Lie Group Methods

Follow this topic Ask about this topic

Momentum

5 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

Jul 13Week of Sep 28

Latest papers 28

Sep 27, 2026cs.SC

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

Sep 15, 2026cs.RO

LiLi: Lie Theory Based 3D LiDAR Scan Alignment Degeneracy Detection

In this paper, we study 3D LiDAR scan alignment in challenging scenarios with degeneracies, such as straight corridors or flat fields, where the alignment solution is not unique and compromises localization and mapping accuracy. Existing degeneracy detection methods that neglect the potential for reassociating data points are prone to being sensitive to noise and complex degeneracies. Therefore, we propose LiLi - a novel method that leverages Lie theory to identify the full set of degenerate transformations within the SE(3) Lie group of rigid transformations. The method employs perturbations of the optimized solution and compares the resulting optimized poses to ensure robust detection of degeneracies. By leveraging generators from the Lie algebra se(3), the method provides a systematic approach to describing the set of degenerate transformations. Quantitative evaluations on synthetic data show significant improvement over the state-of-the-art Hessian-based method, reducing alignment error by 50%, with more significant improvements for datasets featuring noise. In the real-world degenerate datasets, the proposed method integrated into LiDAR-based odometry yields superior localization performance compared to the reference solution based on the Hessian-based degeneracy detector on a 260 m long trajectory, and succeeds on a 430 m long round-trip tunnel trajectory where the reference fails.

Vsevolod Hulchuk, Jan Bayer, Jan Faigl

Faculty of Electrical Engineering, Czech Technical University, Technická 2, 166 27, Prague, Czech Republic

LiDAR Odometry Lie Group Methods Point Cloud Registration LiDAR Localization

Sep 14, 2026cs.RO

LieSpline-DP: Lie-Group B-Spline Diffusion Policy for Smooth Robot Manipulation

Diffusion Policy (DP) is a powerful Learning from Demonstration (LfD) method for robotic manipulation, yet it suffers from discontinuous and non-smooth trajectories. Spline-based action representations promote smooth motion within individual action chunks, but existing spline-based methods neither guarantee cross-chunk

C^2

continuity nor account for the group structure of

\mathrm{SE}(3)

. We therefore propose LieSpline-DP, a Lie-group B-spline diffusion policy that generates end-effector trajectories directly on

\mathrm{SE}(3)

and couples consecutive plans by sharing their boundary control poses, ensuring

C^2

continuity throughout the entire planned trajectory. Across three real-robot tasks, LieSpline-DP produces lower trajectory jerk and higher task success rates than the DP baseline. The gains are particularly pronounced in real-world tasks involving liquids and flexible objects: in our real-robot experiments, LieSpline-DP achieved a 100% success rate on both pouring and bucket hooking, whereas the DP baseline achieved only 10% and 30%, respectively.

Erxuan Xie, Bang Liu, Pingyun Nie +3

Shanghai Jiao Tong University

Trajectory Generation Diffusion Policy Lie Group Methods Robot Motion Generation Robotic Manipulation Robot Imitation Learning

Sep 14, 2026cs.LG

$\mathbb{SL}(n)$ Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition

Mixed-curvature representation learning seeks to capture rich geometric structures that cannot be adequately modeled by a single curvature regime. Existing approaches largely rely on product manifolds, which require manually specifying how different curvature spaces are combined and separate their curvature contributions across factors. We introduce the

\mathbb{SL}(n)

space, a representation geometry defined by the simple

\det(A)=1

constraint and a left invariant Schatten-

p

Finsler structure. Despite this minimal construction,

\mathbb{SL}(n)

exhibits pointwise negative, zero, and positive flag curvature around a common flagpole, while its mixed-curvature and curvature-coupling capacities are asymptotically maximal relative to the intrinsic geometric upper bound. Beyond geometry, its noncommutative group structure provides inherent order sensitivity, and its non-nilpotent Lie algebra admits nonzero nested Lie brackets at arbitrary depth, enabling deep order-aware composition. Empirically,

\mathbb{SL}(n)

consistently outperforms a broad range of representation manifold baselines across graph benchmarks at different scales. It reduces average distortion over the strongest baselines by

44.3\%

on KEGG and

40.5\%

on HumanCyc, and improves Hits@20 by

42.8\%

on OGBL-PPA. Experiments on Flickr30k-Order further support its ability to capture higher order dependencies from ordered composition. Together, these results show how a seemingly simple structural constraint can yield unexpectedly rich geometry, capacity, and composition within a unified representation space.

