Vanishing geodesic distances in infinite dimensions can be created.
problem Vanishing geodesic distances in infinite-dimensional spaces.
method Constructing a weak Riemannian metric in a Hilbert manifold.
result Vanishing geodesic distances can be engineered.
The paper connects geodesic nets to distance function critical points.
problem Understanding the relationship between geodesic nets and distance function critical points.
method Established a relationship between geodesic nets and critical points of the distance function.
result Bounded the number of balanced points and the length of certain minimizing geodesic nets.
Bounds on geodesic distances on Stiefel manifold derived from new metrics.
problem Improving geodesic computation algorithms and understanding Stiefel manifold.
method New geometric insights and Lipschitz constants for geodesic distances.
result Explicit bounds on geodesic distances and conditions for attaining bounds.
Geodesic distance vanishes for certain Sobolev metrics on diffeomorphisms.
problem Analyzing geodesic distance in diffeomorphism groups for various Sobolev norms.
method Study of right-invariant Sobolev metrics on compactly supported diffeomorphisms.
result Geodesic distance vanishes identically for s<min{n/p,1}, and is positive otherwise. This work proposes a model for geodesic distances and flows on manifolds.
problem Geodesic distances and flows on differentiable manifolds.
method Manifold-augmented Eikonal equation solutions.
result Geodesic flow provides globally length-minimizing curves.
New distance function proves rigidity in geodesic lamination space.
problem Proving rigidity in geodesic lamination space.
method Introduced left Hausdorff distance function and proved rigidity result.
result Extended mapping class group is isomorphic to bijections preserving left Hausdorff convergence.
The L2-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N,g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for…
This paper introduces a new method for SAR imagery region discrimination using geodesic distances.
problem Region discrimination in monopolarized SAR imagery.
method Geodesic distance between GI0 models. result Advantages of using geodesic distance over stochastic distances.
Geodesic rays and chordal distances link algebraic and geometric properties of positive metrics.
problem Understanding the geometry of the space of positive metrics at infinity.
method Using Monge-Ampère equations and test configurations, algebraic descriptions of geodesic rays and chordal distances are derived.
result The Mabuchi chordal distance between geodesic rays associated with ample test configurations equals the spectral distance between their filtrations.
New method estimates geodesic distances using spherelets.
problem Accurately estimating geodesic distances on unknown manifolds.
method Uses spherelets to locally approximate unknown subspaces and estimate geodesic distances.
result Lower error for many manifolds, validated through simulations and real data.
URerF learns geodesic distances in noisy manifolds.
problem Learning geodesic distances in noisy high-dimensional data.
method Unsupervised random forest (URerF) with Bayesian Information Criterion.
result URerF outperforms other methods in estimating geodesic distances on noisy data.
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
The distance function ϱ(p,q) (or d(p,q)) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over Rn, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no F…
The Virasoro-Bott group endowed with the right-invariant L2-metric (which is a weak Riemannian metric) has the KdV-equation as geodesic equation. We prove that this metric space has vanishing geodesic distance.
Circle's metric is at least π/4 away from any simply connected geodesic space.
problem Comparing simply connected geodesic spaces to the circle.
method Using Gromov-Hausdorff distance and topological properties.
result The Gromov-Hausdorff distance between circle and any simply connected geodesic space is at least π/4.
Study geodesic curvature in Heisenberg group, interpreting it as distance correction.
problem Interpreting geodesic curvature in the Heisenberg group.
method Analyzing smooth horizontal curves in the Heisenberg group, interpreting curvature as distance correction.
result Geodesic curvature in Heisenberg group is the first term in distance expansion.
Assigns compact set distance-like functions to non-compact geodesic spaces.
problem Assigning distance-like functions to compact sets in non-compact geodesic spaces.
method Assigns each compact set a distance-like function and studies the pseudo-metric on the space of compact subsets.
result Obtains a pseudo-metric on the space of compact subsets that is less than the Hausdorff distance.
