Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the Poisson's equation, which has a subalgebra isomorphic to the dimensional special Euclidean group or group of rigid motions of . Looking the adjoint representation of ${\rm SE}(3)…
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48 results for “Lie group SE(2)”
Study on Langevin dynamics on planar motion group, highlighting geometric mechanism.
problem Understanding Langevin dynamics on the Lie group SE(2).
method Intrinsic formulation on SE(2), using invariant vector fields and natural projection.
result Effective macroscopic diffusion on R^2 emerges through averaging.
On dual quaternions, dual split quaternions and Cartan-Schouten metrics on perfect Lie groupsmath.DG
The paper explores metrics on Lie groups and their connections to dual quaternions.
problem Understanding metrics on Lie groups and their geometric properties.
method Analyzing Cartan-Schouten metrics on perfect Lie groups and their connections to dual quaternions.
result Biinvariant metrics on perfect Lie groups are shown to be Cartan-Schouten metrics.
Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.
Study cohomological equation for robotic screw motions on SE(3).
problem Understanding obstruction phenomena in robotic rigid-body motion.
method Combining Fourier analysis and Peter-Weyl theory, reduce to finite-dimensional linear transport systems.
result Explicit screw motion illustrates resonance conditions and finite-dimensional obstructions.
Characterizes stably elliptic elements in Lie groups and their properties.
problem Understanding stably elliptic elements in Lie groups and their geometric and algebraic properties.
method Characterization through fixed point algebra and Weyl group action; relates to maximal invariant cones and compactness of order intervals.
result Connected components of stably elliptic elements can be described using Weyl group action on a compactly embedded Cartan subalgebra.
This paper summarizes closed-form relations for SE(3) maps and their derivatives.
problem Closed-form expressions for SE(3) maps and their derivatives are scattered in the literature.
method Summarizes and provides proofs for relevant closed-form relations of the exponential and Cayley map on SE(3).
result Provides an implicit generalized-alpha scheme for rigid/flexible multibody systems using the Cayley map.
The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integra…
Closed-form relations and approximations for SE(3) derivatives for robust numerical simulations.
problem Deriving closed-form derivatives and approximations for SE(3) for robust numerical simulations.
method Avoiding block partitioning, deriving higher-order approximations for differential, first and second derivatives, Jacobian, and Hessian.
result Compact and numerically robust closed-form relations for SE(3) derivatives.
Attention tokens are group elements, negated algebra norms.
problem Attention mechanism for matrix Lie groups
method Lie-Algebra Attention
result Score matches learned kernel on invariant poses, outperforms vector-token baseline
Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometrymath.OC
The paper solves optimal control problems for various convex sets using convex trigonometry.
problem Optimal control problems with 2D convex compact sets.
method Using convex trigonometry to derive extremals for various problems.
result Geodesics in multiple sub-Finsler problems are derived.
The paper studies symmetry reduction and optimal control on Riemannian manifolds.
problem Symmetry reduction and optimal control on Riemannian manifolds.
method Derivation of reduced equations of motion for variational problems on Lie groups and Riemannian homogeneous spaces.
result Derivation of geodesic equations and reduced equations of motion for various applications.
Researchers find metric lines in SE(2) using Hamilton-Jacobi theory.
problem Identifying metric lines in the Special Euclidean group on the plane.
method Alternative proof using Hamilton-Jacobi theory.
result Metric lines in SE(2) are identified.
We propose a notion of distance between two parametrized planar curves, called their discrepancy, and defined intuitively as the minimal amount of deformation needed to deform the source curve into the target curve. A precise definition of discrepancy is given as follows. A curve of transformations in the special Eucli…
We show that an equivariantly embedded Hermitian symmetric space in a projective space, which contains neither a projective space nor a hyperquadric as a component, is characterized by their fundamental forms as a local submanifold of the projective space. Using some invariant-theoretic properties of the fundamental fo…
Algorithm finds connected components on Lie groups for multi-orientation image analysis.
problem Finding connected components in complex, multi-orientation images.
method Iterative algorithm using morphological dilations and Hamilton-Jacobi-Bellman kernels on Lie groups.
result Algorithm converges in finitely many steps and can differentiate between crossing and aligned structures.
Noether's First Theorem yields conservation laws for Lagrangians with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. In recent work the authors showed the mathematical structure behind both th…
A new diffusion model generates novel protein backbones without relying on pretrained networks.
problem Generating novel protein backbones without relying on pretrained networks.
method Developed a SE(3) invariant diffusion model on multiple frames, called FrameDiff.
result Generated designable protein monomers up to 500 amino acids without pretrained networks.