Xingrun Li, Yusuke Mukuta, Xin Yang +2

The University of Tokyo · Stanford University

Graph Representation Learning Geometric Representation Learning Lie Group Methods

Sep 14, 2026cs.RO

Trajectory Bundle Method in SE(3) for Black-Box Fixed-Wing Aircraft Trajectory Optimization

Dynamically feasible trajectory optimization for rigid-body systems is naturally formulated on the special Euclidean group SE(3) but is challenging when dynamics are available only as black-box computations without derivatives. This paper formulates the Trajectory Bundle Method (TBM) for motion planning implicitly on SE(3). Bundles are constructed in the Lie algebra and propagated through nonlinear rigid-body dynamics using exponential and logarithmic maps, enabling derivative-free planning of non-Euclidean trajectories. We show that Euclidean TBM interpolation error is bounded quadratically by bundle diameter and extend this result to SE(3), where the bound additionally depends on a local Lipschitz constant of the Log map. Numerical experiments corroborate these bounds. Finally, we demonstrate SE(3) TBM by optimizing an acrobatic, collision-free fixed-wing maneuver through a rotated aperture without explicit models or derivatives of the vehicle dynamics, aerodynamics, or collision model.

Matthew D. Osburn, Cameron K. Peterson, John L. Salmon

Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602, USA · Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA

Trajectory Optimization Trajectory Generation Black-Box Optimization Lie Group Methods Motion Planning Derivative-Free Optimization

Sep 2, 2026stat.ML

Schrödinger Bridges on Lie Group Manifolds for Probabilistic Intrinsic Generation

Generative modeling directly on geometric manifolds can avoid errors introduced by flattening non-Euclidean data, repeated ambient projection, and coordinate inconsistency in Euclidean representations. Schrodinger bridges provide a probabilistic generative framework for entropy-regularized transport between prescribed endpoint distributions. We study Schrodinger bridges for kinetic dynamics on Lie group manifolds with state X_t = (g_t, xi_t) in G x g, allowing endpoint observations to constrain only the variables that are actually measured. In particular, the entropy projection determines the conditional law of the unobserved endpoint velocities. For the same observed endpoint bridge, we develop two computational realizations: Wrapped-Kernel Bridge Calibration (WKBC) uses an explicit periodized kinetic kernel on compact Abelian groups, whereas Reciprocal Conditional-Control Bridge Matching (RCCBM) handles compact non-Abelian groups through two-sided endpoint calibration and mollified conditional-control matching. The canonical teacher-mixture path law is itself a Markov reciprocal law, so forward generation uses a calibrated initial law and one learned Doob controller. Moreover, we establish a modular error bound in the bounded-Lipschitz path metric that provides a clean separation of errors due to endpoints, control regression, initialization, discretization, and related approximations. Experiments on multiple Lie group manifold datasets validate the feasibility and consistency of our proposed method, covering protein and RNA torsions, SO(3), U(n), and the Protein Conformational Transition Pathway Generation task using mdCATH trajectories in a compact reduced representation. The source code is publicly available at https://github.com/cafferyzhang12/Schr-dinger_Bridge_on_LieGroup.

