Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

TL;DR

Across structured tasks, including the double pendulum and top quark tagging, it is demonstrated that spectral sparsity reliably reveals one-parameter symmetries, position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.

Abstract

Continuous symmetries are fundamental to many scientific and learning problems, yet they are often unknown a priori. Existing symmetry discovery approaches typically search directly in the space of transformation generators or rely on learned augmentation schemes. We propose a fundamentally different perspective based on spectral structure. We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT). Our central observation is that invariance to a subgroup induces structured sparsity in the spectral decomposition of a function across irreducible representations. Instead of optimizing over generators, we detect symmetries by identifying this induced sparsity pattern in the spectral domain. We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis through their irreducible representations. Across structured tasks, including the double pendulum and top quark tagging, we demonstrate that spectral sparsity reliably reveals one-parameter symmetries. These results position spectral analysis as a principled and interpretable alternative to generator-based symmetry discovery.

Figures (6)

• Figure 1: Symmetry discovery and learning framework. The input is first aligned via a learnable orthogonal transformation $Q\in SO(n)$ and decomposed into two-dimensional blocks, which are converted to polar coordinates to obtain radii and torus angles. Primitive torus Fourier features $U(x)$, together with radial features $R(x)$, are fed to the predictor $\phi_w$. Training minimizes the prediction loss along with a spectral resonance regularizer that promotes support only on frequency directions satisfying $\langle m,\lambda\rangle = 0$, thereby enabling recovery of the latent one-parameter symmetry generator.
• Figure 2: Recovered vs. True Rotational Generators. Comparison of the learned (left) and ground-truth (right) generators for (a) the Double Pendulum system (diagonal $\Delta(SO(2))$ generator) and (b) the Top Tagging task. Both learned generators demonstrate near-perfect alignment with the physical ground truth (cosine similarity $= 0.9999$).
• Figure 3: Robustness to noise (noise sweep). Performance comparison between Spectral Discovery and Augerino on the 6D double pendulum task across increasing additive noise levels. Left: Mean cosine similarity between the learned and ground-truth symmetry generators. Right: Mean test loss. Spectral Discovery maintains near-perfect generator recovery under low and moderate noise and degrades gracefully at high noise, while Augerino exhibits lower alignment and higher variability.
• Figure 4: Sample efficiency (sample sweep). Performance comparison between Spectral Discovery and Augerino as the number of training samples increases. Left: Mean cosine similarity between the learned and ground-truth symmetry generators. Right: Mean test mean-squared error. Spectral Discovery achieves near-perfect symmetry recovery with fewer samples and consistently lower predictive error, while Augerino improves more gradually with data.
• Figure 5: LieGAN comparison: sample sweep (mean cosine similarity). Mean cosine similarity between the recovered and ground-truth symmetry generators on the 6D double pendulum task as the number of training samples increases. Spectral Discovery achieves higher generator alignment across all sample sizes, reaching near-perfect recovery at larger sample regimes.

Theorems & Definitions (6)

• Proposition 5.0: Spectral characterization of left-invariance along $\exp(tB)$
• Corollary 5.0: Resonance condition on the maximal torus
• Proposition B.0: Spectral characterization of left-invariance along $\exp(tB)$
• proof
• Corollary B.0: Resonance condition on the maximal torus
• proof