Symmetry Discovery

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21 papers4 papers in the last four weeks, up 33% on the four weeks before. 0.0% of all new papers.

Oct 12, 2026

Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry

Continuous symmetry discovery is typically formulated as a separate estimation problem for each function or dataset. We instead investigate whether the map from a function to its symmetry structure can be learned across a family of functions from finite observations. Using the infinitesimal differential characterization of continuous symmetries, we formulate this map as an inverse symmetry operator, analogous to an inverse PDE problem. For global Lie-group symmetries, we establish a representation-theoretic population characterization: the symmetry Lie algebra is the nullspace of a positive-semidefinite operator constructed from representation coefficients and the infinitesimal group action. We also show that the inverse symmetry operator is equivariant under coordinate changes through the adjoint action. This characterization separates the geometric symmetry map from finite-sample estimation, motivating a hybrid approach that amortizes statistically difficult components while retaining analytic structure. On SO(5) benchmarks, amortized equivariant estimation of harmonic coefficients improves symmetry recovery from sparse observations, whereas ordinary least squares suffices for SL(2,R). We extend this viewpoint to local symmetries, where symmetry directions form smoothly varying tangent distributions rather than a fixed Lie algebra. Under the Lie-bracket consistency condition, Frobenius' theorem guarantees local integral submanifolds along which observables remain unchanged. We recover this geometry using a hybrid estimator that learns observation weights while preserving analytic differential and geometric recovery. Across global, non-compact, local, and physically motivated synthetic benchmarks, our results support learning the statistically difficult finite-sample components while preserving the known geometric structure of symmetry recovery.

Pavan Karjol, Scindhiya Laxmi, Prathosh AP

Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India · Department of Mathematics and Computing, IIT (ISM) Dhanbad, India

Symmetry Discovery Operator Learning Lie Group Methods

Oct 7, 2026

KASALv2: Fully Automatic 3D Rotational Symmetry Classification and Axis Localization

Rotational symmetry is an important prior in 6D pose estimation, improving pose accuracy and supporting symmetry-aware evaluation. However, current symmetry annotations for 3D objects remain largely manual or semi-automatic, often requiring predefined types or orders, which limits scalability. This work introduces a fully automatic, reference-free framework for symmetry-type classification, rotational-order identification, and full-axis localization across all eight canonical 3D rotational symmetry types. The method localizes a dominant high-order axis, infers its rotational order through self-consistency analysis, and reconstructs the complete symmetry structure under a hierarchy-guided formulation. A texture-aware extension further models appearance-induced reductions in rotational order while preserving axis orientations. Experiments on idealized and real-world datasets demonstrate strong accuracy and generalization, achieving 94.75% accuracy on 438 symmetric objects in GSO. Training FoundationPose with these priors improves accuracy by up to 0.9% across five BOP datasets, showing that automatically estimated rotational priors improve downstream 6D pose estimation. Code is available at https://github.com/WangYuLin-SEU/KASAL.

Mengxin Zhang, Yulin Wang, Chen Luo +2

School of Mechanical Engineering, Southeast University Nanjing, China

Symmetry Discovery Symmetry-Aware Object Pose Estimation

Sep 27, 2026

LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery

Discovering underlying symmetries from data has emerged as a crucial challenge in scientific discovery. Existing data-driven methods for symmetry discovery fail to determine the exact number and mathematical form of unknown infinitesimal generators. Recent explicit methods represent generators using a predefined function library and identify them through algebraic optimization, but they often struggle to capture complex symmetries involving high-order polynomials or transcendental functions. To address this limitation, we formulate symmetry discovery as a joint optimization problem over the function library and coefficients. We propose a novel framework that leverages an encoder-decoder architecture to dynamically generate symbolic expressions and expand the library. This generation process is optimized via reinforcement learning, which accelerates the exploration of the symbolic search space through step-wise rewards. Experiments demonstrate that LieDiscover can successfully uncover open-form infinitesimal generators involving high-order polynomials or transcendental functions, which remain intractable for existing methods. The discovered symmetries also improve performance in downstream PDE solving and discovery tasks.