Shizhe Zhang, Mingyang Zhao, Lei Ma

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

Schrödinger Bridges Generative Modeling Lie Group Methods Molecular Generation

Aug 3, 2026cond-mat.stat-mech

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 \textit{LieStoNet}, an end-to-end, \emph{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

Jul 29, 2026cs.LG

SE(3)-MeanFlow: Few-Step Protein Backbone Generation on Lie Groups

Generative modeling of protein backbones promises the de novo design of proteins with prescribed structural and functional properties. Existing diffusion and flow-matching models produce high-quality backbones on SE(3)^N, but inference requires numerically integrating an ODE over hundreds of network evaluations, each involving a Lie group exponential map - a bottleneck for high-throughput design campaigns. We introduce SE(3)-MeanFlow, a few-step generative framework that extends MeanFlow from Euclidean space to the Lie group geometry of protein frames. Working natively in the Lie algebra so(3) and in R^3, we derive closed-form average-velocity identities for rotations and translations, giving simulation-free training targets. We further introduce an SE(3) alpha-Flow objective that removes the Jacobian-vector product from the rotation branch and serves as a warm-up stage, after which training switches to a small-t stabilized MeanFlow loss that is used for the remainder of pretraining and for rectification-based post-training. In protein backbone generation, SE(3)-MeanFlow matches or exceeds flow-matching baselines that use several times more sampling steps, and its advantage widens in the few-step regime, where rectification lets it lead at every matched budget - at a modest cost in diversity.

Yikun Bai, Binghang Lu, Yikai Liu +7

Department of Mathematics, Purdue University · School of Electrical and Computer Engineering, Purdue University · Department of Biochemistry, Purdue University +5

Flow Matching Lie Group Methods Generative Modeling 3D Molecular Generation Mean Flow Molecular Generation+2

Jul 28, 2026cs.CV

Quasi-SVD: Learning a Lie-constrained matrix factorisation for real-time imaging

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in imaging pipelines. This study introduces a fully parallelized matrix factorization framework for GPUs by enforcing matrix orthogonality on left singular vectors via Lie-parametrised algebra and recovering the remaining components through soft constraints. This asymmetric constraint design enables an efficient parallel and provably valid decomposition, achieves high reconstruction fidelity and substantially accelerates computation relative to the exact SVD, with real-time throughput exceeding standard video frame rates. Performance is evaluated on multiple imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices, and (3) an MNIST denoising benchmark at an intermediate scale with known ground truth. Across regimes and instruments, the proposed framework demonstrates robust domain transfer at various matrix scales, sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, the proposed SVD framework makes structured matrix factorisation practical for real-time processing.

Christopher Hahne

University of Bern 3008 Bern, Switzerland

Lie Group Methods Computational Imaging Singular Value Decomposition Matrix Factorization

Jul 27, 2026cs.RO

A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. Inmultibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group SO (3)R3, although rigid body motions form the semidirect product Lie group SE (3). It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of SE (3) have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that SE (3) is the appropriate cspace for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

A. Müller

Institute of Robotics, Johannes Kepler University, Altenbergerstr. 69, 4040 Linz, Austria

Dynamical Systems Multibody Dynamics Lie Group Methods

Jul 14, 2026cs.LG

Learning Forced Multibody Dynamics on Lie Groups

We propose an architecture for learning the dynamics of mechanical systems based on discrete forced Euler-Lagrange equations on Lie groups using only position data. By formulating the dynamics directly on manifold-valued configuration spaces, the method naturally respects the geometric structure of the systems and preserves geometric invariants and conservation laws. The reliance on position measurements alone makes the framework applicable in settings where velocity data are unavailable or noisy. The approach extends naturally to multibody systems, accommodates external control inputs, and demonstrates strong performance on both synthetic and real-world datasets.

Martine Dyring Hansen, Marta Ghirardelli, Elena Celledoni +2

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

Robot Dynamics Identification Dynamical Systems Multibody Dynamics Nonlinear System Identification Lie Group Methods

Jul 9, 2026cs.RO

Learning Adaptive Solvers for Distributed Factor Graph Optimization on Matrix Lie Groups

Modern robotic perception increasingly involves large-scale geometric optimization problems distributed across multiple robots or sessions. However, existing distributed solvers often depend on brittle hand tuning and primarily target rigid body pose graphs. To address this, we present DeepCORD, a learning-augmented framework for distributed factor graph optimization on general matrix Lie groups. By unfolding a parallel and accelerated Riemannian optimizer into differentiable iterations, DeepCORD learns a self-supervised feedback policy that dynamically adapts solver parameters according to the optimization phase and communication status. The resulting method enables adaptive distributed optimization over matrix Lie groups under both synchronous and asynchronous communication regimes. Extensive experiments on real-world

\mathrm{SE}

(3) pose graph optimization and

\mathrm{SL}

(4) projective submap alignment show that our method achieves lower objective values than existing distributed baselines on most benchmarks across realistic operating scenarios.

