Lie Group Methods

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Lie group methods study transformations such as rotations and rigid-body motions using groups and their tangent Lie algebras, preserving composition and geometric constraints. They support robotics, state estimation, trajectory optimization, simulation, and symmetry-aware learning through tools such as exponential and logarithm maps, invariant metrics, geometric integrators, and manifold optimization. Current approaches also incorporate group structure into neural operators, attention, diffusion models, and symmetry discovery, with tests spanning robot control, pose graphs, physical dynamics, and molecular conformations. A central challenge is to preserve geometric guarantees and global validity while keeping derivatives, optimization, and sampling efficient, particularly near singularities or on complex groups.

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42 papers3 papers in the last four weeks, down 50% on the four weeks before. 0.0% of all new papers.

Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs
Oct 25, 2026Matt L. Wiemman, Peter MelchiorSymmetry Discovery Lie Group Methods
Princeton University
Learning Operators for Symmetry Discovery: From Lie Groups to Local Symmetry Geometry
Oct 12, 2026Pavan Karjol, Scindhiya Laxmi, Prathosh APSymmetry Discovery Operator Learning
Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India · Department of Mathematics and Computing, IIT (ISM) Dhanbad, India
Non-Commutative Lie-Algebraic State-Space Models: Finite-Division Variety Laws, Semidirect Cascades, and Gradient Saddle Escape
Sep 29, 2026Zeyu JiaState Space Models Sequence Modeling
School of Biomedical Engineering and Technology, Tianjin Medical University, Medical School, Tianjin University · School of Biomedical Engineering and Technology, Tianjin Medical University · Medical School, Tianjin University
LieDiscover: Adaptive Symbolic Library Construction for Explicit Open-form Symmetry Discovery
Sep 27, 2026Xinxin Li, Jianming Ma, Xingyu Cui +3Lie Group Methods Symmetry Discovery
East China Normal University · Shanghai Jiao Tong University · Institute of Applied Physics and Computational Mathematics +2
LiLi: Lie Theory Based 3D LiDAR Scan Alignment Degeneracy Detection
Sep 15, 2026Vsevolod Hulchuk, Jan Bayer, Jan FaiglLiDAR Odometry Lie Group Methods
Faculty of Electrical Engineering, Czech Technical University, Technická 2, 166 27, Prague, Czech Republic
LieSpline-DP: Lie-Group B-Spline Diffusion Policy for Smooth Robot Manipulation
Sep 14, 2026Erxuan Xie, Bang Liu, Pingyun Nie +3Robot Motion Generation Diffusion Policy
Shanghai Jiao Tong University
$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition
Sep 14, 2026Xingrun Li, Yusuke Mukuta, Xin Yang +2Finsler Geometry Graph Representation Learning
The University of Tokyo · Stanford University
Trajectory Bundle Method in SE(3) for Black-Box Fixed-Wing Aircraft Trajectory Optimization
Sep 10, 2026Matthew D. Osburn, Cameron K. Peterson, John L. SalmonTrajectory Optimization Trajectory Generation
Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602, USA · Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA
Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation
Sep 2, 2026Shizhe Zhang, Mingyang Zhao, Lei MaSchrödinger Bridges Generative Modeling
1National Biomedical Imaging Center, College of Future Technology, Peking University, Beijing, China · 2State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
Dual Park-Ravani Interpolation of Rigid Motions: Acceleration-Field Continuity and Holonomic Hermite Repair
Aug 28, 2026Daniel ConduracheLie Group Methods
“Gheorghe Asachi” Technical University of Ia¸i, Romania
Arbitrary-Order Hermite Interpolation of Rigid-Motion Jets via Hyper-Multidual Quaternions
Aug 27, 2026Daniel ConduracheLie Group Methods Representation Geometry
Technical University of Iasi, D. Mangeron Street no. 59, 700050 Iasi, Romania · Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Memorandumului 14, 400114 Cluj-Napoca, Romania
LieStoNet: Learning Lie Symmetries from Spatiotemporal Data for Stochastic Dynamical Systems
Aug 3, 2026Shida Liu, Abhishek Gupta, Sumit Sinha +1Stochastic Differential Equations Lie Group Methods
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.
SE(3)-MeanFlow: Few-Step Protein Backbone Generation on Lie Groups
Jul 29, 2026Yikun Bai, Binghang Lu, Yikai Liu +7Flow Matching Lie Group Methods
Department of Mathematics, Purdue University · School of Electrical and Computer Engineering, Purdue University · Department of Biochemistry, Purdue University +5
Quasi-SVD: Learning a Lie-constrained matrix factorisation for real-time imaging
Jul 28, 2026Christopher HahneLie Group Methods Computational Imaging
University of Bern 3008 Bern, Switzerland
A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body
Jul 27, 2026A. MüllerDynamical Systems Multibody Dynamics
Institute of Robotics, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria
Learning Forced Multibody Dynamics on Lie Groups
Jul 14, 2026Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni +2Robot Dynamics Identification Dynamical Systems
Department of Mathematics and Cybernetics, SINTEF Digital Forskningsveien 1, 0373 Oslo, Norway · Department of Mathematical Sciences Norwegian University of Science and Technology (NTNU) Alfred Getz’ vei 1, 7034 Trondheim, Norway · Instituto de Ciencias Matematicas (CSIC) Calle Nicolás Cabrera 13-15, Cantoblanco, 28048 Madrid, Spain
Learning Adaptive Solvers for Distributed Factor Graph Optimization on Matrix Lie Groups
Jul 9, 2026Jaeho Shin, Maani Ghaffari, Yulun TianFactor Graph Optimization Riemannian Optimization
University of Michigan
Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems
Jul 3, 2026Antonia Hager, Torleiv H. BryneKalman Filtering Error-State Kalman Filtering
Airbus Central Research and Technology, Taufkirchen, Germany · Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway
The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups
Jun 18, 2026Przemyslaw MusialskiLie Group Methods Attention Mechanisms
New Jersey Institute of Technology
What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry
Jun 17, 2026Zachary P. BradshawGeometric Representation Learning Lie Group Methods
QodeX Quantum, Inc. Chicago, IL
LieBN: Batch Normalization over Lie Groups
Jun 13, 2026Ziheng Chen, Yue Song, Rui Wang +2Geometric Representation Learning Lie Group Methods
Department of Information Engineering and Computer Science, University of Trento, Trento, Italy · Computing and Mathematical Sciences, Caltech, CA, USA · School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China
LieIPM: Lie Group Interior Point Method for Direct Trajectory Optimization of Rigid Bodies
Jun 9, 2026Sangli Teng, Ruiqi Zhang, Tzu-Yuan Lin +5Trajectory Optimization Riemannian Optimization
University of California, Berkeley · MIT · Ohio University +1
Improved Representation of Matrix Lie Group Operations through Tensor Notation
Jun 9, 2026Clark TaylorLie Group Methods
Air Force Institute of Technology WPAFB, OH, USA 45433 · WPAFB, OH, USA 45433
Efficient Computation of Distance Functions for Navigation Vector Fields in Lie Groups
Jun 3, 2026Vinicius M. Gonçalves, João Baião, Felipe Bartelt +4Robotic Control Robotics
The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space
Jun 1, 2026Bing-Cheng Chuang, I-Hsuan Chu, Bor-Jiun Lin +3Lie Group Methods Robotic Manipulation
Department of Computer Science and Information Engineering & Artificial Intelligence Center of Research Excellence, National Taiwan University, Taipei, Taiwan · Institute of Artificial Intelligence Innovation, National Yang Ming Chiao Tung University, Hsinchu, Taiwan · Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan.
Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning
May 24, 2026Tianwei Wang, Bryan Chen, Qian Zuo +3Dynamic Neural Networks Equivariant Neural Networks
School of Informatics University of Edinburgh · Bio-inspired Computing and Machine Learning (BCML) Lab · School of Mathematics University of Edinburgh +2
SE3Kit: A Lightweight Python Library for Specialized Geometric Primitives in Robotics
May 21, 2026Daniyal Maroufi, Omid Rezayof, Farshid AlambeigiRobotics Lie Group Methods
1 1D. Maroufi, O. Rezayof and F. Alambeigi are with the Walker Department of Mechanical Engineering and Texas Robotics at The University of Texas at Austin, Austin, TX, 78712, USA.
Learning Orthonormal Bases for Function Spaces
May 19, 2026Hamidreza Kamkari, Mohammad Sina Nabizadeh, Justin SolomonLie Group Methods
MIT CSAIL Cambridge, MA
Noise scheduling and linear dynamics in diffusion models on Lie groups
May 17, 2026Javad KomijaniLie Group Methods High-Energy Physics
Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland
Interpretable Machine Learning for Spatial Science: A Lie-Algebraic Kernel for Rotationally Anisotropic Gaussian Processes
May 11, 2026Kane Warrior, Dalia ChakrabartyGaussian Processes Lie Group Methods
University of York, Department of Mathematics, York, United Kingdom

