Paper deep dive
Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms
Pavan Karjol, Kumar Shubham, Prathosh AP
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 94%
Last extracted: 3/13/2026, 12:32:09 AM
Summary
The paper introduces a spectral framework for discovering continuous one-parameter subgroups of SO(n) by leveraging the Generalized Fourier Transform (GFT). It demonstrates that invariance to a subgroup induces structured sparsity in the spectral domain, specifically satisfying a resonance condition ⟨m,λ⟩=0. By aligning inputs to invariant planes and penalizing off-resonant Fourier modes, the method enables joint learning of target functions and latent symmetry generators, offering an interpretable alternative to generator-based optimization.
Entities (5)
Relation Signals (3)
Generalized Fourier Transform → enables → Symmetry Discovery
confidence 95% · We introduce a framework for discovering continuous one-parameter subgroups using the Generalized Fourier Transform (GFT).
Resonance Constraint → induces → Spectral Sparsity
confidence 95% · a regularization mechanism based on the resonance constraint (Corollary 5.2), which encourages functions consistent with subgroup invariance by penalizing off-resonant Fourier modes and thereby promoting sparsity
Maximal Torus → simplifies → Generalized Fourier Transform
confidence 90% · We develop symmetry detection procedures on maximal tori, where the GFT reduces to multi-dimensional Fourier analysis
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
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.
Tags
Links
- Source: https://arxiv.org/abs/2603.07299v1
- Canonical: https://arxiv.org/abs/2603.07299v1
Trouble viewing inline? Open PDF directly →
Full Text
61,739 characters extracted from source content.
Expand or collapse full text
Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms Pavan Karjol Department of Electrical Communication Engineering Indian Institute of Science Bengaluru, Karnataka Kumar Shubham Department of Electrical Communication Engineering Indian Institute of Science Bengaluru, Karnataka Prathosh AP 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. 1 Introduction Symmetry is a fundamental structural principle in machine learning. When a target function is invariant or equivariant under a group of transformations, incorporating this structure into the model can substantially improve generalization, reduce sample complexity, and enhance interpretability. Translation equivariance in convolutional neural networks Cohen2016, kondor2018generalization, cohen2019general and permutation invariance in Deep Sets Zaheer2017 are canonical examples in which explicit knowledge of the underlying symmetry group enables the design of parameter-efficient models with improved generalization performance. However, these approaches rely on explicit knowledge of the underlying symmetry group, an assumption that often does not hold in practical settings. In many scientific and real-world problems, the relevant symmetry is not known a priori. Such situations commonly arise in physical simulations, molecular modeling, and dynamical systems, where symmetry is implicit rather than directly observable. Automatically identifying the active symmetry from data, therefore, remains a central challenge. To make this challenge more tractable, one can focus on structured families of symmetries that admit analytic characterization. Among continuous symmetries, one-parameter subgroups occupy a particularly important role. They capture structured rotational or continuous transformation behaviors while remaining analytically tractable, and higher-dimensional subgroups can often be understood as compositions of these fundamental components. Discovering such one-parameter subgroups provides a natural entry point for understanding more complex transformation structure. In this work, we propose a spectral framework for discovering continuous symmetries based on the Generalized Fourier Transform (GFT). Our central observation is that invariance to a one-parameter subgroup of SO(n)SO(n) induces structured sparsity in the spectral decomposition of a function restricted to that subgroup. Rather than searching directly in the space of generators, we analyze how a learned function decomposes across irreducible representations in the frequency domain. Spectral concentration patterns then serve as signatures of invariance. To build intuition, consider a function on the plane that is invariant to all rotations. In angular Fourier coordinates, this full rotational invariance forces all nonzero angular frequencies to vanish, leaving spectral mass only at the zero mode. More generally, invariance to a one-parameter subgroup imposes a weaker constraint: rather than collapsing entirely to the zero frequency, the spectrum is restricted to a structured subset of admissible frequencies determined by the subgroup. Our approach detects symmetry by identifying this induced spectral sparsity pattern. We develop symmetry detection procedures on maximal tori, where the GFT reduces to a multi-dimensional Fourier transform with directly interpretable frequency structure. This setting allows us to translate representation-theoretic constraints into concrete spectral patterns that can be detected computationally. Why spectral rather than generator search? Recent approaches to symmetry discovery typically parameterize candidate generators and optimize over transformation actions in the input space. Methods such as Augerino benton2020learning learn augmentation distributions that promote invariance, while LieGAN yang2023liegan learns continuous transformation groups through adversarial objectives. These approaches explicitly model transformations and enforce invariance via data-level perturbations or learned group actions. In contrast, our approach detects symmetry through spectral structure. Instead of optimizing over candidate generators, we examine the frequency content of a learned function and identify invariance via structured sparsity across irreducible representations. This yields an interpretable and representation-theoretic criterion for symmetry discovery that avoids direct optimization over transformation parameters. To our knowledge, prior symmetry discovery methods have not explicitly leveraged representation-level sparsity patterns in the Generalized Fourier Transform domain as a primary mechanism for identifying continuous subgroups. We integrate this spectral reasoning into the learning pipeline in two complementary ways: (i) a model architecture that incorporates maximal-torus Fourier features, and (i) a regularization mechanism based on the resonance constraint (Corollary 5.2), which encourages functions consistent with subgroup invariance by penalizing off-resonant Fourier modes and thereby promoting sparsity in the spectral coefficients. Empirically, we demonstrate that spectral sparsity reliably reveals one-parameter subgroups and that incorporating this structure improves learning performance across multiple tasks. 