Jaeho Shin, Maani Ghaffari, Yulun Tian

University of Michigan

Factor Graph Optimization Riemannian Optimization Learning to Optimize Pose Graph Optimization Lie Group Methods

Jul 3, 2026cs.RO

Derivations of Error-State Kalman Filter Kinematics for Globally Applicable Aided Inertial Navigation Systems

Global navigation systems require state estimation algorithms that handle Earth's curvature, Earth's rotation, and gravitational variations. These factors can typically be neglected in local navigation algorithms for robots, drones, etc. In classical error-state Kalman Filtering (ESKF) the error state dynamics are trajectory-dependent. Invariant ESKFs utilize Lie Group symmetries to represent the error, which can render error propagation trajectory-independent for group-affine systems. Choosing between a standard filter (where position and velocity errors are defined additively in the navigation frame), a left-invariant filter (where errors are represented in the body frame) and a right-invariant filter (where errors are represented in the navigation/world frame) depends on system dynamics and sensor configuration. This note presents the mathematical formulas for four classical and invariant ESKFs for globally applicable aided inertial navigation systems. It is intended to serve as a systematic reference for comparison and implementation.

Antonia Hager, Torleiv H. Bryne

Airbus Central Research and Technology, Taufkirchen, Germany · Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway

Kalman Filtering Lie Group Methods Invariant EKF Inertial Navigation

Jun 18, 2026cs.LG

The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups

We place the attention token on the group: a token is an element

g_i

of a matrix Lie group

G

-- a bare transformation, with no feature payload and no external action

ρ(g)

carrying it. To our knowledge this is the first attention construction whose tokens are bare matrix Lie group elements: their score is the closed-form algebra norm of the relative pose rather than a learned kernel, and it reaches the affine full-frame groups that every irrep- or surjective-exp-based method must exclude. We call it Lie-Algebra Attention. Once tokens are group elements, the rest follows with none of the usual representation-theoretic machinery. The relative geometry of a pair is canonical,

g_i^{-1} g_j

, so the pairwise invariant

w_{ij} = \log(g_i^{-1} g_j)

is intrinsic rather than designed; equivariance under the diagonal

G

-action is tautological, and the cocycle condition holds automatically. The attention score is the negative squared algebra norm,

s_{ij} = -\|\log(g_i^{-1} g_j)\|_λ^2/τ

: the canonical proximity kernel under a block-weighted Frobenius inner product, with no irreducible representations, spherical harmonics, Clebsch-Gordan products, or learned kernel. The construction applies to any matrix Lie group on a chosen logarithm chart containing the relative poses, including the non-compact non-abelian affine groups with scale and shear that no vector-token attention method reaches: neither the irrep tradition nor surjective-exp methods. Three sequence-completion experiments, on SE(2), SO(3), and Aff(2), bear this out: the closed-form score matches a learned MLP kernel on the same invariant and outperforms it on SE(2), using 50 to 80x fewer score parameters, while a vector-token baseline breaks invariance by five to twelve orders of magnitude.

Przemyslaw Musialski

New Jersey Institute of Technology

Lie Group Methods Attention Mechanisms

Jun 17, 2026cs.LG

What Your Model Threw Away and Why You'll Want It Back: Masking, Fingerprinting, and Privacy from Discarded Geometry

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation

π

of a Lie group

G

on a space

V

and a learned function

f\colon V \to \mathbb{R}

, we define two objects measuring the symmetry invisible to

f

. The null fiber at a point

x \in V

is the set

N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}

of group elements whose inverse action on

x

is undetectable by

f

. When

N_G(f,x)

is independent of

x

, it coincides with the stabilizer

\mathrm{Stab}_G(f)

, the largest subgroup of

G

under which

f

is invariant. For smooth maps to

\mathbb{R}

, the preimage theorem guarantees that null fibers have dimension at least

\dim G - 1

at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of

f

. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under

\mathrm{SO}(3)

and spherical image classification under the Möbius group

\mathrm{PSL}(2, \mathbb{C})

. The framework applies uniformly to classical neural networks and variational quantum circuits.