Xinxin Li, Jianming Ma, Xingyu Cui +3

East China Normal University · Shanghai Jiao Tong University · Institute of Applied Physics and Computational Mathematics +2

Lie Group Methods Symmetry Discovery

Sep 27, 2026

Discovering Symmetries in Neural Network Parameter Spaces

Parameter space symmetries are important for understanding neural networks' loss landscape, training dynamics, and generalization. However, systematically identifying these symmetries remains a challenge. In this paper, we formalize data-dependent parameter symmetries and characterize loss invariance and the group-action axioms through infinitesimal conditions, which provide objectives for jointly learning group generators and nonlinear action maps. Our framework systematically uncovers parameter symmetries, including previously unknown ones. To study larger networks, we establish conditions under which subnetwork symmetries extend to the full model. The same construction gives an explicit family of finite-batch symmetries, providing both analytical examples and a foundation for discovery through small subnetworks. Using the infinitesimal characterization and subnetwork construction, we implement a framework for automated discovery of parameter symmetries, and successfully uncovered symmetries in various architectures, including pretrained transformer models.

Bo Zhao, Nima Dehmamy, Robin Walters +1

Harvard University · IBM Research · Northeastern University +1

Scientific Discovery Symmetry Discovery

Sep 18, 2026

Emergent Intelligence: Resonant Oscillators Produce Proactive Adaptive Behavior

Most artificial neural systems are built to map given inputs to outputs. Adaptive agents face a prior problem: they must act without enough evidence, seek encounters with the world, and revise behavior when evidence appears. We propose another starting point for intelligent neural networks: proactive search without signals, curiosity at its most basic. We ask whether it can come from a minimal untrained circuit. The spiking unit studied here inverts its response to input: with no signal in its window it fires faster; once signals arrive it switches to a slower, inverted regime. Search needs three or more such oscillators in counter-phase, each reading the same input in a different time window. With no training, supervision, parameter tuning, or controller, the composite switches on its own between exploratory spiral search and exploitative tracking, finding both first-degree symmetry and second-degree groups. The switch comes from temporal disagreement between its fast and slow readings of the same signal. We view the circuit as evolutionarily trained: its abilities come from structure, not experience. Ablation over 63 configurations and 63,000 trials shows the switch needs both temporal staggering and counter-phase opposition, neither enough alone: the behavior is emergent, not programmed. The spiral persists at zero rotational diffusion, so it is structural, and degrades gently under perturbation. More oscillators improve spiral regularity but cut resource capture, so the smallest sufficient circuit wins. We propose that this principle underlies search in simple organisms, navigation and decisions in complex ones, and, being so simple and common, goes unnoticed unless you strip the logic bare. Eventually, networks of such proactive primitives may offer another foundation for AI architectures that explore our world rather than merely predict the next symbol in a sequence.

Alex Fedosov, Maxim Yakimenko, Sander Stepanov

FoundAItion Inc · FoundAItion Inc.

Symmetry Discovery Oscillatory Neural Networks Exploration-Exploitation Tradeoff Spiking Neural Networks

Sep 11, 2026

Symmetry Discovery in Quantum Learning: Observable-Level and Task-Level Inference from Finite Measurements

Symmetry reduces the capacity of a quantum learning model, but the imposed group must match both the measured information and the label transformation. We establish a finite-measurement theory for inferring this group from candidate transformations. The central structural result identifies observable-invisible transformations with the stabilizer of a projected state whenever the probe span is invariant. It turns recovered generators into a valid subgroup and identifies the continuous invisible space with its Lie algebra. For finite dictionaries, an unbiased shadow statistic distinguishes zero from positive squared expectation discrepancies with an inverse-gap measurement rate, improving the inverse-square-gap rate of uniform discrepancy estimation. A commuting qubit lower bound proves the gap dependence optimal at fixed snapshot scale, and simultaneous intervals support data-dependent tolerances. Task validation then tests either the joint distribution through a characteristic kernel or its encoded mean through a classical--quantum discrepancy. An exact group-average identity relates the latter to joint-state asymmetry and specifies its conversion to binary task breaking mass. Projection bias quantifies the cost of excessive symmetry, while an

\ell_1

readout bound quantifies the capacity gained by relaxing it. At an invariant pure-state backbone, retained and nontrivial breaking sectors are Fisher-orthogonal. Ising-chain calculations connect finite-shot recovery, label-dependent symmetry, and physical sector drift. These results determine which symmetry the measurements support and provide the statistical and geometric basis for a subsequent release decision.

