Geometric methods study 3-manifold splittings.
problem Studying Heegaard splittings of 3-manifolds.
method Geometric approaches.
result Recent advances in geometric methods.
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48 results for “Geometric Methods”
Spray theory aids geometric hydrodynamics methods.
problem Geometric methods in hydrodynamics need a spray theory.
method Use convenient calculus to complete Michor and Kriegl's approach.
result A unitary approach to geometric hydrodynamics established.
GeoAdaLer enhances geometric understanding of Adam for stochastic optimization.
problem Understanding geometric principles behind Adam's success in stochastic optimization.
method Introduces GeoAdaLer, an adaptive learning method based on geometric properties.
result Extends interpretability and effectiveness in complex optimization scenarios.
Discusses new probabilistic morphisms and geometric methods in machine and statistical learning.
problem Addressing challenges in statistical, machine, and manifold learning.
method Introduces category of probabilistic morphisms and geometric methods.
result New insights and applications in various learning fields.
Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
A geometric Laplace methodmath.DG
Geometrically reformulates the Laplace method for optimal transport.
problem Approximating integrals using the Laplace method without geometric interpretation.
method Introduces the Kim-McCann Riemannian metric to give a geometric formulation of the Laplace method.
result Expresses the first-order term of the Laplace method using geometric objects.
Based on Nielsen fixed point theory and Gröbner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.
New method proves length spectrum rigidity in various geometric settings.
problem Length spectrum rigidity in geometric settings.
method Combination of dynamical systems and geometric group theory.
result Provides concise proofs and extends classical results.
Only Bayes should learn a manifold (on the estimation of differential geometric structure from data)stat.ML
We investigate learning of the differential geometric structure of a data manifold embedded in a high-dimensional Euclidean space. We first analyze kernel-based algorithms and show that under the usual regularizations, non-probabilistic methods cannot recover the differential geometric structure, but instead find mostl…
New approach combines geometric and probabilistic methods to estimate manifold dimension in high-dimensional data.
problem Estimating the dimension of manifolds in high-dimensional data.
method Combines a modified box-counting algorithm (geometric) and a new probabilistic method (nearest neighbor distance analysis).
result The combined method is robust, fast, and effective in estimating manifold dimension.
We introduce two constructions in geometric deep learning for 1) transporting orientation-dependent convolutional filters over a manifold in a continuous way and thereby defining a convolution operator that naturally incorporates the rotational effect of holonomy; and 2) allowing efficient evaluation of manifold convol…
Proves conjecture for hyperbolic-by-cyclic groups using geometric methods.
problem Proving Farrell-Jones Conjecture for specific group structures.
method Geometric methods and structure theory of mapping tori.
result Proved Farrell-Jones Conjecture for mapping tori of automorphisms of hyperbolic-by-cyclic groups.
In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the…
In this paper, we are concerned with the 2D and 3D geometric shape generation by prescribing a set of characteristic values of a specific geometric body. One of the major motivations of our study is the 3D human body generation in various applications. We develop a novel method that can generate the desired body with c…
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
A new geometric method for clustering SPD data improves upon Euclidean and Riemannian approaches.
problem Skewed interpretations of SPD data in Euclidean analysis and computational inefficiency of Riemannian methods.
method Proposes a geometric method based on the Thompson metric for unsupervised clustering of SPD data.
result Demonstrates improved clustering results using inductive midrange centroid computation.
Geometric methods solve sampling, optimisation, inference, and adaptive decision-making.
problem Efficient solutions for sampling, optimisation, inference, and adaptive decision-making.
method Derive algorithms exploiting geometric structures of Hamiltonian systems, Hilbertian subspaces, and information geometry.
result Wide range of geometric theories emerge in these fields, enabling efficient solutions.