1.1 Contributions Our main contributions are as follows: • A spectral framework for continuous symmetry discovery. We propose a Generalized Fourier Transform (GFT)-based methodology for identifying continuous one-parameter subgroups through structured spectral sparsity. On maximal tori, where the GFT reduces to multi-dimensional Fourier analysis, invariance manifests as interpretable concentration patterns across irreducible components. • Spectral architectures for symmetry-aware learning. We incorporate spectral reasoning directly into the model design through architectures built on maximal-torus Fourier features. These architectures expose interpretable frequency structure and enable symmetry discovery to emerge jointly with predictive learning. • Resonance-based regularization for invariant structure. We introduce a regularization mechanism derived from the resonance constraint (Corollary 5.2) that penalizes off-resonant Fourier modes and induces structured spectral sparsity. This provides a principled, representation-theoretic bias toward functions consistent with one-parameter subgroup invariance. • Comprehensive empirical evaluation. We validate the proposed framework across diverse synthetic and structured tasks, demonstrating that spectral sparsity reliably reveals latent one-parameter subgroups and improves generalization when incorporated into the learning process. 2 Related Work 2.1 Symmetries and Neural Network Symmetries and invariances arise naturally across physical, chemical, and biological systems, where they often determine fundamental structural and dynamical properties. Recent advances in areas such as drug discovery stark2022equibind, corso2022diffdock, nguyen2025equicpi and subatomic particle physics gross1996role, hwang2010symmetries, gutowski2007symmetry have demonstrated how explicitly leveraging symmetry can guide scientific reasoning and enable new discoveries. In deep learning, symmetry principles have become central to geometric deep learning bronstein2021geometric. For example, the success of convolutional neural networks (CNNs) in image processing lecun1998gradient, sanborn2022bispectral can largely be attributed to their built-in equivariance to two-dimensional translations sanborn2022bispectral. Building on this idea, recent works have incorporated broader group symmetries into neural architectures, including G-equivariant neural networks cohen2016group, steerable CNNs weiler2019general, weiler2018learning, and Lie group based constructions such as LieConv finzi2020generalizing and SO(3)SO(3)-equivariant models kondor2018clebsch. These approaches demonstrate that encoding symmetry can significantly enhance representation efficiency and generalization. However, most of the existing methods assume explicit knowledge of the underlying symmetry group. In many scientific and real-world problems, the symmetry structure is implicit and not known a priori yang2023latent, desai2022symmetry, limiting the applicability of such approaches. This gap motivates the development of methods that can discover and exploit latent symmetries directly from data. 2.2 Automatic symmetry discovery Automatic symmetry discovery seeks to infer latent group structure directly from data symmetries sanborn2022bispectral. Early approaches parameterize Lie algebra generators to model continuous transformations rao1998learning, chau2020disentangling, cohen2014learning, but the recovered structure is often implicit and difficult to interpret. sanborn2022bispectral employ spectral methods, though their framework is limited to discrete subgroups. Other methods, including LieGAN yang2023liegan and LieGG moskalev2022liegg, learn generators adversarially or extract them post hoc from trained networks, typically separating symmetry discovery from the predictive objective. Augerino benton2020learning enables joint learning of generators and predictors, yet the resulting invariance or equivariance of the model remains non-interpretable. In contrast, our approach unifies prediction and symmetry discovery within a single training framework while ensuring that the predictor’s invariance is explicitly characterized. By embedding spectral analysis directly into learning, we obtain a transparent description of the underlying continuous group, so that both the discovered symmetry and the model’s invariance structure are analytically interpretable and aligned with the predictive task. 3 Problem Formulation Figure 1: Symmetry discovery and learning framework. The input is first aligned via a learnable orthogonal transformation Q∈SO(n)Q∈ 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)U(x), together with radial features R(x)R(x), are fed to the predictor ϕw _w. Training minimizes the prediction loss along with a spectral resonance regularizer that promotes support only on frequency directions satisfying ⟨m,λ⟩=0 m,λ =0, thereby enabling recovery of the latent one-parameter symmetry generator. Let the dataset be =(xi,yi)i=1T,D=\(x_i,y_i)\_i=1^T, where each input xi∈ℝnx_i ^n, target yi∈ℝmy_i ^m and SO(n)SO(n) denote the special orthogonal group. Assume the data are generated by an underlying function :→ℝm, f:X ^m, so that yi=(xi)y_i= f(x_i) for all i, where the domain ⊂ℝnX⊂R^n is closed under the action of SO(n)SO(n). Further, suppose that f is invariant under an unknown one-parameter subgroup ℋ⊂SO(n)H⊂ SO(n), i.e., (h⋅x)=(x),∀h∈ℋ,x∈. f(h· x)= f(x), ∀\,h∈H,\;x∈X. Given access only to the training dataset D, the primary objectives of our work are: 1. To learn a predictor that accurately approximates the true target function f, 2. To identify the unknown one-parameter subgroup ℋ⊂SO(n)H⊂ SO(n) under which f is invariant. To formalize the second objective, we briefly review the structure of one-parameter subgroups of SO(n)SO(n). 4 Preliminaries 4.1 One-Parameter Subgroups of SO(n)SO(n) A one-parameter subgroup of SO(n)SO(n) is a continuous family of rotations ℋ=exp(tB):t∈ℝ,H=\ (tB):t \, where B∈(n)B∈ so(n). The Lie algebra of SO(n)SO(n) is (n)=B∈ℝn×n∣B⊤=−B, so(n)=\B ^n× n B =-B\, the space of skew-symmetric matrices. Every B∈(n)B∈ so(n) generates a one-parameter subgroup via the matrix exponential, and conversely every one-parameter subgroup of SO(n)SO(n) arises in this way. Hence, learning a one-parameter subgroup of SO(n)SO(n) is equivalent to identifying its skew-symmetric generator. 