Zeyu Chen

Symmetry Discovery Quantum Machine Learning Scientific Discovery Symmetry Breaking

Sep 7, 2026

Efficient Learning and Symmetry Discovery under Exact Invariances

Learning with group invariances is central to many scientific and geometric learning problems, yet its computational foundations remain poorly understood. Even for classical supervised regression settings, it has been unclear whether one can efficiently compute a regression function that is exactly invariant to a given group action. Recent work showed that exact invariance can be enforced in polynomial time when the underlying group is finite and known, but left open the cases of infinite groups and unknown symmetries. In this paper, we resolve both challenges. First, we present the first polynomial-time algorithm for learning with exact group invariances that applies uniformly to finite and infinite groups. The runtime is polynomial in the data dimension and sample size, and independent of the group, while achieving strong generalization guarantees. This provides a computational explanation for the empirical success of invariant and equivariant methods in geometric machine learning and partially answers a recent open question in the literature. Second, we study learning in the symmetry discovery setting, where the invariance group is unknown. Focusing on the subgroup lattice of a finite group, we show that exact symmetries can be identified from data and exploited for learning in polynomial time. For regression over finite-dimensional feature spaces, our algorithm provably recovers the underlying symmetry, matches the minimax-optimal sample complexity of the known-symmetry setting, and runs in time polynomial in the data dimension and sample size. Our analysis relies on tools from random Cayley graphs and expander theory, which may be of independent interest.

Ashkan Soleymani, Behrooz Tahmasebi, Patrick Jaillet +1

MIT Laboratory for Information and Decision Systems (LIDS) · Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA · Technical University of Munich (CIT, MCML, MDSI) and MIT Computer Science and Artificial Intelligence Laboratory (CSAIL)

Symmetry Discovery Invariant Representation Learning Statistical Learning Theory

Aug 8, 2026

Adaptive Symmetry Discovery for Dynamical System Identification

Dynamical systems model trajectory data generated by fixed underlying dynamics, with applications ranging from biology to physics. Especially in scientific settings, dynamical systems are not generic but often exhibit symmetries imposed by physical laws, formalized through equivariance with respect to group actions. The identification problem concerns recovering the parameters of a system from observed trajectories. In this work, we study adaptive symmetry discovery for dynamical system identification and address how a system can be identified from a single trajectory when it is equivariant with respect to an unknown symmetry group. To this end, we first show that for known symmetries, the system can be identified from a significantly shorter single trajectory than in the generic setting, and we precisely characterize this improvement. We then consider the automatic symmetry discovery setting, proposing a method to learn the symmetry group directly from a single trajectory and incorporate it into the identification procedure, achieving the same optimal trajectory length as in the known-symmetry case. Our analysis relies on tools from group representation theory and the expander properties of Cayley graphs, and may be of independent interest for the study of symmetries in dynamical systems.

Behrooz Tahmasebi, Melanie Weber

Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA.

Dynamical Systems Nonlinear System Identification System Identification Symmetry Discovery

Aug 3, 2026

LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce LieStoNet, an end-to-end, template-free framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at this link.

Shida Liu, Abhishek Gupta, Sumit Sinha +1

School of Engineering and Applied Sciences (SEAS), Harvard University, Cambridge, MA, USA. · Tracelink, Wilmington, MA, USA. · SEAS, Physics and OEB, Harvard University, Cambridge, MA, USA.

Stochastic Differential Equations Lie Group Methods Symmetry Discovery

Jul 3, 2026

Observable- and Positional-Encoding-Dependent Symmetry Readout from Neural Network Weights

Post-hoc analysis of trained neural network weights often seeks to recover geometric structure directly from the parameters. We show that, for positional-encoding-equipped neural fields, the symmetry visible from weights is not the true symmetry group itself, but an observable symmetry set determined by the trained parameters, the positional encoding (PE), and readout observable. We formulate this dependence through an exact observability hierarchy,

G_{\mathrm{obs}}^{\mathrm{exact}} \subseteq G_{\mathrm{lift}}^{\mathrm{exact}}(φ) \cap G_{\mathrm{true}}

, where

G_{\mathrm{lift}}^{\mathrm{exact}}(φ)

is the set of input transformations that the PE can exactly lift to the feature space. The hierarchy implies that even when a target function has a geometric symmetry, that symmetry may be structurally invisible to weight-level observables if the PE does not represent the corresponding transformation. We test this prediction using MLPs trained on two-dimensional signed distance functions with multiple shape symmetry groups, positional encodings, and Gram-based observables. The results show a consistent PE-dependent pattern: DyadicAxisPE supports

D_4

-sensitive readout but structurally suppresses

D_3

rotations, TriAxisPE yields lower

D_3

/

D_6

readout scores under the tested Gram observables by replacing coordinate axes with three 120-degree-separated axes, and random Fourier features mainly exhibit a

π

-rotation response under these readouts. These findings show that PE design affects not only approximation behavior but also which structures are accessible to post-hoc weight-level readouts. This provides a basis for a principled observable-dependent symmetry readout.