Geometry of Datamath.MG
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Randomized Geometric Algebra for Convex Neural Networks Optimizes Transfer Learning.
problem Training neural networks to global optimality via convex optimization.
method Randomized algorithms in Clifford's Geometric Algebra for hypercomplex vector spaces.
result Convex optimization and geometric algebra improve LLMs' robustness and reliability in transfer learning.
Proposes IIKL for preserving geometric properties of non-Euclidean data.
problem Loss of geometric information in non-Euclidean data representation.
method IIKL method builds Riemannian manifold and isometrically induces metric.
result Preserves geometric structure of original data in 3D and high-dimensional datasets.
In this paper, we extend the geometric descent method recently proposed by Bubeck, Lee and Singh to tackle nonsmooth and strongly convex composite problems. We prove that our proposed algorithm, dubbed geometric proximal gradient method (GeoPG), converges with a linear rate and thus achieves the optimal …
The ABP method is used to prove geometric inequalities for submanifolds and tensors.
problem Establishing geometric inequalities for submanifolds and tensors.
method Application of the Alexandrov-Bakelman-Pucci (ABP) method.
result Logarithmic Sobolev inequality and Sobolev-type inequality for submanifolds and tensors.
Optimal tracking of nonholonomic systems using geometric methods.
problem Tracking a trajectory for nonholonomic mechanical systems.
method Geometric optimal control, Pontryagin Maximum Principle, variational approach.
result Optimal control solutions for nonholonomic systems validated by examples and simulations.
We use the methods of geometric control theory to study extremal trajectories of vertical rolling disk. We focus on the role of symmetries of the underlying geometric structures. We demonstrate the computations in the CAS Maple package DifferentialGeometry.
Symmetry classification and invariant solutions of the classical geometric mean reversion processmath.DS
Investigates geometric mean reversion process using Lie symmetry method.
problem Describes dynamics of short-term interest rates.
method Lie symmetry method and optimal system of invariant solutions.
result Constructs an optimal system of invariant solutions.
The paper integrates dissipative and curl forces using geometric methods.
problem Incorporating dissipative forces into curl forces for non-conservative systems.
method Geometric metriplectic approach, Herglotz principle, generalized Euler-Lagrange equation, Galley's method.
result Natural formulations for Lagrangian and Hamiltonian dynamics of non-conservative systems.
Geometric method improves uncertainty estimation in real-time.
problem Improving uncertainty estimation in machine learning models.
method Geometric distance from training inputs for uncertainty estimation, post-hoc calibration.
result Method yields better uncertainty estimations than existing approaches.
PackIt creates a virtual space for testing geometric planning skills.
problem Evaluating geometric planning abilities in virtual environments.
method Developed a virtual environment, PackIt, for geometric planning tasks.
result Demonstrated the effectiveness of various methods for geometric planning.
The paper proposes a method to determine the geometric priors of relational data.
problem Identifying geometric structure in heterogeneous, high-dimensional data.
method Combinatorial approach analyzing nearest-neighbor structures and local neighborhood growth rates.
result The method can identify the geometric priors of suitable embedding spaces for relational data.
Machine learning methods struggle with geometric data, but shape space analysis provides a framework for studying and analyzing geometric variability.
problem Machine learning methods struggle with geometric data
method Shape space analysis provides a mathematical and computational framework
result Characterizes shape variability, compares geometric objects, and analyzes structural trajectories
This paper develops a general method for constructing Poisson integrators.
problem Lack of a general theory for Poisson integrators due to geometric challenges.
method Adapting structural results about symplectic realizations to create geometric approximations.
result Developed a general approach for constructing geometric integrators on Poisson manifolds.
Geometric Ergodicity in Modified Variations of Riemannian Manifold and Lagrangian Monte Carlostat.ME
This work investigates a mixture of LMC and RMHMC with MMALA for geometric ergodicity.
problem Lack of geometric ergodicity study in Riemannian manifold and Lagrangian Monte Carlo methods.
method Investigates a mixture of LMC and RMHMC with MMALA to achieve geometric ergodicity.
result Demonstrates geometric ergodicity in the mixture of LMC and RMHMC with MMALA.
