0.9979 | Helped |
| Approximate the non-polynomial ground-truth vector field with degree 2 polynomial coefficients | Our method · Non-polynomial symmetry, f=sin(x)-cos(y) | similarity: 0.62 | Mixed |
Surprising findings
- LieGAN similarity on the Gaussian task at N=200 is highly variable (0.7636 ± 0.2862), and LieGAN fails entirely on the non-affine polynomial case (0.034).
- On a non-polynomial trigonometric ground truth, the method's polynomial vector field model only reaches similarity 0.62.
Limitations
- Polynomial coefficient assumption limits expressiveness, and the number of terms grows quickly with dimension.
- Vector fields X and hX are indistinguishable in the search, causing unwanted freedom; elbow-curve selection of number of components is heuristic.
- Similarity score is naive (arithmetic mean of component-wise cosine similarities) and not defined for collections of vector fields with metrics.
Experimental setup
Datasets: synthetic 2D Gaussian data synthetic non-affine polynomial data Bear Lake Weather dataset synthetic trigonometric level-set data simulated discrete rotation data from prior work
Benchmarks: Gaussian affine symmetry detection (LieGAN comparison) non-affine symmetry detection Bear Lake Weather real data non-polynomial level set discrete rotation infinitesimal isometries
Metrics: similarity score (cosine-type between generators) computation time (s)
Baselines: LieGAN
Builds on: LieGAN McTorch Riemannian Adagrad
Closest papers by meaning
- Learning Infinitesimal Generators of Continuous Symmetries from Data (NeurIPS 2024 · similarity 0.90)
- Discovering Symmetry Groups with Flow Matching (ICML 2026 · similarity 0.90)
- A Generative Model of Symmetry Transformations (NeurIPS 2024 · similarity 0.88)
- AtlasD: Automatic Local Symmetry Discovery (ICML 2025 · similarity 0.87)
- Explicit Discovery of Nonlinear Symmetries from Dynamic Data (ICML 2025 · similarity 0.86)
- Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution (ICML 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.