4.2 Canonical Structure of Skew-Symmetric Generators Every B∈(n)B∈ so(n) admits a real orthogonal block decomposition. There exists Q∈SO(n)Q∈ SO(n) such that Q⊤BQ=⨁k=1⌊n/2⌋λkJ⊕ 0nmod2,J=[0−110],Q BQ= _k=1 n/2 _kJ\; \;0_\,n 2, J= bmatrix0&-1\\ 1& -0 bmatrix, (1) where λk∈ℝ _k . Thus, any skew-symmetric generator decomposes into independent planar rotation generators acting on mutually orthogonal two-dimensional invariant subspaces. If n is even, the decomposition consists entirely of 2×22× 2 rotation blocks; if n is odd, an additional one-dimensional trivial block appears. For simplicity, we assume n is even. Under this decomposition, exp(tB)=Q[⨁k=1n/2R(λkt)]Q⊤, (tB)=Q [ _k=1^n/2R( _kt) ]Q , where R(θ)R(θ) denotes a planar rotation matrix. Key Idea. Learning B⟺B\; \; (i) learning an orthogonal matrix Q that defines the invariant planes, and (i) learning rotation rates λ1,…,λn/2 _1,…, _n/2 governing the dynamics within each plane. 5 Methodology Building on the canonical structure described in the preliminaries, our objective is to jointly learn the target function f and the latent one-parameter subgroup generated by B. Rather than optimizing directly over arbitrary skew-symmetric matrices, we parameterize the generator through its real canonical decomposition B=Q(⨁k=1n/2λkJ)Q⊤,B=Q ( _k=1^n/2 _kJ )Q , where the orthogonal matrix Q determines the orientation of invariant two-dimensional planes and the parameters λk\ _k\ specify the angular velocities within each plane. During training, Q and λk\ _k\ are learned end-to-end together with the predictor. Given an input x, we first transform it via z=Q⊤xz=Q x, aligning the coordinates with the invariant planes so that the subgroup action decomposes into independent planar rotations. Our key observation is that invariance under the subgroup generated by B induces specific structure in the generalized Fourier transform (GFT) of the function expressed in this aligned domain. In particular, symmetry leads to spectral sparsity and resonance conditions among admissible frequencies determined by λk\ _k\. Motivated by this structure, we construct Fourier features in the transformed domain and feed them into a function approximator ϕφ such as neural network. To encourage recovery of the underlying symmetry, we introduce regularization that promotes sparsity and resonance-consistent frequency components. As training progresses, dominant spectral modes emerge whose structure characterizes the latent generator, thereby enabling simultaneous discovery of the symmetry parameters and efficient learning of the target function. Please refer to Figure 1 for further details about the architecture. Key Idea. Continuous symmetries manifest as structured sparsity and resonance constraints in an appropriate Fourier domain. By aligning inputs with invariant planes through a learnable transformation, constructing Fourier features in this aligned basis, and enforcing sparsity and resonance-consistent structure in the spectral coefficients, we recover the latent generator from data. The dominant frequency components encode the underlying symmetry, while the resulting symmetry-adapted representation facilitates more stable and efficient function learning. 5.1 Spectral Characterization of Invariance To analyze the symmetry behavior of any function f under different group action we first lift the function to the group G<SO(n)G<SO(n) such that for any fixed probe data x0∈ℝnx_0 ^n F(g):=f(g⋅x0)F(g):=f(g· x_0) Intuitively, for any given group transformation, the function F captures the change in the target function for any given probe vector (x0x_0). If f is invariant under one-parameter subgroup ℋ<SO(n)H<SO(n), then F(g) is also invariant under same group action i.e., F(h⋅g)=F(exp(tB)⋅g)=F(g),∀h∈ℋ,g∈GF(h· g)=F( (tB)· g)=F(g), ∀ h∈H,g∈ G Therefore, symmetry of f translates into left-invariance of F along the subgroup direction. This invariance imposes strong structural constraints on the spectral decomposition of F, which we now analyze using the Group Fourier Transform. 5.1.1 Group Fourier Transform Let G be a compact Lie group equipped with normalized Haar measure dgdg. For an irreducible unitary representation ρπ:G→ℂdπ×dπ _π:G ^d_π× d_π, the Group Fourier Transform of F∈L2(G)F∈ L^2(G) at π is defined as F^(π)=∫GF(g)ρπ(g)∗g, F(π)= _GF(g)\, _π(g) \,dg, (2) where F^(π)∈ℂdπ×dπ F(π) ^d_π× d_π denotes the matrix of Fourier coefficients associated with the representation π. The collection F^(π)π∈G^\ F(π)\_π∈ G provides a complete spectral description of F. Crucially, group invariance properties of F translate into algebraic constraints on these matrix-valued Fourier coefficients. In particular, left-invariance along a one-parameter subgroup generated by B imposes infinitesimal conditions involving the derived representation dρπ(B)d _π(B). The following proposition makes this relationship precise and forms the basis of our spectral characterization. Proposition 5.1 (Spectral characterization of left-invariance along exp(tB) (tB)). Let G be a compact Lie group with normalized Haar measure dgdg and Lie algebra g. Fix B∈B∈ g and define left-translation (LtF)(g):=F(exp(tB)g)(L_tF)(g):=F( (tB)g). For F∈L2(G)F∈ L^2(G) and an irreducible unitary representation ρπ:G→ℂdπ×dπ _π:G→C^d_π× d_π, define F^(π):=∫GF(g)ρπ(g)∗dg∈ℂdπ×dπ. F(π):= _GF(g)\, _π(g) \,dg ∈C^d_π× d_π. If F is left-invariant along exp(tB) (tB), i.e. LtF=FL_tF=F for all t∈ℝt , then for every π∈G^π∈ G, F^(π)dρπ(B)∗=0. F(π)\,d _π(B) =0. (3) In particular, if dρπ(B)d _π(B) is invertible then F^(π)=0 F(π)=0. Interpretation. equation 3 forces F^(π) F(π) to lie in the nullspace of dρπ(B)∗d _π(B) . 