GD-VAEs learn dynamics from observations using geometric and topological information.
problem Learning parsimonious representations of nonlinear dynamics from observations.
method Develops data-driven methods incorporating geometric and topological information using Variational Autoencoders (VAEs).
result GD-VAEs provide methods for learning reduced dimensional representations of nonlinear dynamics.
New geometrical method optimizes portfolio with minimal risk.
problem Optimizing portfolios with risk aversion and constraints.
method Computing generalized Euclidean distance to a simplex.
result Determines portfolios with minimal risk and high returns.
Second-order generalized Monge--Ampère equations on a plane and its geometric singular solutionsmath.DG
Study geometric singular solutions of generalized Monge-Ampère equations.
problem Solving generalized Monge-Ampère equations on a plane.
method Using exterior differential systems and Cauchy characteristics.
result Criteria for geometric singular solutions to be equivalent to specific types.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
Introduces quantum cohomology and helices in geometric methods.
problem Quantum cohomology and helices.
method Isomonodromic approach and Dubrovin's conjecture.
result Recent results on quantum cohomology and helices.
We derive a representation formula for the tensorial wave equation $\Box_\bg φ^I=F^I$ in globally hyperbolic Lorentzian spacetimes $(\M^{2+1}, \bg)$ by giving a geometric formulation of the method of descent which is applicable for any dimension.
We consider the hyperbolic geometric flow introduced by Kong and Liu [KL]. When the Riemannian metric evolve, then so does its curvature. Using the techniques and ideas of S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection and the curvature…
Paper compares stock price prediction models using Heston and Geometric Brownian Motion.
problem Predicting stock prices accurately.
method Developed Heston and Geometric Brownian Motion models using Ito's lemma and Euler-Maruyama methods.
result Models outperform statistical indicators in predicting stock prices.
Survey on geometric foundations of data reduction methods.
problem High-dimensional data with intrinsic nonlinear structure.
method Spectral manifold learning methods.
result Derivation and convergence analysis of spectral manifold learning.
Geometric interpretation improves VAE performance and robustness.
problem Improving Variational Autoencoder performance and robustness.
method Introducing a geometric perspective on VAEs, sampling from the Riemannian latent space.
result Improved generation and interpolations with competitive or better performance on benchmark datasets.
New method improves matrix completion with functional maps.
problem Matrix completion with geometric structure.
method Functional map regularization for geometric matrix completion.
result Significant performance improvement over state-of-the-art methods.
Study characterizes Einstein manifolds in almost Ricci solitons.
problem Characterizing Einstein manifolds in almost Ricci solitons.
method Geometric dynamics and geometric analysis methods.
result Characterizes Einstein manifolds in the class of complete almost Ricci solitons.
Enhances quantum circuit synthesis using deep learning and geometric methods.
problem Optimizing quantum circuits for time efficiency.
method Combining deep learning with geometric control techniques.
result Improved time-optimal control in quantum circuit synthesis.
In this paper, a type of integrable evolution equation--the generalized Landau-Lifshitz equation into is considered. We deal with this equation from a geometric point of view by rewriting it in a geometric form. Through the geometric energy method, we show the global well-posedness of the corresponding Cauchy pro…
Geometric Variational Inference improves efficiency in complex probability distributions.
problem Efficiently accessing information in non-linear and high-dimensional probability distributions.
method Geometric Variational Inference (geoVI) uses Riemannian geometry and the Fisher information metric to construct a coordinate transformation.
result geoVI provides a more efficient variational approximation by a normal distribution, demonstrated on various problems.
In this paper, an unsupervised machine learning method for geometric constellation shaping is investigated. By embedding a differentiable fiber channel model within two neural networks, the learning algorithm is optimizing for a geometric constellation shape. The learned constellations yield improved performance to sta…