Method approximates Riemannian barycenter on manifolds.
problem Computing the exact Riemannian barycenter is computationally expensive.
method Uses under- and over-approximations of Riemannian distance to compute an approximate barycenter.
result Approximation method is more efficient than exact methods and steepest descent.
Study approximates sub-Riemannian structures with Riemannian metrics and analyzes spectral convergence.
problem Approximating sub-Riemannian structures for analysis.
method Constructing Riemannian metrics tailored to sub-Riemannian structures and studying spectral convergence.
result Riemannian volumes converge to Popp's volume and spectral convergence of Laplace operators is studied.
Study approximates Riemannian manifolds using polyhedra.
problem Understanding Tullio Regge's approximation theorem.
method Proof of Regge theorem using polyhedra approximation.
result Integral of scalar curvature approximated by polyhedral curvature.
The study provides conditions for approximating Riemannian manifolds with polyhedral metrics.
problem Approximating Riemannian manifolds with polyhedral metrics.
method Conditions on curvature tensors for Lipschitz and local polyhedral approximations.
result Conditions are sufficient for local polyhedral approximations, conjectured to be sufficient for global approximations.
Paper develops a finite dimensional approximation scheme for Riemannian manifolds.
problem Integration on Riemannian manifolds.
method New finite dimensional approximation scheme motivated by categorical colimit.
result Establishes a generalization for L1-functionals on Riemannian manifolds. Optimal algorithms for Riemannian optimization with reduced complexity.
problem Stochastic optimization on Riemannian manifolds with limited data.
method Zeroth-order Riemannian Averaging Stochastic Approximation algorithms using Riemannian moving-average estimators and novel geometric conditions.
result Achieves optimal sample complexities for generating approximate first-order stationary solutions.
Riemannian stochastic gradient descent approximates a diffusion process called Riemannian stochastic modified flow.
problem Improving convergence rate of Riemannian stochastic gradient descent.
method Using stochastic differential geometry, the paper shows RSGD can be approximated by the Riemannian stochastic modified flow (RSMF).
result RSGD can be approximated by the solution to the RSMF driven by an infinite-dimensional Wiener process, increasing the order of approximation.
Length metrics can be closely approximated by conformally flat metrics.
problem Approximating length metrics with conformally flat metrics.
method Uniform approximation of length metrics by conformally flat Riemannian metrics.
result Any length metric on \(\mathbb{R}^d\) can be uniformly approximated by conformally flat Riemannian metrics.
Bayesian neural networks approximate Gaussian, this method adapts to non-Gaussian posteriors.
problem Bayesian neural networks struggle with non-Gaussian posteriors, leading to poor performance.
method Proposes a Riemannian Laplace approximation to adapt to the shape of the true posterior.
result Consistently improves over conventional Laplace approximation across tasks.
Approximates distances on Riemannian manifolds efficiently.
problem High computational cost of pairwise distances on large Riemannian manifolds.
method Approximates distances using a two-dimensional model space with constant curvature.
result Linear number of geodesic boundary value problems required for approximation.
Proves theorem for Riemannian manifolds, extending previous work.
problem Proving Quantitative Fatou Theorem on Riemannian manifolds.
method Extending ε-approximation lemma to manifold setting.
result Proves Quantitative Fatou Theorem for Lipschitz domains on Riemannian manifolds.
Improves Laplace approximation for Bayesian inference on Riemannian manifolds.
problem Inaccurate Gaussian approximations for complex targets and finite-data posteriors.
method Develops alternative variants of the Laplace approximation using a Riemannian metric.
result Exact approximations at the limit of infinite data, improving practical performance.
The Heisenberg group's curvature and Gauss-Bonnet theorem are explored using Riemannian approximation.
problem Defining curvature in the Heisenberg group for smooth surfaces and curves.
method Using a Riemannian approximation scheme to define sub-Riemannian Gaussian and signed geodesic curvatures.
result Proved a Heisenberg version of the Gauss-Bonnet theorem.
Long spacelike embeddings can be approximated by isometric ones.
problem Approximating long embeddings to isometric embeddings in Lorentzian spaces.
method Proving approximation by constructing C1 isometric embeddings. result Long spacelike embeddings can be C0-approximated by C1 isometric embeddings. Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solution…
Smooth approximations of Lipschitz maps via Ehresmann fibrations and Reeb sphere theorem for functions.
problem Approximating Lipschitz maps and understanding singular points in Riemannian manifolds.
method Using Ehresmann fibrations and Reeb's sphere theorem for Lipschitz functions.
result A Lipschitz map can be approximated by a smooth map via Ehresmann fibrations.
Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.
problem Analyzing stochastic processes on surfaces in contact sub-Riemannian manifolds.
method Employing Riemannian approximations, a second order partial differential operator is derived on the surface. The stochastic process moves along the characteristic foliation induced by the contact distribution.
result Elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from separatrices.
Smooth approximation of integral cycles mod 2 in Riemannian manifolds.
problem Approximating mod 2 integral cycles by smooth submanifolds.
method Approximation of mod 2 integral cycles by smooth submanifolds with controlled singularities.
result Every mod 2 integral cycle can be approximated by a smooth submanifold with a controlled singular set.
Study on extending curves in sub-Riemannian manifolds with compatibility conditions.
problem Validating Whitney extension property for horizontal curves in sub-Riemannian manifolds.
method Analyzing equiregular and singular sub-Riemannian manifolds, using nilpotent approximation and Lusin-like approximation.
result Extension property holds for horizontal curves in sub-Riemannian manifolds with specific conditions.
Graphs approximate Laplacian spectra on manifolds.
problem Approximating Laplacian spectra on complex manifolds.
method Graph Laplacians on proximity graphs.
result Spectra of graph Laplacians approximate the Laplacian spectra of manifolds.
Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be thos…
The paper introduces a method for dimension reduction using sub-Riemannian geometry.
problem Dimension reduction for manifold learning and surface reconstruction.
method Combining local linear approximations of a point cloud to obtain lower dimensional bundles.
result Sub-Riemannian geodesics can successfully be applied to problems like constructing an approximating submanifold and computing distances.
Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Study random walks on sub-Riemannian manifolds using retractions.
problem Modeling random walks on sub-Riemannian manifolds.
method Use retractions to approximate normal geodesics and study convergence to Brownian motion.
result Convergence of geodesic random walks defined with different connections.
{\em Riemannian cubics} are curves in a manifold M that satisfy a variational condition appropriate for interpolation problems. When M is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are know…
Graphs approximate semigroups for diffusion on Riemannian manifolds.
problem Approximating semigroups for diffusion on Riemannian manifolds.
method Discretized approximation using random walks on proximity graphs.
result Quantitative error estimates for convergence of discrete semigroups to continuous semigroups.
Variational approximations for curve flows on Riemannian manifolds.
problem Approximating solutions to curvature and elastic flow problems on Riemannian manifolds.
method Variational formulations, finite element approximations, piecewise linear elements, stability analysis.
result Derived schemes can compute rotationally symmetric self-shrinkers and geodesics.
Constructs Riemannian manifolds to approximate metric spaces.
problem Approximating a metric space with a Riemannian manifold.
method Constructs a Riemannian manifold to approximate a metric space (X,dX). result Characterizes metric spaces that can be approximated by Riemannian manifolds.
Derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous.
problem Smoothness of sub-Riemannian geodesics
method Proving Lp-Hölder continuity of derivatives result Derivatives of sub-Riemannian geodesics are Lp-Hölder continuous Smooth approximation of integral cycles in manifolds.
problem Approximating integral cycles in Riemannian manifolds.
method Approximation of integral cycles by smooth submanifolds with controlled area and singularities.
result Integral cycles can be approximated by smooth submanifolds with controlled area and singularities.
Formulas for mean curvature in various spaces.
problem Understanding mean curvature in different geometric settings.
method First and second variation formulas for arbitrary variations in Riemannian and sub-Riemannian manifolds.
result Formulas derived for mean curvature in Heisenberg group via approximation.
We improve Riemannian metrics for constrained systems control.
problem Controlling mechanical systems with configuration constraints.
method Constructing complete Riemannian metrics by modifying incomplete ones.
result A controller can be found to satisfy a design criterion.
Develops a curvature-corrected tangent space method for manifold-valued data.
problem Generalizing real-valued data approximation to manifold-valued data.
method Systematic approach to developing global-geometry aware, computationally feasible approximation schemes.
result Proposes CC-tHOSVD for low-rank approximation of manifold-valued data.