5.1.2 GFT on the Maximal Torus While Proposition 5.1 holds for a general compact Lie group G, in our analysis we specialize to the maximal torus of SO(n)SO(n). The maximal torus is a compact, abelian subgroup consisting of simultaneous planar rotations in mutually orthogonal invariant planes. This choice is particularly convenient because its irreducible representations are one-dimensional characters, leading to a simple and computationally efficient Fourier analysis. Importantly, any one-parameter subgroup of SO(n)SO(n), after alignment to its invariant planes via an appropriate orientation matrix Q, lies inside a maximal torus. Thus, performing the GFT on the maximal torus captures the spectral structure induced by any one-parameter subgroup, while significantly simplifying both the representation theory and the resulting computational framework. We now specialize the general spectral constraint in Proposition 5.1 to a choice of G that yields a particularly simple and computationally efficient Fourier analysis. Concretely, we take G to be a maximal torus of SO(n)SO(n) (with n even for simplicity). After aligning coordinates to the invariant planes of the generator via an orientation matrix Q, the corresponding one-parameter subgroup exp(tB)\ (tB)\ acts as independent rotations within each 22-dimensional plane and therefore lies inside a maximal torus. Since maximal tori are compact and abelian, all irreducible representations are one-dimensional characters, so the GFT reduces to a standard Fourier series on n/2T^n/2. This retains the spectral signature of any one-parameter subgroup (up to the learned orientation) while greatly simplifying the representation-theoretic machinery. Torus parametrization and characters. Let r:=n/2r:=n/2. A maximal torus of SO(n)SO(n) can be identified (after a suitable choice of basis) with r≅(S1)r,θ=(θ1,…,θr)∈[0,2π)r,T^r (S^1)^r, θ=( _1,…, _r)∈[0,2π)^r, where θk _k is the rotation angle in the kkth invariant plane. The irreducible unitary representations of rT^r are the characters indexed by m∈ℤrm ^r (ρπ=χm _π= _m): χm(θ):=ei⟨m,θ⟩=exp(i∑k=1rmkθk). _m(θ):=e^i m,θ = \! (i _k=1^rm_k _k ). Accordingly, the GFT coefficients of F become scalars F^(m)∈ℂ F(m) . Corollary 5.2 (Resonance condition on the maximal torus). Assume n is even and write the generator in canonical form B=Q(⨁k=1rλkJ)Q⊤,r=n/2,B=Q ( _k=1^r _kJ )Q , r=n/2, with λ=(λ1,…,λr)∈ℝrλ=( _1,…, _r) ^r. Consider the maximal torus rT^r in the aligned coordinates and let F^(m) F(m) denote the scalar Fourier coefficient of F at character χm _m, m∈ℤrm ^r. If F is left-invariant along exp(tB) (tB), then F^(m)⟨m,λ⟩=0,∀m∈ℤr. F(m)\, m,λ =0, ∀ m ^r. (4) In particular, if ⟨m,λ⟩≠0 m,λ ≠ 0 then F^(m)=0 F(m)=0. Equivalently, nonzero spectral mass can occur only on the resonant set ℛ(λ)=m∈ℤr:⟨m,λ⟩=0.R(λ)=\m ^r: m,λ =0\. (5) Key Insight. Left-invariance along exp(tB) (tB) forces the Fourier spectrum to concentrate only on frequencies m satisfying the linear resonance condition ⟨m,λ⟩=0 m,λ =0. Thus, symmetry reduces the effective spectral support to a lower-dimensional subset, and the surviving dominant frequencies directly encode the generator parameters λ (and, via alignment, the invariant planes). Table 1: Comparison of symmetry discovery frameworks. We compare Spectral Discovery (ours) with Augerino benton2020learning, LieGAN yang2023liegan, LieGG moskalev2022liegg, InfGen ko2024learning, BeyondAffine shaw2024symmetry, and Bispectral Neural Networks (BNNs) sanborn2022bispectral. Methods are evaluated along four axes: (i) whether symmetry discovery and prediction are learned in a joint training framework, (i) whether the resulting predictor is explicitly invariant in an interpretable manner, (i) the class of symmetry groups supported, and (iv) whether the method is used as a baseline in this work. Method Joint Framework Interpretable Predictor Scope Baseline Ours: Spectral Discovery Yes Yes 11D 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 11D Lie groups (incl. non-affine) – BNNs Yes Yes Discrete groups – 5.2 Symmetry Discovery and Learning Framework The resonance condition in Corollary 5.2 provides the central principle behind our symmetry discovery framework. It shows that left-invariance along exp(tB) (tB) restricts the Fourier spectrum to frequencies satisfying the linear constraint ⟨m,λ⟩=0 m,λ =0. However, this characterization holds only when the generator is expressed in its canonical block-diagonal form, i.e., when the coordinate system is aligned with the invariant rotational planes. In general, these planes are unknown, and therefore the resonance condition alone reveals information about λ only after an appropriate alignment. Learning invariant planes via orthogonal alignment. To uncover the latent symmetry structure, we introduce a learnable orthogonal transformation Q∈ℝn×nQ ^n× n that aligns the input space with the invariant planes of the generator. Since B∈(n)B∈ so(n) admits a block-diagonal decomposition into 2×22× 2 rotational blocks, learning Q allows us to represent the generator in this canonical form. We therefore transform the input features as Z(x):=Qx∈ℝn,Z(x):=Qx ^n, (6) where n is assumed even for simplicity. The transformed representation Z(x)Z(x) is decomposed into n/2n/2 two-dimensional blocks: Z(x)=⨁k=1n/2zk,zk=(z2k−1,z2k)⊤∈ℝ2,Z(x)= _k=1^n/2z_k, z_k=(z_2k-1,z_2k) ^2, (7) where ⨁ denotes concatenation of blocks. Each block zkz_k corresponds to a candidate invariant rotational plane. By learning the orthogonal matrix Q jointly with the model parameters, we effectively search for a coordinate system in which the latent one-parameter subgroup acts as independent planar rotations. In this aligned domain, the resonance condition derived earlier becomes directly applicable for identifying the generator parameters. Torus Fourier Features. Each two-dimensional block zk=(z2k−1,z2k)⊤z_k=(z_2k-1,z_2k) can be equivalently represented as a complex number uk:=z2k−1+iz2k∈ℂ.u_k:=z_2k-1+iz_2k . Under the aligned subgroup action, each block undergoes planar rotation, uk↦eiλktuk,u_k\; \;e^i _ktu_k, so that the action becomes diagonal in the complex representation. Writing uk=rkeiθk,rk=|uk|,u_k=r_ke^i _k,r_k=|u_k|, we obtain angular variables θ=(θ1,…,θn/2)θ=( _1,…, _n/2) that parametrize a maximal torus. Motivated by the GFT analysis on this torus, we construct Fourier features indexed by integer frequency vectors m=(m1,…,mn/2)∈ℤn/2m=(m_1,…,m_n/2) ^n/2: Z^m(x)=exp(i⟨m,θ(x)⟩)=exp(i∑k=1n/2mkθk(x)). Z_m(x)= (i m,θ(x) )= (i _k=1^n/2m_k _k(x) ). These correspond exactly to the characters of the maximal torus and form the spectral basis in which the resonance condition ⟨m,λ⟩=0 m,λ =0 arises naturally. Bandwidth Truncation and Fundamental Frequencies. In practice, we restrict attention to a finite bandwidth parameter ℬB such that mk∈−ℬ,…,ℬ,m_k∈\-B,…,B\, which yields (2ℬ+1)n/2(2B+1)^n/2 candidate frequency components. However, many of these frequencies lie along the same lattice direction and therefore encode redundant symmetry information. Primitive Directions and Frequency Rays. The index set ℤn/2Z^n/2 forms an integer lattice. Each nonzero index m determines a one-dimensional frequency direction, since all integer multiples of m correspond to harmonics of the same angular combination. To isolate the fundamental direction associated with m, we define its primitive representative: prim(m)=mgcd(|m1|,…,|mn/2|).prim(m)= m (|m_1|,…,|m_n/2|). (8) By