Oct 25, 2026J/K move · Enter open · S save

Data-driven Discovery of Lie Symmetries and First Integrals with Latent ODEs

Matt L. Wiemman, Peter Melchior

Princeton University

Symmetry Discovery Lie Group Methods Neural ODEs Latent Dynamics Modeling

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We present LatentLieODE, a novel neural ODE method to discover linear Lie symmetries and first integrals of the learned dynamics from timeseries observations such as video inputs. It starts with an autoencoder, which learns a low dimensional representation of the observed system and then maps trajectories to decorrelated latent coordinates

z \in \mathbb{R}^d

. A bank of linear infinitesimal generators

\mathbf{A}_k \in \mathbb{R}^{d \times d}

is fitted jointly against a differentiable Lie-bracket residual

\mathbf{J}_f(z)\mathbf{A}z - \mathbf{A} f(z)

, while an MLP optimises the latent ODE

f(z) = \mathrm{d}z/\mathrm{d}t

to reconstruct the observation-space timeseries. After suitable generators and

f

are found, we fit a third network

H_φ

to generate the first integral of the system. Across six benchmark systems (harmonic oscillator, simple and twin pendulums, Lotka-Volterra, pixel pendulum, sphere geodesics) we find each system's relevant symmetries, and the discovered invariants match ground-truth conserved quantities at

|ρ| \geq 0.978

. On the sphere, the pipeline recovers a non-abelian

\mathfrak{so}(3)

, certified by the negative-definite Killing form of its recovered structure constants. This work presents significant advances in data-driven discovery of symmetries and provides a new pathway for neural surrogates of dynamical systems.