An iterative SE(3)-Transformer model is developed for graph data.
problem Manipulating three-dimensional data with rotational and translational symmetries.
method Iterative SE(3)-equivariant attention mechanism applied to graph data.
result Iterative SE(3)-Transformer outperforms single-pass version on a toy problem.
Study uses multidimensional SE-NBD process to analyze default portfolios and identify shock amplification.
problem Analyzing interactions and shock propagation in default portfolios with multiple sectors.
method Applied multidimensional self-exciting negative binomial distribution (SE-NBD) process to 13 sectors.
result Identified upstream and downstream sectors, showing shock amplification in default portfolios.
Speech enhancement (SE) aims to reduce noise in speech signals. Most SE techniques focus only on addressing audio information. In this work, inspired by multimodal learning, which utilizes data from different modalities, and the recent success of convolutional neural networks (CNNs) in SE, we propose an audio-visual de…
Speech enhancement (SE) aims to reduce noise in speech signals. Most SE techniques focus only on addressing audio information. In this work, inspired by multimodal learning, which utilizes data from different modalities, and the recent success of convolutional neural networks (CNNs) in SE, we propose an audio-visual de…
Feature Selection with the R Package MXM: Discovering Statistically-Equivalent Feature Subsetsstat.ML
The statistically equivalent signature (SES) algorithm is a method for feature selection inspired by the principles of constrained-based learning of Bayesian Networks. Most of the currently available feature-selection methods return only a single subset of features, supposedly the one with the highest predictive power.…
Proposes SE(3) equivariant graph neural networks with local frames for efficient geometric approximation.
problem Equivariance in deep learning for arbitrary transformations, especially in physics.
method Introduces SE(3) equivariant graph neural networks with complete local frames to efficiently approximate geometric quantities.
result Achieves best or competitive performance in Newton mechanics modeling and equilibrium molecule conformation generation.
This paper develops a new method for constructing splines on Lie groups using Poisson equation solutions.
problem Existing methods for constructing splines on Lie groups have limitations and assumptions that may not reflect actual curves.
method The paper introduces a new approach using solutions of the Poisson equation on Lie groups to construct splines.
result The new method allows for global splines with arbitrary initial conditions, improving curve reconstruction.
Study on geodesic distances on SE(3)/SO(2) in machine learning.
problem Investigating the efficiency of computationally efficient sections in selecting geodesic distances.
method Analyzing geodesic distances on reductive homogeneous spaces, proving the efficiency of minimal distance sections.
result Minimal distance sections are not always geodesic minimizers, but minimal horizontal geodesics are.
Smooth SE structures on Sasaki-joins and Bott orbifolds constructed.
problem Constructing smooth Sasaki-Einstein structures on Sasaki-joins.
method Categorical relationship between Sasaki-joins and Bott orbifolds, explicit construction.
result Infinitely many smooth SE structures up to dimension eleven, conjecture for all odd dimensions.
Proposes HypCSE for enhanced hierarchical clustering.
problem Challenges in existing hierarchical clustering methods.
method Hyperbolic Continuous Structural Entropy (HypCSE) neural networks.
result Superior performance on seven datasets.
Group equivariant neural networks simplify complex tasks with group representation theory.
problem Challenging tasks requiring input transformations like rotations.
method Group representation theory, non-commutative harmonic analysis, differential geometry.
result A neural network is group equivariant if and only if it has a convolutional structure.
Sep-SpectralNet improves SE for broader applicability and scalability.
problem Three main drawbacks of current SE implementations: generalizability, scalability, and eigenvectors separation.
method Sep-SpectralNet extends SpectralNet with an eigenvector separation post-processing step.
result Sep-SpectralNet achieves consistent SE approximation and generalization, enhancing scalability and applicability.