construction, prim(m)prim(m) has coprime entries and is the minimal integer vector in the same direction as m. We therefore retain only the set of primitive frequency directions: ℳ=prim(m):mk∈−ℬ,…,ℬ,m≠0.M= \prim(m)\;:\;m_k∈\-B,…,B\,\;m≠ 0 \. (9) Model Architecture and Objective. Let U(x)=Z^m(x)m∈ℳU(x)=\ Z_m(x)\_m denote the collection of primitive Fourier features and let R(x)=(r1,…,rn/2)R(x)=(r_1,…,r_n/2) denote the radii of the aligned 2D blocks. We concatenate these features and feed them to a function approximator ϕw:(U(x),R(x))↦y^, _w:(U(x),R(x)) y, parameterized by weights w. In addition to learning w and the alignment matrix Q, we learn the generator parameters λ by enforcing the resonance condition derived in Corollary 5.2. Recall that symmetry requires nonzero Fourier coefficients only when ⟨m,λ⟩=0 m,λ =0. To promote this structure, we penalize violations of resonance weighted by the corresponding feature coefficients. Let CmC_m denote the first-layer weight in ϕw _w associated with feature Z^m Z_m. The overall objective is minw,λ,Qℒ(y,ϕw(U(x),R(x)))+μ∑m∈ℳ(Cm⟨m,λ⟩)2, _w,\,λ,\,Q\;L\! (y, _w(U(x),R(x)) )\;+\;μ _m (C_m\, m,λ )^2, (10) where μ>0μ>0 controls the strength of the resonance regularization. The first term ensures predictive accuracy, while the second term encourages spectral support to concentrate on resonant frequency directions. After training, the learned parameters (Q,λ)(Q,λ) determine the estimated Lie algebra generator B=Q(⨁k=1n/2λkJ)Q⊤,B=Q ( _k=1^n/2 _kJ )Q , and ϕw _w provides the final prediction. 6 Experiments Objective. We evaluate the proposed spectral discovery framework for jointly learning a predictive mapping and recovering the latent one-parameter continuous symmetry underlying the data. Our emphasis is on accurately identifying both the symmetry orientation and the subgroup direction while maintaining strong predictive performance. Benchmarks. We consider two representative tasks spanning synthetic physical dynamics and real-world data. First, we evaluate on a 6D double pendulum system with spring coupling, where the target mapping is invariant under a latent one-parameter rotation acting simultaneously across multiple 2D planes. This task assesses recovery of a shared subgroup direction in the presence of nuisance coordinates. Second, we consider the Top Quark Tagging classification task using jet constituent four-momenta. The underlying physical laws admit Lorentz symmetries, and we focus on discovering a rotational one-parameter subgroup while learning the classification mapping. Ablations on noise robustness (Figure 3) and sample efficiency (Figure 4) are provided in Appendix D. Baseline. We compare against Augerino [benton2020learning], which jointly learns a predictive model and a distribution over continuous transformations to encourage invariance during training (see Table 1 for further details). This provides the closest comparison in a joint supervised learning and symmetry discovery setting. Complete training details are provided in Appendix C.1. Comparison with additional baselines is provided in Appendix D.1. Figure 2: Recovered vs. True Rotational Generators. Comparison of the learned (left) and ground-truth (right) generators for (a) the Double Pendulum system (diagonal Δ(SO(2)) (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=0.9999). Table 2: Double Pendulum (6D), 32K samples. Comparison of predictive accuracy and symmetry recovery. Results are reported as mean ± standard deviation. Lower is better for Test MSE and Invariance Error; higher is better for Cosine Similarity. Method Test MSE ↓ Inv. Error ↓ Cosine Similarity ↑ _tab Spectral Discovery 0.00298 ± 0.00024 0.00070 ± 0.00027 0.9999 ± 0.00001 _tab Augerino 0.01053±0.000400.01053± 0.00040 0.00232±0.000250.00232± 0.00025 0.9233±0.07080.9233± 0.0708 Table 3: Top Quark Tagging (Elliptic Regime). Classification accuracy and generator recovery. Results are reported as mean ± standard deviation. Higher is better for both metrics. Method Accuracy (%) ↑ Cosine Similarity ↑ _tab Spectral Discovery 84.87 ± 0.39 0.9999 ± 0.00002 _tab Augerino 74.9±2.474.9± 2.4 0.994±0.0030.994± 0.003 Evaluation Metrics We evaluate performance using three metrics: (M1) Test MSE, computed on a held-out test set; (M2) Invariance Error, (M3) Cosine similarity between the learned and true generators. Further details about evaluation metrics are provided in the Appendix C.2. 6.1 Results Double Pendulum (6D). On the double pendulum benchmark (Figure 2 and Table 2), Spectral Discovery achieves substantially stronger symmetry recovery while maintaining competitive predictive accuracy. The learned generator aligns nearly perfectly with the ground-truth rotational symmetry, with significantly lower invariance error than Augerino. While both methods fit the regression mapping, augmentation-based invariance does not reliably recover the true infinitesimal generator, whereas spectral discovery yields stable and interpretable generator identification. Top Quark Tagging. On the Top Quark Tagging task (Table 3), Spectral Discovery improves classification accuracy while consistently recovering the underlying rotational subgroup, as evidenced by high generator alignment. These results demonstrate that explicit symmetry discovery in the spectral domain can jointly enhance predictive performance and structural interpretability in real-world high-energy physics settings. 7 Discussion An important observation emerging from our analysis is the structural similarity between our formulation and Equivariant Multi-Layer Perceptrons (EMLP) finzi2021practical. For a known symmetry group, EMLP enforces equivariance via the fixed-point condition ρ(g)v=vρ(g)v=v, which for continuous groups reduces to the Lie algebra constraint dρ(A)v=0dρ(A)v=0 for all A∈A∈ g. To implement this, EMLP constructs equivariant linear layers by vectorizing the weights as v=vec(W)v=vec(W) in a tensor-product representation space (e.g., Vout⊗Vin∗V_out V_in^*) and obtaining equivariant solutions as elements of the nullspace of the induced Lie algebra action. In our framework, each Fourier coefficient F^(π)=∫GF(g)ρπ(g)∗g∈End(Vπ)≅Vπ⊗Vπ∗ F(π)= _GF(g) _π(g)^*\,dg (V_π) V_π V_π^* resides in the same type of tensor-product space. The infinitesimal constraint F^(π)dρπ(B)∗=0 F(π)\,d _π(B)^*=0 imposes an identical Lie-algebra nullspace condition. Thus, both approaches reduce symmetry to nullspace constraints in representation space: EMLP constrains vectorized weights, while our method constrains Fourier blocks. When a layer maps Vπ→VπV_π→ V_π, its weight lies in End(Vπ)End(V_π), the same space as the corresponding