Naoya Chiba, Satoshi Sugiyama, Yuki Uranishi

D3 Center, The Unversity of Osaka

Neural Representation Geometry Implicit Neural Representations Symmetry Discovery Positional Encoding Weight-Space Analysis

Jun 16, 2026

Blind Recovery of Latent Domains via Unsupervised Symmetry Discovery

Primary motivation in blind inverse problems is to recover signals of interest from corrupted observations without knowing the obfuscating mechanism. Blind deconvolution is a prominent approach when the corruption is convolutional, but it is not applicable when general linear transformations obfuscate the domain structure. In this work, we propose an unsupervised framework for recovering latent domains and signals by discovering symmetries of the data distribution. Our framework models observations as linear measurements of signals sampled from a latent random field, and optimizes a shallow group-convolutional network by imposing stationarity and locality regularization at the model output. The model learns a latent symmetry action and an appropriate filter, thereby mapping unstructured observations to a symmetry-based representation that reveals latent signals. Experiments on stochastic processes, Ising models, shuffled and bit-scrambled images, and neural recordings show that the method recovers latent domains and signals from unstructured observations, suggesting symmetry discovery as a new direction for unsupervised structure learning and blind inverse problems.

Onur Efe, Arkadas Ozakin

Physics Department, Bogazici University

Symmetry Discovery Inverse Problems Image Inverse Problems

May 28, 2026

Attention-based optimizer for symmetry finding

Finding symmetries is crucial for understanding physical models. In this work, we present an optimization framework that searches Pauli symmetries of Hamiltonians, merging the fields of machine learning with automated symmetry finding. Built on a Set-Transformer architecture, our framework uses self-attention to encode the pairwise and higher-order correlations among the Pauli-Strings. The relations are then decoded as a candidate, which is further optimized with a custom commutation-based objective, and mapped to a symmetry of the input Hamiltonian. We apply our method to random Pauli Hamiltonians, periodic one and two dimensional transverse-field Ising model and the Toric code. We show that for physical Hamiltonians (Ising and Toric), our framework succeeds with near-deterministic probability while providing substantial advantage compared to state-of-the-art strategies. For random Pauli Hamiltonians, we estimate the required computational resources, specifically the number of parallel starts and the number of GPUs, to find a symmetry with high success probability under fixed design specifications.

Shreya Banerjee, Vinodh Raj Rajagopal Muthu, Charlie Nation +5

Department of Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom · Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada · QuAOS collaboration

Set Transformer Quantum Many-Body Physics Neural Combinatorial Optimization Symmetry Discovery

May 19, 2026

Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules

Symmetry determines which physical responses of molecules and crystals are allowed and underlies many models used throughout the physical sciences. Yet several widely used machine-learned models learn symmetry only approximately from data, raising the question of whether approximate symmetry is sufficient for scientific prediction. Using the Orb-v3 force field, we show that it is often not: the relaxed space group of a crystal depends on the orientation of the input for

47\%

of a

150

-crystal subsample, and relaxation lowers the space group for half of all

399

crystals studied. To characterize, quantify, and correct such failures, we introduce a tensor algebra,

\star_G

, defined by convolution over a finite symmetry group and applicable to tensors with noncommuting entries. We prove that the symmetry group determines the algebra's block-diagonal transform and use this structure to analyze model errors on symmetric targets. The resulting projections decompose the average squared error into orthogonal components arising from selection-rule violations, inconsistencies across symmetry-related inputs, and residual error. The removable components reveal how much error arises from broken symmetry, provide reference-free lower bounds on prediction error, and can be eliminated without sacrificing accuracy. For operators, projection followed by blockwise truncation yields the optimal symmetric low-rank approximation. Applying these projections to force predictions reduces both rates of space-group change to at most

2\%

. Exact selection rules, reference-free error bounds, and symmetry-preserving corrections are therefore available for any trained model without retraining.