Singularity, Sasaki-Einstein manifold, Log del Pezzo surface and AdS/CFT correspondence: Part Ihep-th
A five dimensional Sasaki-Einstein (SE) manifold provides a AdS/CFT pair for four dimensional SCFT, and those pairs are very useful in studying field theory and AdS/CFT correspondence. The space of known SE manifolds is increased significantly in the last decade, and we initiated the study of various fi…
Improving speech system performance in noisy environments remains a challenging task, and speech enhancement (SE) is one of the effective techniques to solve the problem. Motivated by the promising results of generative adversarial networks (GANs) in a variety of image processing tasks, we explore the potential of cond…
One of the ways to train deep neural networks effectively is to use residual connections. Residual connections can be classified as being either identity connections or bridge-connections with a reshaping convolution. Empirical observations on CIFAR-10 and CIFAR-100 datasets using a baseline Resnet model, with bridge-c…
SE(3)-Transformers maintain equivariance for 3D data under rotations and translations.
problem Ensuring stable and predictable performance in 3D data under transformations.
method Introducing a self-attention module that is equivariant under continuous 3D roto-translations.
result The SE(3)-Transformer outperforms non-equivariant and non-attention models on real-world datasets.
Tiled Squeeze-and-Excite improves channel attention with local spatial context.
problem Improving channel attention mechanisms in neural networks.
method Proposes tiled squeeze-and-excite (TSE) framework for channel attention.
result Local context of 7 rows or columns is sufficient for matching global context performance.
We prove a general extrinsic rigidity theorem for homogeneous varieties in . The theorem is used to show that the adjoint variety of a complex simple Lie algebra (the unique minimal G orbit in ) is extrinsically rigid to third order. In contrast, we show that the ad…
EDGI improves sample efficiency and generalization in tasks with spatial and temporal symmetries.
problem Sample inefficiency and poor generalization in tasks with geometric symmetries.
method Equivariant Diffuser framework, SE(3)xZxSn-equivariant diffusion model.
result EDGI is more sample efficient and generalizes better than non-equivariant models.
SE-RRMs solve structured problems like Sudoku and ARC-AGI by enforcing symbol equivariance.
problem Structured reasoning problems like Sudoku and ARC-AGI.
method Symbol-equivariant recurrent reasoning models enforcing permutation equivariance.
result SE-RRMs outperform prior RRMs on 9x9 Sudoku and generalize to larger and smaller instances.
Most recent studies on deep learning based speech enhancement (SE) focused on improving denoising performance. However, successful SE applications require striking a desirable balance between denoising performance and computational cost in real scenarios. In this study, we propose a novel parameter pruning (PP) techniq…
A new metric mav offers a practical alternative to costly Riemannian distance.
problem Efficiently compute Riemannian distance on SE(3) invariant metrics.
method Propose mav distance, defined as Riemannian length of a curve.
result Mav distance offers a trainable invariant for geometric deep learning.
P-SE explains model decisions with minimal feature subsets and fast estimators.
problem Explain model decisions in regression and classification.
method Probabilistic Sufficient Explanations (P-SE) with random Forests for conditional probability estimation.
result Consistent and efficient explanations for regression and classification models.
Artificial neural network (ANN) provides superior accuracy for nonlinear alternating current (AC) state estimation (SE) in smart grid over traditional methods. However, research has discovered that ANN could be easily fooled by adversarial examples. In this paper, we initiate a new study of adversarial false data injec…
We use self-report and electrodermal activity (EDA) wearable sensor data from 77 nights of sleep on six participants to test the efficacy of EDA data for sleep monitoring. We used factor analysis to find latent factors in the EDA data, and causal model search to find the most probable graphical model accounting for sel…
Replication study shows Deep-SE still not as effective as previously thought for agile effort estimation.
problem Improving accuracy in estimating agile software development effort.
method Close replication of Deep-SE using additional data and comparison with multiple baselines.
result Deep-SE outperforms only a few cases, suggesting more work is needed.
New neural network processes 3D volumes with improved equivariance.
problem Improving neural network performance on 3D volumes with symmetries.
method Equivariant neural network using moving frames approach.
result Trained model outperforms benchmarks in medical volume classification.
MAPS and MAPS-SE learn from multiple suboptimal experts to improve policies efficiently.
problem Learning from multiple suboptimal experts in reinforcement learning.
method Active policy improvement from multiple black-box oracles.
result MAPS and MAPS-SE achieve sample efficiency and state-wise imitation learning.
Improved scaffold generation for protein motifs using SE(3) flow matching.
problem Lack of structural diversity in generated scaffolds for protein motifs.
method Extended FrameFlow for motif-scaffolding with motif amortization and motif guidance.
result 2.5 times more designable and unique motif-scaffolds compared to state-of-the-art.
Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?physics.soc-ph
A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent close to 3. We revisit this results and use standard tests as well as develop a battery of new non-parametric and parametric tests (in particular with stretche…