GFT coefficient, making the latter one of the valid solutions of the weight. Limitations and Future Work Our framework currently targets one-parameter subgroups of SO(n)SO(n) and is optimized for recovering a single dominant generator. Extending it to jointly discover higher-dimensional or multiple commuting subgroups remains future work. Moreover, the analysis is performed on the maximal torus of SO(n)SO(n), leveraging its simple irreducible representations for efficiency. Incorporating broader compact-group representations beyond the maximal torus would expand the range of discoverable symmetries. References Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms (Supplementary Material) Table 4: Notation used in the method. Symbol Meaning =(xi,yi)i=1TD=\(x_i,y_i)\_i=1^T Training dataset of T input–target pairs. T Number of training samples. xi∈ℝnx_i ^n Input feature vector for sample i. yi∈ℝmy_i ^m Target/output vector for sample i. n,mn,m Input and output dimensions, respectively. ⊂ℝnX ^n Input domain (assumed closed under the action of SO(n)SO(n)). :→ℝm f:X ^m Unknown ground-truth target function generating y=(x)y= f(x). SO(n)SO(n) Special orthogonal group (rotations in ℝnR^n). (n) so(n) Lie algebra of SO(n)SO(n): skew-symmetric matrices B⊤=−B =-B. ℋ⊂SO(n)H⊂ SO(n) Unknown one-parameter subgroup under which f is invariant. h⋅xh· x Group action of h∈SO(n)h∈ SO(n) on x∈ℝnx ^n (rotation). ℋ=exp(tB):t∈ℝH=\ (tB):t \ One-parameter subgroup generated by B∈(n)B∈ so(n). B Skew-symmetric Lie-algebra generator of the latent symmetry. exp(tB) (tB) Matrix exponential mapping generator B to a group element in SO(n)SO(n). Q∈SO(n)Q∈ SO(n) Learnable orthogonal alignment matrix (orients invariant 22D planes). J=[0−110]J= bmatrix0&-1\\ 1&0 bmatrix Canonical 2×22× 2 rotation generator. r=n/2r=n/2 Number of invariant 22D planes (assuming n even). λ=(λ1,…,λr)∈ℝrλ=( _1,…, _r) ^r Rotation rates / angular velocities in each invariant plane. B=Q(⨁k=1rλkJ)Q⊤B=Q ( _k=1^r _kJ )Q Real canonical decomposition/parameterization of the generator. z=Q⊤xz=Q x or Z(x)=QxZ(x)=Qx Aligned coordinates after orthogonal transformation (notation varies in text). Z(x)=⨁k=1rzkZ(x)= _k=1^rz_k Block decomposition of aligned coordinates into r two-dimensional blocks. zk=(z2k−1,z2k)⊤∈ℝ2z_k=(z_2k-1,z_2k) ^2 The k-th 22D block (candidate invariant plane coordinates). uk=z2k−1+iz2k∈ℂu_k=z_2k-1+iz_2k∈C Complex representation of the k-th 22D block. uk=rkeiθku_k=r_ke^i _k Polar form: radius rkr_k and angle θk _k. rk=|uk|r_k=|u_k| Radius (magnitude) of block k. θk∈[0,2π) _k∈[0,2π) Torus angle (phase) in block k. θ=(θ1,…,θr)θ=( _1,…, _r) Vector of torus angles; parameterizes r≅(S1)rT^r (S^1)^r. rT^r Maximal torus in aligned coordinates (product of r circles). m=(m1,…,mr)∈ℤrm=(m_1,…,m_r)∈Z^r Integer frequency (character index) on the torus. ⟨m,θ⟩ m,θ Standard inner product ∑k=1rmkθk _k=1^rm_k _k. χm(θ)=ei⟨m,θ⟩ _m(θ)=e^i m,θ Torus character / Fourier basis function. Z^m(x)=exp(i⟨m,θ(x)⟩) Z_m(x)= (i m,θ(x) ) Torus Fourier feature indexed by m. ℬB Bandwidth truncation; restricts mk∈−ℬ,…,ℬm_k∈\-B,…,B\. prim(m)prim(m) Primitive direction: m/gcd(|m1|,…,|mr|)m/ (|m_1|,…,|m_r|). ℳM Set of primitive frequency directions retained after bandwidth truncation. U(x)=Z^m(x)m∈ℳU(x)=\ Z_m(x)\_m Collection of primitive torus Fourier features. R(x)=(r1,…,r)R(x)=(r_1,…,r_r) Vector of radii from aligned 22D blocks. ϕw _w Predictor / function approximator with parameters (weights) w. y^=ϕw(U(x),R(x)) y= _w(U(x),R(x)) Model prediction. ℒ(y,ϕw(⋅))L(y, _w(·)) Prediction loss (task-dependent). CmC_m First-layer weight in ϕw _w associated with Fourier feature indexed by m. μ>0μ>0 Regularization strength for resonance penalty. ℛ(λ)=m∈ℤr:⟨m,λ⟩=0R(λ)=\m∈Z^r: m,λ =0\ Resonant frequency set implied by invariance. ⟨m,λ⟩=0 m,λ =0 Resonance condition (nonzero spectral mass only on resonant frequencies). Appendix A Identifiability of the Alignment and Generator Parameters Our model parameterizes a one-parameter subgroup generator in canonical form B=Q(⨁k=1rλkJ)Q⊤,J=(0−110),r=n/2,B\;=\;Q ( _k=1^r _kJ )Q , J\;=\; pmatrix0&-1\\ 1&0 pmatrix, r=n/2, (11) where Q∈SO(n)Q∈ SO(n) aligns the ambient coordinates with r invariant 22-planes and λ=(λ1,…,λr)∈ℝrλ=( _1,…, _r) ^r are the rotation rates within each plane. Since Q and λ are learned from data through spectral structure, it is important to clarify what aspects of (Q,λ)(Q,λ) are identifiable. A.1 Identifiability of Q up to block-wise rotations and permutations The decomposition in equation 11 is not unique: each invariant 22-plane admits an arbitrary choice of orthonormal basis. Concretely, let S=⨁k=1rR(αk),R(α)=(cosα−sinαsinαcosα)∈SO(2),S\;=\; _k=1^rR( _k), R(α)\;=\; pmatrix α&- α\\ α& α pmatrix∈ SO(2), (12) be a block-diagonal change of basis that rotates each 22-plane independently. Because R(α)JR(α)⊤=JR(α)JR(α) =J for all α, we have S(⨁k=1rλkJ)S⊤=⨁k=1rλkJ.S ( _k=1^r _kJ )S \;=\; _k=1^r _kJ. (13) Therefore, replacing Q with Q′=QSQ =QS leaves B unchanged: Q′(⨁k=1rλkJ)Q′⊤=Q(⨁k=1rλkJ)Q⊤=B.Q ( _k=1^r _kJ )Q \;=\;Q ( _k=1^r _kJ )Q \;=\;B. (14) Hence, Q is identifiable only up to right-multiplication by block-diagonal rotations equation 12 (i.e., a gauge freedom corresponding to basis choice within each invariant plane). In addition, if we permute the ordering of the invariant planes, the generator is unchanged up to a corresponding permutation of λ. Let P be a 2×22× 2-block permutation matrix (i.e., P permutes the r blocks), so that P(⨁k=1rλkJ)P⊤=⨁k=1rλπ(k)JP ( _k=1^r _kJ )P \;=\; _k=1^r _π(k)J (15) for some permutation π of 1,…,r\1,…,r\. Then B is also invariant under (Q,λ)↦(QP,π(λ)).(Q,λ)\; \;(QP,\,π(λ)). (16) Combining the two, the alignment matrix is identifiable only up to the equivalence class Q∼QSP,Q\; \;QSP, (17) where S is block-diagonal with SO(2)SO(2) blocks and P is a 22-block permutation matrix. Finally, if some rates coincide (e.g., λi=λj _i= _j), then additional mixing between the corresponding 22-planes may be possible without changing B, enlarging the non-identifiability group beyond equation 17. In practice, we interpret successful recovery of Q as recovery of the invariant subspaces (planes), rather than a unique orthonormal basis within each plane. A.2 Identifiability of λ from surviving (resonant) frequencies On the maximal torus in aligned coordinates, invariance along exp(tB) (tB) implies the resonance constraint F^(m)⟨m,λ⟩=0,m∈ℤr, F(m)\, m,λ =0, m ^r, (18) so nonzero spectral mass can occur only on frequency indices m satisfying ⟨m,λ⟩=0. m,λ =0. (19) Let ℳsurv⊂ℤrM_surv ^r denote the set of surviving (detected) frequency directions, e.g., those for