Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

Pavan Karjol; Kumar Shubham; Prathosh AP

cs.LGMar 7, 2026

Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

Authors: Pavan Karjol, Kumar Shubham, Prathosh AP

Organizations: Department of Electrical Communication Engineering , Indian Institute of Science , Bengaluru, Karnataka

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 & tables

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.

Method	Joint Framework	Interpretable Predictor	Scope	Baseline
Ours: Spectral Discovery	Yes	Yes	$1$ D Lie groups	–
Augerino	Yes	No	Generic Lie groups	✓
LieGAN	No (discovery only)	No	Generic Lie groups	–
LieGG	No (post-hoc discovery)	No	Generic Lie groups	–
InfGen	No (post-hoc discovery)	No	Generic Lie groups	–
BeyondAffine	No (post-hoc discovery)	No	$1$ D Lie groups (incl. non-affine)	–

Table 1: Comparison of symmetry discovery frameworks. We compare Spectral Discovery (ours) with Augerino Benton et al. [2020] , LieGAN Yang et al. [2023b] , LieGG Moskalev et al. [2022] , InfGen Ko et al. [2024] , BeyondAffine Shaw et al. [2024] , and Bispectral Neural Networks (BNNs) Sanborn et al. [2022] . Methods are evaluated along four axes: (i) whether symmetry discovery and prediction are learned in a joint training framework , (ii) whether the resulting predictor is explicitly invariant in an interpretable manner, (iii) the class of symmetry groups supported, and (iv) whether the method is used as a baseline in this work.

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 $=0.9999$ ).

Method	Test MSE $\downarrow$	Inv. Error $\downarrow$	Cosine Similarity $\uparrow$
Spectral Discovery	0.00298 $\pm$ 0.00024	0.00070 $\pm$ 0.00027	0.9999 $\pm$ 0.00001
Augerino	$0.01053\pm 0.00040$	$0.00232\pm 0.00025$	$0.9233\pm 0.0708$

Table 2: Double Pendulum (6D), 32K samples. Comparison of predictive accuracy and symmetry recovery. Results are reported as mean

\pm

standard deviation. Lower is better for Test MSE and Invariance Error; higher is better for Cosine Similarity.

Method	Accuracy (%) $\uparrow$	Cosine Similarity $\uparrow$
Spectral Discovery	84.87 $\pm$ 0.39	0.9999 $\pm$ 0.00002
Augerino	$74.9\pm 2.4$	$0.994\pm 0.003$

Table 3: Top Quark Tagging (Elliptic Regime). Classification accuracy and generator recovery. Results are reported as mean

\pm

standard deviation. Higher is better for both metrics.

Symbol	Meaning
${\mathcal{D}}=\{(x_{i},y_{i})\}_{i=1}^{T}$	Training dataset of $T$ input–target pairs.
$T$	Number of training samples.
$x_{i}\in\mathbb{R}^{n}$	Input feature vector for sample $i$ .
$y_{i}\in\mathbb{R}^{m}$	Target/output vector for sample $i$ .
$n,m$	Input and output dimensions, respectively.
${\mathcal{X}}\subset\mathbb{R}^{n}$	Input domain (assumed closed under the action of $SO(n)$ ).

Table 4: Notation used in the method.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Configuration
Double Pendulum samples	32,000
Top Tagging samples	64,000
Epochs	40
Optimizer	Adam
Learning rate	$2\times 10^{-3}$
Predictor $\phi_{w}$	MLP (3 hidden layers, ReLU)

Appendix

Table 5: Summary of experimental settings used across all tasks.

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.

Figure 6: LieGAN comparison: noise sweep (mean cosine similarity). Mean cosine similarity between the recovered and ground-truth symmetry generators on the 6D double pendulum task as additive noise increases. Spectral Discovery maintains near-perfect alignment for low and moderate noise and degrades gracefully at high noise, while LieGAN exhibits lower alignment throughout.

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Abstract

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