We find an upper bound for geodesic distances associated to monotone Riemannian metrics on positive definite matrices and density matrices.
New geodesics for surfaces with boundary, extending Thurston's work.
problem Extending Thurston's Lipschitz distance to surfaces with boundaries.
method Constructing geodesics in the arc distance metric space.
result Teichmueller space of surfaces with boundaries is a geodesic metric space.
Geodesic distance vanishes for critical Sobolev norms on diffeomorphism groups.
problem Analyzing geodesic distance in diffeomorphism groups for critical Sobolev norms.
method Combining techniques from [JM19] and [BHP18]
result Geodesic distance vanishes for Ws,n/s norms when s∈(0,1) and sp≤n. Study develops geodesic theory for foliations, proving Laplacian comparison theorems.
problem Comparing Laplacians on totally geodesic Riemannian foliations.
method Variational theory of geodesics, limit of Riemannian distance approximations.
result Sharp comparison theorems for sub-Riemannian distance in Sasakian foliations.
We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for all distances accessible by computer. Our method gives a new preferred finite set …
The paper shows how to recover true node positions from a graph or similarity matrix.
problem Recovering true distances and positions from a graph or similarity matrix.
method Two steps: matrix factorisation followed by nonlinear dimension reduction.
result Nonlinear dimension reduction can recover latent positions close to a manifold where geodesic distance is encoded.
Proves conjecture about geodesic foliations in Riemannian planes.
problem Geodesic foliations with bounded distance in non-flat Riemannian planes.
method Analyzes total curvature and visibility properties to prove conjecture.
result Proves conjecture in two specific cases.
Study shows vanishing distance in fluid dynamics equations.
problem Understanding the geometric origins of fluid dynamics equations.
method Analyzing geodesic distances on diffeomorphism and symplectomorphism groups.
result Modified Constantin-Lax-Majda and surface quasi-geostrophic equations arise from metrics with vanishing geodesic distance.
Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.
problem Regularity of distance maps on geodesically complete spaces with curvature bound above.
method Define and prove regularity of distance maps as Hurewicz fibrations.
result Sphere theorem for geodesically complete CAT(1) spaces.
Researchers define geodesic curvature for 3D sub-Riemannian curves, improving distance calculations.
problem Improving sub-Riemannian distance calculations for 3D curves.
method Introducing geodesic curvature kζ for smooth horizontal curves in 3D contact sub-Riemannian manifolds. result Geodesic curvature appears as the first corrective term in the Taylor expansion of sub-Riemannian distance.
Efficiently represents large geodesic distance matrices for MDS analysis.
problem Quadratic growth of geodesic distance matrices for large point sets.
method Sparse biharmonic interpolation to learn a subset of points for efficient approximation.
result 2x faster and 20x less memory usage than current methods, enabling analyses of large point sets.
Here we study geodesics connecting two given points on odd-dimensional spheres respecting the Hopf fibration. This geodesic boundary value problem is completely solved in the case of 3-dimensional sphere and some partial results are obtained in the general case. The Carnot-Carathéodory distance is calculated. We also p…
We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length minimizing geodesics are deduced using simple methods based on the calculus of varia…
New formulas for geodesics on Stiefel and flag manifolds using trust-region method.
problem Computing geodesics and logarithms on Stiefel and flag manifolds.
method Closed-form geodesic formulas, trust-region solver, Fréchet derivatives.
result Efficient computation of geodesic distance and logarithm map.
Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
problem Exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.
method Combining Varadhan's formula, Loewner's theorem, and the method of stationary phase.
result Characterization of squared sub-Riemannian distance and cut locus on generalized Heisenberg-type groups and star graphs.
Asymptotic geodesics in convex polygons are convex for large distances.
problem Understanding convexity of geodesics in Hilbert geometry.
method Analyzing the distance function between asymptotic geodesics for large t.
result The distance function between asymptotic geodesics is convex for sufficiently large t.