Zachary P. Bradshaw

QodeX Quantum, Inc. Chicago, IL

Geometric Representation Learning Lie Group Methods Privacy-Preserving ML Invariant Representation Learning

Jun 13, 2026cs.LG

LieBN: Batch Normalization over Lie Groups

Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left- and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Extensive experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Ziheng Chen, Yue Song, Rui Wang +2

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

Geometric Representation Learning Lie Group Methods

Jun 9, 2026cs.RO

LieIPM: Lie Group Interior Point Method for Direct Trajectory Optimization of Rigid Bodies

Designing dynamically feasible trajectories for rigid bodies is a fundamental problem in robotics. While direct methods are widely used, the existing constrained optimizers typically operate in Euclidean space and ignore the manifold structure of rigid body motions. This mismatch may introduce singularities or lead to poorly conditioned optimization problems. To bridge this gap, we develop a structure-aware framework for constrained trajectory optimization directly on matrix Lie groups. Our approach is based on the second-order rigid body models utilizing Lie group structures, which enables efficient Newton-type updates while preserving the underlying geometry. Building on this model, we propose a line-search Lie Group Interior Point Method (LieIPM) to handle constraints on the manifolds. We instantiate the framework for rigid body motion planning using Lie group variational integrators and derive closed-form intrinsic derivatives that exploit group symmetries. The LieIPM preserves the topology of rotation motions by construction and avoids singularities. Numerical results demonstrate superior robustness and faster convergence compared to general-purpose solvers and structure-exploiting optimal control methods.

Sangli Teng, Ruiqi Zhang, Tzu-Yuan Lin +5

University of California, Berkeley · MIT · Ohio University +1

Trajectory Optimization Riemannian Optimization Lie Group Methods Constrained Optimization Robot Motion Planning

Jun 9, 2026cs.RO

Improved Representation of Matrix Lie Group Operations through Tensor Notation

Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.

Clark Taylor

Air Force Institute of Technology WPAFB, OH, USA 45433 · WPAFB, OH, USA 45433

Lie Group Methods

Jun 3, 2026cs.RO

Efficient Computation of Distance Functions for Navigation Vector Fields in Lie Groups

Vector-field-based methods are widely used for robot control and are often applied to the path-tracking problem. Some vector field approaches require repeatedly computing the distance between the robot configuration and the curve, as well as the corresponding closest point. Recently, vector fields have been extended to Lie Groups. In this case, this computation can be expensive, especially when performed at high control frequencies on embedded platforms. This paper proposes a method for efficiently computing the distance between a point and a curve represented as what is called a G-polynomial curve, which is a curve representation that generalizes polynomial curves to matrix Lie groups. The proposed approach exploits the structure of these curves to reduce the problem to a small number of polynomial root-finding computations. Simulation results show that the method significantly reduces computation time while maintaining accuracy compared to existing optimization-based approaches. Practical formulas are also provided for the case of the group SE(3), and the method is validated experimentally on a robotic manipulator. The methodology is implemented in a computational package, available online.

Vinicius M. Gonçalves, João Baião, Felipe Bartelt +4

Robotic Control Robotics Lie Group Methods

Jun 1, 2026cs.RO

The Lie We Tell: Correcting the Euclidean Fallacy in Vision Language Action Policies via Score Matching on Tangent Space

Diffusion-based Vision-Language-Action policies achieve remarkable success in robotic manipulation, yet commit a fundamental geometric error we term the

\textbf{Euclidean Fallacy}

: representing SE(3) poses as flat

\mathbb{R}^{12}

vectors. This approximation induces (1) manifold drift violating SO(3) constraints, (2) broken equivariance under coordinate transformations, and (3) non-geodesic trajectories with excessive kinematic cost. We introduce