which the learned spectral coefficients are non-negligible. Then λ must satisfy the homogeneous linear system ⟨m,λ⟩=0,∀m∈ℳsurv. m,λ =0, ∀ m _surv. (20) Stacking constraints yields Mλ=0,M∈ℤ|ℳsurv|×r,Mλ=0, M ^|M_surv|× r, (21) where each row of M is m⊤m for m∈ℳsurvm _surv. Existence. The system equation 21 always admits the trivial solution λ=0λ=0; however λ=0λ=0 corresponds to a degenerate subgroup action. In our setting, we seek nontrivial λ≠0λ≠ 0, which exists iff dimker(M)≥ 1. (M)\;≥\;1. (22) Equivalently, rank(M)≤r−1rank(M)≤ r-1. In practice, approximate invariance leads to approximate constraints ⟨m,λ⟩≈0 m,λ ≈ 0, and λ can be estimated as a (regularized) right-singular vector associated with the smallest singular value of M. Uniqueness up to scale (and sign). Because equation 20 is homogeneous, if λ is feasible then so is cλcλ for any c∈ℝc , so λ can be at best identifiable up to scale (and thus sign). A standard normalization (e.g., ‖λ‖2=1\|λ\|_2=1) fixes this degree of freedom. After normalization, λ is unique iff the nullspace is one-dimensional: dimker(M)=1⟺rank(M)=r−1. (M)=1 (M)=r-1. (23) This condition depends on the number and linear independence of surviving frequency directions. Intuitively, each independent m contributes one linear constraint on λ; to determine λ up to scale in ℝrR^r, we need r−1r-1 independent constraints. Role of the types of surviving frequencies. If the surviving set ℳsurvM_surv lies in a low-dimensional subspace (e.g., many m are collinear or concentrated on a few coordinates), then rank(M)rank(M) may be small and ker(M) (M) high-dimensional, yielding non-identifiability. Conversely, a diverse set of linearly independent m vectors tightens the constraints and can make λ identifiable (up to scale). This motivates using primitive frequency directions and promoting sparsity patterns that reveal multiple independent resonant m’s. Practical estimation. Given an estimated ℳsurvM_surv, one may recover λ by solving minλ∈ℝr‖Mλ‖22s.t.‖λ‖2=1, _λ ^r\ \|Mλ\|_2^2 .t. \|λ\|_2=1, (24) which returns the unit-norm right-singular vector of M associated with the smallest singular value. When the constraints are exact and rank(M)=r−1rank(M)=r-1, this recovers λ uniquely up to sign. A.3 Summary of identifiability Overall, the generator B is identifiable, while (Q,λ)(Q,λ) are identifiable only up to natural symmetries: (i) Q up to independent SO(2)SO(2) rotations within each invariant plane and permutations of the planes, and (i) λ up to scale/sign, with uniqueness after normalization governed by the rank condition rank(M)=r−1rank(M)=r-1 determined by the number and linear independence of surviving resonant frequencies. What ultimately matters. While the alignment matrix Q is only identifiable up to block-wise rotations and permutations (Section A.1), these ambiguities do not affect the recovered one-parameter subgroup. Indeed, all equivalent choices of (Q,λ)(Q,λ) related by the transformations in equation 17 and scaling of λ induce the same Lie algebra generator B in equation 11 and therefore the same subgroup exp(tB)\ (tB)\. Consequently, the practically meaningful question is whether the underlying generator direction is correctly identified. This reduces primarily to identifiability of λ. If there exists a nontrivial solution λ≠0λ≠ 0 to equation 20 and the nullspace is one-dimensional (so λ is unique up to scale/sign), then the induced generator B=Q(⨁k=1rλkJ)Q⊤B\;=\;Q ( _k=1^r _kJ )Q (25) is uniquely determined (up to the inherent gauge symmetries above) and correctly captures the underlying one-parameter subgroup. In summary, ambiguities in Q correspond only to basis choices within invariant planes and are largely irrelevant for symmetry discovery. The central requirement for successful recovery is that λ be identifiable (up to scaling), which in turn depends on the number and linear independence of the surviving resonant frequency directions. Appendix B Proofs See 5.1 Proof. Fix an irrep ρπ:G→ℂdπ×dπ _π:G→C^d_π× d_π and write ut:=exp(tB)∈Gu_t:= (tB)∈ G. We first compute how the group Fourier coefficient transforms under left translation. Step 1: Fourier coefficient of a left translate. By definition, LtF^(π)=∫G(LtF)(g)ρπ(g)∗g=∫GF(utg)ρπ(g)∗g. L_tF(π)= _G(L_tF)(g)\, _π(g) \,dg= _GF(u_tg)\, _π(g) \,dg. Use the change of variables h=utgh=u_tg (equivalently, g=ut−1hg=u_t^-1h). Since Haar measure is left-invariant, dg=dhdg=dh. Thus LtF^(π) L_tF(π) =∫GF(h)ρπ(ut−1h)∗h = _GF(h)\, _π(u_t^-1h) \,dh =∫GF(h)(ρπ(ut−1)ρπ(h))∗h = _GF(h)\, ( _π(u_t^-1)\, _π(h) ) \,dh =∫GF(h)ρπ(h)∗ρπ(ut−1)∗h = _GF(h)\, _π(h) \, _π(u_t^-1) \,dh =F^(π)ρπ(ut−1)∗. = F(π)\, _π(u_t^-1) . Because ρπ _π is unitary, ρπ(ut−1)=ρπ(ut)∗ _π(u_t^-1)= _π(u_t) , hence ρπ(ut−1)∗=ρπ(ut) _π(u_t^-1) = _π(u_t). Therefore, LtF^(π)=F^(π)ρπ(ut)∀t∈ℝ. L_tF(π)= F(π)\, _π(u_t) ∀\,t . (26) Step 2: Use left-invariance. If F is left-invariant along exp(tB) (tB), then LtF=FL_tF=F for all t, hence LtF^(π)=F^(π) L_tF(π)= F(π) for all t. Combining with equation 26, F^(π)ρπ(ut)=F^(π)∀t∈ℝ. F(π)\, _π(u_t)= F(π) ∀\,t . (27) Equivalently, F^(π)(ρπ(ut)−I)=0∀t∈ℝ. F(π)\, ( _π(u_t)-I )=0 ∀\,t . (28) Step 3: Differentiate at t=0t=0. The map t↦ρπ(ut)=ρπ(exp(tB))t _π(u_t)= _π( (tB)) is smooth (indeed analytic), so we may differentiate equation 27 at t=0t=0: 0=dt|t=0(F^(π)ρπ(exp(tB)))=F^(π)dt|t=0ρπ(exp(tB))=F^(π)dρπ(B),0= ddt |_t=0 ( F(π)\, _π( (tB)) )= F(π)\, ddt |_t=0 _π( (tB))= F(π)\,d _π(B), where dρπ(B):=dt|t=0ρπ(exp(tB))d _π(B):= ddt |_t=0 _π( (tB)) is the derived (Lie algebra) representation. Finally, since ρπ _π is unitary, dρπ(B)d _π(B) is skew-Hermitian: ρπ(exp(tB))∗ρπ(exp(tB))=I⇒dt|t=0(ρπ(exp(tB))∗ρπ(exp(tB)))=0⇒dρπ(B)∗+dρπ(B)=0. _π( (tB)) _π( (tB))=I\ \ ddt |_t=0 ( _π( (tB)) _π( (tB)) )=0\ \ d _π(B) +d _π(B)=0. Hence dρπ(B)∗=−dρπ(B)d _π(B) =-d _π(B), and the identity F^(π)dρπ(B)=0 F(π)\,d _π(B)=0 is equivalent to F^(π)dρπ(B)∗=0, F(π)\,d _π(B) =0, which is exactly equation 3. Step 4: Invertible case. If dρπ(B)d _π(B) is invertible, then multiplying F^(π)dρπ(B)∗=0 F(π)\,d _π(B) =0 on the right by (dρπ(B)∗)−1(d _π(B) )^-1 gives F^(π)=0 F(π)=0. This holds for every π∈G^π∈ G, completing the proof. ∎ See 5.2 Proof. We apply Proposition 5.1 with G=rG=T^r, whose irreducible unitary representations are the characters ρπ=χm _π= _m. Left-invariance along exp(tB) (tB) means F(exp(tB)g)=F(g)F( (tB)g)=F(g) for all t, hence Proposition 5.1 gives F^(m)dχm(B)∗=0. F(m)\,d _m(B) =0. It remains to compute dχm(B)d _m(B). In the aligned coordinates, exp(tB) (tB) corresponds to torus angles θ(t)=(λ1t,…,λrt)θ(t)=( _1t,…, _rt), so χm(exp(tB))=χm(θ(t))=exp(i⟨m,λ⟩t). _m( (tB))= _m(θ(t))= \! (i m,λ t ). Differentiating at t=0t=0 yields dχm(B)=dtχm(exp(tB))|t=0 d _m(B)= . ddt _m( (tB)) |_t=0 =i⟨m,λ⟩, =i m,λ , dχm(B)∗ d _m(B) =−i⟨m,λ⟩. =-i m,λ . Substituting into the spectral constraint gives F^(m)(−i⟨m,λ⟩)=0⟺F^(m)⟨m,λ⟩=0, F(m)\,(-i m,λ )=0 F(m)\, m,λ =0, Thus, nonzero Fourier coefficients F^(m) F(m) occur, only if ⟨m,λ⟩=0 m,λ =0. ∎ Appendix C Experiments C.1 Training Protocol Datasets. For the double pendulum experiment, we use 32,00032,000 samples. For the Top Quark Tagging task, we use 64,00064,000 samples. In both cases, we follow the same training protocol described below. Model architecture. The predictor ϕw _w is implemented as a multilayer perceptron (MLP) with three hidden layers and ReLU activations. Unless otherwise stated, the spectral and radial features described in Section 5 are concatenated and fed into this network. Optimization. All models are trained for 4040 epochs using the Adam optimizer with learning rate 2×10−32× 10^-3. Mini-batch training is used throughout. Resonance regularization schedule. To stabilize training, we employ a warm-up phase for the resonance regularization coefficient μ in the objective equation 10. For the first 1010 epochs, μ is set to 0.10.1. After the warm-up period, μ is gradually increased during training up to a maximum scaling factor of 2.02.0. This schedule allows the predictor to first learn a reasonable fit before strongly enforcing the spectral resonance constraints. Model selection. Unless otherwise specified, we select the best checkpoint based on validation performance and report the mean and standard deviation across multiple random seeds. Table 5: Summary of experimental settings used across all tasks. Component Configuration Double Pendulum samples 32,000 Top Tagging samples 64,000 Epochs 40 Optimizer Adam Learning rate 2×10−32× 10^-3 Predictor ϕw _w MLP (3 hidden layers, ReLU) Warm-up epochs 10 Initial resonance weight μ 0.1 Maximum μ scaling 2.0 Model selection Best validation checkpoint C.2 Evaluation Metrics We report the following metrics: (M1) Test MSE. Mean squared error evaluated on the held-out test set. (M2) Invariance Error. When the ground-truth generator XgtX_gt is available, we measure invariance by sampling transformations R(t)=exp(tXgt)R(t)= (tX_gt) and computing InvErr=x,t[‖fθ(x)−fθ(R(t)x)‖22].InvErr=E_x,t [\|f_θ(x)-f_θ(R(t)x)\|_2^2 ]. (M3) Generator Recovery. We measure generator recovery using cosine similarity between the learned generator matrix XθX_θ and the ground-truth generator XgtX_gt: CosSim(Xθ,Xgt)=⟨vec(Xθ),vec(Xgt)⟩‖vec(Xθ)‖2‖vec(Xgt)‖2+ϵ.CosSim(X_θ,X_gt)= (X_θ),vec(X_gt) \|vec(X_θ)\|_2\|vec(X_gt)\|_2+ε. Appendix D Sensitivity Analysis 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. We compare Spectral Discovery and Augerino on the 6D double pendulum system, evaluating: (i) recovery of the latent one-parameter symmetry generator via cosine similarity, and (i) predictive accuracy via test loss. Noise robustness. Figure 3 presents performance as Gaussian noise increases from 0.10.1 to 1.01.0. Spectral Discovery maintains near-perfect cosine similarity for low and moderate noise levels and exhibits only moderate degradation at the highest noise level. In contrast, Augerino consistently recovers a less aligned generator across all noise levels. Both methods experience increasing test loss as noise grows, as expected. However, Spectral Discovery preserves substantially stronger symmetry alignment throughout the entire noise regime. This indicates that frequency-based symmetry identification remains stable even when the signal-to-noise ratio decreases. Sample efficiency. Figure 4 evaluates performance as the number of training samples increases from 88k to 6464k. Spectral Discovery achieves near-perfect generator recovery even at moderate sample sizes and maintains this alignment as more data are provided. Augerino improves with additional samples but remains less accurate in identifying the underlying subgroup direction. In terms of predictive performance, Spectral Discovery consistently attains lower test error across all sample sizes. This suggests that explicitly exploiting spectral structure leads to more efficient use of available data. Overall interpretation. Across both sweeps, Spectral Discovery demonstrates: • stronger recovery of the latent symmetry generator, • improved robustness to noise, and • better sample efficiency in predictive learning. These findings support the claim that structured spectral analysis provides a principled mechanism for automatic discovery of continuous one-parameter symmetries, enabling reliable structural identification alongside strong predictive performance. D.1 Comparison with LieGAN 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. We compare Spectral Discovery with LieGAN on the 6D double pendulum benchmark, focusing exclusively on recovery of the latent one-parameter symmetry generator. Since LieGAN is designed purely for symmetry discovery and does not include a downstream predictive model, we report only mean cosine similarity between the recovered and ground-truth generators (please refer Table 1 for further details). Sample efficiency. Figure 5 shows generator recovery as the number of training samples increases. Spectral Discovery consistently achieves higher cosine similarity across all sample sizes and approaches near-perfect alignment in larger data regimes. LieGAN improves with more samples but remains less aligned, particularly at higher sample counts. This indicates that exploiting structured spectral concentration enables more accurate identification of the subgroup direction than adversarially learning generators from data. Noise robustness. Figure 6 evaluates robustness under increasing additive noise. Spectral Discovery maintains near-perfect alignment for low and moderate noise levels and degrades gracefully at high noise. LieGAN also improves up to moderate noise but exhibits lower alignment overall and a more noticeable decline at the highest noise level. These results suggest that frequency-based symmetry identification retains structural information more reliably under perturbations. Overall interpretation. Across both sweeps, Spectral Discovery demonstrates stronger recovery of the latent symmetry generator than LieGAN. Because LieGAN does not learn a predictive mapping, this comparison isolates the symmetry identification component. The results support the central premise of our method: leveraging representation-theoretic structure and spectral concentration provides a principled and reliable mechanism for automatic discovery of continuous one-parameter symmetries.