Paulina Hoyos, Shashanka Ubaru, Dongsung Huh +5

UT Austin · IBM Research · Independent +2

Scientific Discovery Symmetry Discovery Equivariant Neural Networks Tensor Decomposition Equivariant Representation Learning Singular Value Decomposition

May 18, 2026

Mapping Uncharted Symmetries: Machine Discovery in Combinatorics

Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. We successfully apply both methods to a research problem in algebraic combinatorics, discovering a new combinatorial interpretation of the

q,t

-Narayana polynomials arising from representation theory. To our knowledge, this is the first such interpretation based on noncrossing partitions. Using one discovered statistic, we find a combinatorial proof of the symmetry of these polynomials in a previously unsolved case. To streamline verification and reproducibility, we release all code, including a formalization of all the mathematical discoveries of this paper in Lean 4.

Eugenio Cainelli, Lorenzo Luccioli, Alessandro Iraci +2

University of Bologna · Pegaso University · University of Pisa

Symmetry Discovery

May 18, 2026

The Symmetries of Three-Layer ReLU Networks

We develop a framework for analyzing parameter symmetries in deep ReLU networks and obtain a complete characterization of the generic parameter fibers for three-layer bottleneck architectures. Our approach provides explicit semi-algebraic descriptions of these fibers and yields a polynomial time algorithm for deciding functional equivalence of two parameters. The symmetries include discrete and continuous transformations arising from layer composition, and depend on whether deeper layers hide or preserve geometric structure from preceding layers. Finally, we show that some of these symmetries induce local conservation laws along gradient flow, while others do not.

Johanna Marie Gegenfurtner, Moritz Grillo, Guido Montúfar

Technical University of Denmark · Max Planck Institute for Mathematics in the Sciences · UCLA and MPI MiS

Parameter Identifiability ReLU Neural Networks Symmetry Discovery

May 18, 2026

Prism: Structural Symmetry Scanning via Duality-Constrained Laplacian Projection

We introduce Prism, a framework for structural symmetry diagnosis in complex networks. Given a graph Laplacian

L

and a duality operator

P

(a symmetric involution), Prism computes the duality defect

δ(L,P) = \|LP - PL\|_F / \|L\|_F

-- a scalar measuring how far the network deviates from structural self-consistency. When

P

encodes the network's true symmetry,

δ

starts near zero and rises monotonically as structure degrades; an arbitrary

P

gives noise. We prove that the optimal

L'

satisfying

[L', P] = 0

is given by a closed-form block-diagonal projection, and provide an unsupervised alternating optimization that learns

P

from the graph's own Fiedler vector. Experiments on synthetic networks show the true-

P

defect is

3.38\times

more sensitive to structural degradation than an index-reversal baseline and more sensitive than modularity. On Zachary's Karate Club with edge noise, Prism achieves

94.5\%

community detection accuracy at

5\%

noise versus

76.6\%

for the raw Laplacian baseline. Applied to live S&P~500 data (2026-05-17), Prism detects rising structural stress (defect

0.43 \to 0.73

over 90 days) while surface correlations remain low -- a signal invisible to correlation-based methods. In a historical backtest spanning five major stress events (2011--2020), the duality defect exhibits a consistent pattern: it reaches elevated levels before the correlation spike that accompanies each crisis, and sustains high readings during periods of structural fragility that conventional metrics classify as calm. The duality defect is a first-principles structural admissibility condition, requiring no training data and computable in milliseconds.

Jiatong Xie

Independent researcher.

Graph Clustering Spectral Methods Anomaly Detection Symmetry Discovery

Apr 24, 2026

ArchSym: Detecting 3D-Grounded Architectural Symmetries in the Wild

Symmetry detection is a fundamental problem in computer vision, and symmetries serve as powerful priors for downstream tasks. However, existing learning-based methods for detecting 3D symmetries from single images have been almost exclusively trained and evaluated on object-centric or synthetic datasets, and thus fail to generalize to real-world scenes. Furthermore, due to the inherent scale ambiguity of monocular inputs, which makes localizing the 3D plane an ill-posed problem, many existing works only predict the plane's orientation. In this paper, we address these limitations by presenting the first framework for detecting 3D-grounded reflectional symmetries from single, in-the-wild RGB images, focusing on architectural landmarks. We introduce two key innovations: (1) a scalable data annotation pipeline to automatically curate a large-scale dataset of architectural symmetries, ArchSym, from SfM reconstructions by leveraging cross-view image matching; and building on the dataset, (2) a single-view symmetry detector that accurately localizes symmetries in 3D by parameterizing them as signed distance maps defined relative to predicted scene geometry. We validate our symmetry annotation pipeline against geometry-based alternatives and demonstrate that our symmetry detector significantly outperforms state-of-the-art baselines on our new benchmark.