\textbf{Lie Diffuser Actor (LDA)}

, a diffusion framework operating intrinsically on SE(3). Our method injects noise through left-invariant SDEs, predicts scores in the tangent space, and retracts samples via the exponential map. This formulation eliminates manifold drift by construction while guaranteeing coordinate-frame equivariance and geodesic optimality. On CALVIN ABC

\rightarrow

D, LDA improves average task length from

3.27

to

3.51

(

+7.3\%

). We further validate our method on real robot and the results show that our methodology outperforms the baseline on majority tasks.

Bing-Cheng Chuang, I-Hsuan Chu, Bor-Jiun Lin +3

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.

Lie Group Methods Robotic Manipulation Vision-Language-Action Models Diffusion Models

May 24, 2026cs.LG

Planning Neural Dynamics with Lie Group Embedding through Supervised Projective Manifold Learning

We propose Lie group embedded dynamical neural networks (LieEDNN) and the corresponding learning algorithms based on gradient descent and metric projection on smooth manifold, where we treat Lie group as an intrinsic representation for continuous symmetry of manifold geometry. Thereby we achieve learnable and stable dynamics on the underlying manifold for general Lie group, and we are able to utilize the powerful representation capability of Lie group such as SO(3) and SE(3) to solve real world engineering problems in areas such as robotics, graphics, and control. Two core challenges are: (i) General Lie groups are incompatible with addition arithmetic, which is necessary for neural network interactions. (ii) The dynamics evolve in the nonlinear representation space of special algebra rather than the normal Euclidean space, which violates the paradigm of common neural ODEs. To address these two challenges, we firstly introduce adjoint Lie group action on the Lie algebra, which induces a linear mapping and transfer to the block-wise structure of weight matrices, such that addition could operate on the Lie algebra as a vector space. Then we parameterize the Lie algebra and the adjoint action as linear transformation so that the architecture is aligned with neural network perceptrons. Explicitly, this embedding appears as block-wise manifold constraints on weights, and we develop algorithms to learn the equilibrium with stability guarantees of the temporal neural network dynamics. Experiments are implemented on a specific Lie group SE(3), with the application scenario of telescopic manipulators.

Tianwei Wang, Bryan Chen, Qian Zuo +3

School of Informatics University of Edinburgh · Bio-inspired Computing and Machine Learning (BCML) Lab · School of Mathematics University of Edinburgh +2

Dynamic Neural Networks Equivariant Neural Networks Lie Group Methods

May 21, 2026cs.RO

SE3Kit: A Lightweight Python Library for Specialized Geometric Primitives in Robotics

The Python robotics ecosystem faces a challenge: while many libraries exist for rigid body transformations, few are both lightweight and mathematically strict. This paper introduces SE3Kit, a lightweight Python library efficient operations on the Special Euclidean Group SE(3) and the Special Orthogonal Group SO(3). Unlike established frameworks that require heavy dependencies (e.g., SpatialMath, PyPose) or general tools that lack robotics-specific features (e.g., SciPy), SE3Kit targets the gap between these extremes. It is designed for embedded deployment, rapid prototyping, and education while providing rigorous mathematical implementation. It provides a pure-Python, NumPy-only implementation of Lie Group operations, without the overhead of deep learning or other visualization software.

Daniyal Maroufi, Omid Rezayof, Farshid Alambeigi

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.