We improve density-based distances using normalizing flows and score matching.
problem Inaccurate density estimates and poor convergence in graph-based methods for high-dimensional spaces.
method Learn densities with normalizing flows and refine geodesics with a score model.
result Improved density-based distances that scale to high dimensions and improve numerical stability.
Catenaries defined on any Riemannian surface using intrinsic distance.
problem Defining catenaries on Riemannian surfaces.
method Defining catenaries as critical points of a potential functional, calculating potential with intrinsic distance, and characterizing using curvature.
result Characterization of catenaries on various Riemannian surfaces.
Study on Finsler spaces with specific metric changes and reversible geodesics.
problem Characterizing Finsler spaces with reversible geodesics.
method Analyzing a Finsler space with a Randers change of Quartic metric and deriving conditions for reversible geodesics.
result The Finsler metric F induces a generalized weighted quasi-distance on the space.
The study proves properties of geodesics and tubular neighborhoods on Finsler manifolds.
problem Existence and properties of geodesics and tubular neighborhoods on Finsler manifolds.
method Analytical proof of properties of geodesics and tubular neighborhoods.
result Existence of tubular neighborhoods and minimization of orthogonal geodesics.
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.
We study Sobolev-type metrics of fractional order s≥0 on the group $\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M=S1 the geodesic distance on $\Diff_c(S^1)$ vanishes if and only if s≤21. For other manifolds we obtain a partial chara…
Study distance and intersection number in curve graphs of surfaces.
problem Understanding the relationship between distance and intersection number in curve graphs of surfaces.
method Introduced efficient geodesics and studied rectangles called spirals in the cellular decomposition.
result Developed an algorithm to reduce intersection number while preserving distance.
Round spheres are uniquely characterized by half-geodesics.
problem Characterizing round spheres in Riemannian geometry.
method Establishing that Riemannian spheres with specific geodesic properties are round.
result Riemannian spheres with all geodesics closed and many half-geodesics are round.
Proposes a new metric learning method using Lie group geodesics.
problem Improving distance metrics for k-NN classification.
method Geodesic interpolation on Lie transformation group to calculate velocities and produce a diffeomorphic global transformation.
result Effective in synthetic and real datasets, improving k-NN classification.
We find a different approach to define convex functions in the sub-Riemannian setting. A function on a sub-Riemannian manifold is nonholonomically geodesic convex if its restriction to any nonholonomic (straightest) geodesic is convex. In the case of Carnot groups, this definition coincides with that by Danniell-Garofa…
This note improves correlation stress tests using geodesic distance.
problem Improving financial risk management through better covariance stress tests.
method Proposes a new geometrically invariant definition of correlation stress tests.
result Demonstrates a submanifold approach to stress testing covariance matrices.
The paper extends the collar theorem to non-compact surfaces using new comparison theorems.
problem Proving the collar theorem for non-compact surfaces.
method Developed new Toponogov-type triangle comparison theorems.
result Eliminated the compactness hypothesis for the collar theorem.
Gradient descent algorithms on manifolds solve control and mean computation problems.
problem Control and mean computation on positive definite Hermitian matrices.
method Riemannian and natural gradient algorithms applied to geodesic distance.
result Efficient algorithms for control and mean computation demonstrated.
Researchers compare brain connectomes using geodesic distance on manifold for twin pairs.
problem Assessing functional similarity in brain networks between monozygotic and dizygotic twins.
method Using fMRI data, the researchers compared functional networks between mono- and dizygotic twin pairs by measuring similarity with geodesic distance on graph Laplacians.
result Functional networks are more similar in monozygotic twins compared to dizygotic twins, and similarity is higher for task-relevant networks.
Extends manifold learning to non-Euclidean metrics.
problem Applying manifold learning to data in non-Euclidean spaces.
method Generalizes manifold learning to metric spaces and studies conditions for convergence.
result Conditions for the convergence of graph Laplacian in metric spaces.