Hanyu Chen, Ruojin Cai, Steve Marschner +1

Cornell University · Kempner Institute, Harvard University

3D Scene Understanding Symmetry Discovery

Mar 12, 2026

Disentangled Representation Learning through Unsupervised Symmetry Group Discovery

Symmetry-based disentangled representation learning leverages the group structure of environment transformations to uncover the latent factors of variation. Prior approaches to symmetry-based disentanglement have required strong prior knowledge of the symmetry group's structure, or restrictive assumptions about the subgroup properties. In this work, we remove these constraints by proposing a method whereby an embodied agent autonomously discovers the group structure of its action space through unsupervised interaction with the environment. We prove the identifiability of the true symmetry group decomposition under minimal assumptions, and derive two algorithms: one for discovering the group decomposition from interaction data, and another for learning Linear Symmetry-Based Disentangled (LSBD) representations without assuming specific subgroup properties. Our method is validated on three environments exhibiting different group decompositions, where it outperforms existing LSBD approaches.

Barthélémy Dang-Nhu, Louis Annabi, Sylvain Argentieri

Sorbonne Universit´e, CNRS, Institut des Syst`emes Intelligents et de Robotique, ISIR, F-75005 Paris, France

Disentangled Representation Learning Unsupervised Representation Learning Symmetry Discovery Equivariant Representation Learning

Mar 7, 2026

Spectral Discovery of Continuous Symmetries via Generalized Fourier Transforms

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.

Pavan Karjol, Kumar Shubham, Prathosh AP

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

Symmetry Discovery Spectral Methods Lie Group Methods

Feb 28, 2026

Hereditary Geometric Meta-RL: Nonlocal Generalization via Task Symmetries

Meta-Reinforcement Learning (Meta-RL) commonly generalizes via smoothness in the task encoding. While this enables local generalization around each training task, it requires dense coverage of the task space and leaves richer task space structure untapped. In response, we develop a geometric perspective that endows the task space with a "hereditary geometry" induced by the inherent symmetries of the underlying system. Concretely, the agent reuses a policy learned at the train time by transforming states and actions through actions of a Lie group. This converts Meta-RL into symmetry discovery rather than smooth extrapolation, enabling the agent to generalize to wider regions of the task space. We show that when the task space is inherited from the symmetries of the underlying system, the task space embeds into a subgroup of those symmetries whose actions are linearizable, connected, and compact--properties that enable efficient learning and inference at the test time. To learn these structures, we develop a differential symmetry discovery method. This collapses functional invariance constraints and thereby improves numerical stability and sample efficiency over functional approaches. Empirically, on a two-dimensional navigation task, our method efficiently recovers the ground-truth symmetry and generalizes across the entire task space, while a common baseline generalizes only near training tasks.

Paul Nitschke, Shahriar Talebi

P. Nitschke is with the Paulson School Of Engineering And Applied Sciences, Harvard University, Cambridge, US. · S. Talebi is with the UCLA Samueli School of Engineering and Applied Science, University of California, Los Angeles, US.

Symmetry Discovery Meta-Reinforcement Learning Generalization in RL Geometric Representation Learning

Dec 13, 2023

Partial Symmetry Detection for 3D Geometry using Contrastive Learning with Geodesic Point Cloud Patches

Detecting partial extrinsic symmetry in 3D geometry is a fundamental yet persistent challenge in computer vision and graphics, critical for tasks ranging from shape completion to procedural generation. Classical transformation-space voting methods rely on pairwise matching, scaling as O(n^2) and struggling to resolve coherent multi-instance groups. Recent learning approaches advance global symmetry detection but restrict the solution space to reflection planes, failing to capture rotational or translational repetitions such as the legs of a chair or the steps of a staircase. We propose SymCL, a self-supervised contrastive learning framework that detects partial symmetries across rotation, translation, and reflection (with scale-invariant features) and requires no ground truth annotations. By mapping local geodesic patches to a latent space invariant to the Euclidean group, we reformulate symmetry detection as a density-based clustering problem, enabling the simultaneous discovery of multi-instance symmetric relationships in a single forward pass. We evaluate quantitatively on SymPartNet, a new benchmark annotating all PartNet categories with partial symmetry relations, and demonstrate class-agnostic generalization qualitatively on everyday objects outside the training distribution.

Gregor Kobsik, Isaak Lim, Leif Kobbelt

Visual Computing Institute, RWTH Aachen University, Germany

Point Cloud Learning Geometric Representation Learning Symmetry Discovery