Robotics Lie Group Methods

May 19, 2026cs.LG

Learning Orthonormal Bases for Function Spaces

Infinite-dimensional orthonormal basis expansions play a central role in representing and computing with function spaces due to their favorable linear algebraic properties. However, common bases such as Fourier or wavelets are fixed and do not adapt to the structure of a given problem or dataset. In this paper, we aim to represent these bases with neural networks and optimize them. Our key idea is that any target infinite-dimensional orthonormal basis can be viewed either as a point on the Lie manifold of the orthogonal group, or equivalently, as the endpoint of a continuous path on that manifold that connects a reference basis, e.g. Fourier, to that target. Paths on the Lie manifold satisfy ordinary differential equations (ODEs) governed by skew-adjoint integral operators. Using neural networks to define finite-rank generators of such ODEs allows us to parameterize and optimize orthonormal bases in function space. While relying on finite-rank generators to model infinite operators might seem restrictive, we prove a universality result: even with a rank-2 generator, the integrated solutions of the ODE are dense in the orthogonal group under the appropriate operator topology. In other words, for any target orthonormal basis, there exists a path originating from a reference basis and driven by finite-rank generators that gets arbitrarily close to that target basis. We demonstrate the flexibility of our framework by transforming the Fourier basis into the principal components of a functional dataset, eigenfunctions of linear operators, or dynamic modes of energy-preserving physical simulations.

Hamidreza Kamkari, Mohammad Sina Nabizadeh, Justin Solomon

MIT CSAIL Cambridge, MA

Lie Group Methods

May 17, 2026hep-lat

Noise scheduling and linear dynamics in diffusion models on Lie groups

We investigate the role of the noise schedule in diffusion processes on Lie groups, with particular emphasis on applications to lattice gauge theory. We show that a specific noise schedule leads to a linear decay of the expectation value of the Wilson action as a function of diffusion time. We compare this with Euclidean diffusion models, where such behavior requires an explicitly designed drift term, while in the Lie-group setting it arises naturally.

Javad Komijani

Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

Lie Group Methods High-Energy Physics Diffusion Models

May 11, 2026stat.ML

Interpretable Machine Learning for Spatial Science: A Lie-Algebraic Kernel for Rotationally Anisotropic Gaussian Processes

Many three-dimensional spatial fields are anisotropic, with directions of rapid and slow variation that need not align with the coordinate axes. Standard Gaussian process kernels with Automatic Relevance Determination (ARD) capture only axis-aligned anisotropy, while generic full symmetric positive definite (SPD) metrics can represent rotated anisotropy but do not parameterise principal length-scales and directions directly. We introduce an interpretable rotationally anisotropic GP kernel that parameterises a three-dimensional SPD covariance metric using three principal length-scales and an explicit SO(3) rotation. The rotation is represented by an axis-angle vector and mapped to SO(3) via the Lie-algebra exponential map, giving unconstrained Euclidean coordinates for inference while always inducing a valid SPD metric. The construction spans the same family of three-dimensional SPD covariance metrics as a generic full-SPD parameterisation, but exposes the geometry differently: length-scales and orientation are explicit, interpretable, and directly available for prior specification and posterior summaries. We perform Bayesian inference on these quantities using Markov Chain Monte Carlo (MCMC), and characterise the resulting symmetries and weakly identified regimes. On synthetic data with rotated anisotropy, the posterior recovers the generating metric and improves prediction relative to an axis-aligned ARD baseline, while matching the predictive performance of a generic full SPD baseline. When the ground truth is axis-aligned, posterior mass concentrates near the identity rotation and predictive performance matches ARD. On a material-density dataset from a laboratory-fabricated nano-brick, the inferred metric reveals rotated anisotropy that is not captured by axis-aligned kernels.

Kane Warrior, Dalia Chakrabarty

University of York, Department of Mathematics, York, United Kingdom

Gaussian Processes Lie Group Methods Gaussian Process Regression

May 7, 2026cs.RO

Lie Group Formulation of Recursive Dynamics Algorithms of Higher Order for Floating-Base Robots

In this paper, we describe procedures for computing higher-order time derivatives of the Lie-group Newton-Euler, Articulated-Body Inertia, and hybrid dynamics algorithms for floating-base trees, where the base configuration evolves on SE(3) and the attached mechanism is an open kinematic tree with configuration on the (n1+n2)-dimensional manifold T^{n1} \times R^{n2}, using spatial representation of twists. After presenting the algorithms, we collect the resulting recursions into closed-form equations of motion, identifying an admissible Coriolis matrix satisfying the passivity property, and showing that the articulated inertia tensor remains unchanged across all time derivatives. We then apply the developed methods to a 12-DoF aerial manipulator to derive analytical expressions for its geometric forward and inverse dynamics along with their first time derivatives whereas the numerical simulations successfully evaluate these dynamics up to fifth order. Finally, to demonstrate their practical utility, we benchmark the proposed extensions and show that, in the considered tests, their computational cost scales quadratically with the derivative order, whereas the automatic-differentiation baseline exhibits exponential scaling.