The paper extends Laplacian spectra approximations to vector bundles.
problem Approximating the spectrum of the connection Laplacian.
method Extending the graph connection Laplacian to vector bundles and proving spectrum approximation.
result The spectrum of the extended operator approximates the spectrum of the connection Laplacian.
Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds
problem Convergent approximation of Gaussian Whittle-Matern fields on Riemannian manifolds
method Finite Element approximation of SPDEs
result Universal approximation of precision and covariance matrices
Estimates Riemannian simplex coordinates and metric differences.
problem Comparing Riemannian metric to Euclidean on simplices.
method Barycentric coordinates on Riemannian manifolds, Karcher's center of mass.
result Error in metric difference shrinks quadratically with edge length.
Study on curvature in Heisenberg space using Riemannian approximations.
problem Verify if Gaussian and normal curvatures match in Heisenberg space.
method Use Riemannian approximations and limit of curvatures in (R3,gL) to study sub-Riemannian spaces. result Gaussian and normal curvatures do not coincide in Heisenberg space.
Study compares H-type sub-Riemannian manifolds using uniform metrics.
problem Comparing H-type sub-Riemannian manifolds with Riemannian metrics.
method Establishes sub-Hessian and sub-Laplacian comparison theorems for a family of approximating Riemannian metrics.
result Proves a sharp sub-Riemannian Bonnet-Myers theorem.
RSVGD improves SVGD for Bayesian inference on Riemannian manifolds.
problem Bayesian inference on Riemannian manifolds.
method Develops RSVGD, a generalization of SVGD to Riemann manifolds.
result Advantages over SVGD in exploring distribution geometry and particle-efficiency.
Paper develops Riemannian geometry for SPSD matrices with DA applications.
problem Riemannian geometry of SPSD matrices for DA.
method Closed-form expressions, approximations of geodesic path, PT, canonical representation.
result Proposes an algorithm for DA with improved performance.
New proof of Riemannian Penrose Inequality for manifolds with corners
problem Riemannian Penrose Inequality for asymptotically flat manifolds with corners
method Unified argument based on approximate monotonicity
result Positive Mass Theorem and Riemannian Penrose Inequality
New approximative kernels improve PDE-G-CNNs for geometric deep learning.
problem Inaccurate approximations of exact kernels in PDE-G-CNNs.
method Developed new approximative kernels that work regardless of spatial anisotropy.
result New kernels provide better error estimates and maintain reflectional symmetries.
Graph Laplacian approximates manifold eigenvalues with controlled curvature bounds.
problem Approximating eigenvalues of Laplace-Beltrami on manifolds with bounded Ricci curvature.
method Graph discretization of Riemannian manifolds with (ε,ρ)-approximation, proving eigenvalue convergence. result Graph Laplacian eigenvalues converge uniformly to manifold Laplacian eigenvalues as parameters approach zero.
Smooths metrics with nonnegative scalar curvature near singular sets.
problem Approximating metrics with nonnegative scalar curvature near singularities.
method Ricci-DeTurck flow to approximate metrics.
result Approximated metrics converge to the original metric in C∞ away from the singular set. Study on sub-Riemannian manifolds, focusing on conjugate points and caustic stability.
problem Analyzing sub-Riemannian manifolds and their conjugate points.
method Computed sub-Riemannian Hamiltonian flow, approximated conjugate locus, introduced geometric invariant.
result Contact distributions exhibit unique behavior, different from 3D case.
The paper shows how particle movement on a manifold's grid approximates Brownian motion and heat diffusion.
problem Understanding particle movement on curved spaces.
method Analyzing symmetric exclusion process on random grids approximating a Riemannian manifold.
result Empirical density field converges to heat equation solution on the manifold.
Study on heat content for submanifolds in sub-Riemannian geometry.
problem Understanding heat content for submanifolds in sub-Riemannian geometry.
method Existence of smooth tubular neighborhood, definition of relative heat content, approximation via smooth neighborhoods, asymptotic expansion analysis.
result Approximation of relative heat content fails to recover the exact expansion.
Algorithm approximates functions into manifolds with curvature bounds.
problem Approximating functions into manifolds with lower curvature bounds.
method Algorithm using manifold exponential and logarithm, with error bounds based on sectional curvature.
result Error bounds for nonnegative sectional curvature are similar to linear space approximations.