Ahmed Ali, Chiara Gabellieri, Antonio Franchi

Robotics and Mechatronics Department, Electrical Engineering, Mathematics, and Computer Science (EEMCS) Faculty, University of Twente, 7500 AE Enschede, The Netherlands. · Department of Computer, Control and Management Engineering, Sapienza University of Rome, 00185 Rome, Italy.

Multibody Dynamics Robotics Lie Group Methods Inverse Dynamics

May 4, 2026cs.RO

Exact Higher-Order Derivatives for SE(3) via Analytical/AD Methods

Fast prototyping of new SE(3) estimation objectives remains awkward in practice. Modern Lie-group frameworks -- GTSAM, manif, Sophus, SymForce, Ceres -- target first-order workloads through different code-generation and automatic-differentiation strategies, each optimized for a particular seam between hand-derived geometry and generic differentiation. The remaining gap is a compact, AD-safe path from these first-order primitives to exact Hessians, observed-information matrices, and higher-order derivative tensors: the quantities needed for exact Newton steps, observed-information covariance estimates, and covariance correction. This paper presents a hybrid analytical/AD recipe for SE(3) negative log-likelihoods. The practitioner writes the NLL gradient once, generic over a scalar type, and places the analytical/AD seam at the point-action interface y = Tx. Closed-form Lie-group Jacobians are used up to this interface; AD is applied only beyond it. The same source is then instantiated with ordinary floating-point scalars for gradients, vector-seeded dual numbers for exact Hessians in a single forward-mode pass, and nested dual numbers for higher-order derivative tensors. On a representative 6-DoF, 5-landmark SE(3) NLL, the advocated seeded-Hessian path is approximately 5x faster than finite-differencing the AD gradient on this benchmark while matching a nested-AD oracle to machine precision. The implementation adds roughly 70 lines of analytical-Jacobian code over an AD-only baseline. We also identify and fix a removable singularity in the standard SO(3)/SE(3) scalar basis that would otherwise produce NaNs at the origin under seeded AD, and we audit which Lie-group derivative tensors require this stabilized basis. The result is a practical path from rapidly written SE(3) objectives to exact higher-order derivatives, with predictable runtime and no finite-difference tuning.

Frank O. Kuehnel

Sigma Point Labs, Denver, CO, USA

Automatic Differentiation Lie Group Methods Second-Order Optimization Differentiable Optimization

Apr 24, 2026math.GR

Closed Form Relations and Higher-Order Approximations of First and Second Derivatives of the Tangent Operator on SE(3)

The Lie group SE(3) of isometric orientation preserving transformation is used for modeling multibody systems, robots, and Cosserat continua. The use of these models in numerical simulation and optimization schemes necessitates the exponential map, its right-trivialized differential (often referred to as tangent operator), as well as higher derivatives in closed form. The

6\times 6

matrix representation of the differential,

\mathbf{dexp}_{\mathbf{X}}:se\left( 3\right) \rightarrow se\left( 3\right)

, and its first derivative were reported using a

3\times 3

block partitioning. In this paper, the differential, its first and second derivative, as well as the Jacobian and Hessian of the evaluation maps,

\mathbf{dexp}_{\mathbf{X}}\mathbf{Z}

and

\mathbf{dexp}_{\mathbf{X}}^{T}% \mathbf{Z}

, are reported avoiding the block partitioning. For all of them, higher-order approximations are derived. Besides the compactness, the advantage of the presented closed form relations is their numerical robustness when combined with the local approximation. The formulations are demonstrated for computation of the deformation field and the strain rates of an elastic Cosserat-Simo-Reissner rod.

Andreas Mueller

Institute of Robotics, Johannes Kepler University Linz, Austria

Representation Geometry Lie Group Methods