Study shows continuous evolution of curves in Fréchet distance.
problem Continuous evolution of curves under curvature flow.
method Curvature flow and level-set flow, analyzed in Fréchet distance.
result Evolution of curves depends continuously on initial curve.
New network distance based on Laplacian flow captures structure.
problem Measuring similarity between network objects.
method Introducing Laplacian flow to define a new diffusion distance.
result Demonstrated utility and advantage over existing distances.
The paper examines distance function behavior in flows with bounded scalar curvature.
problem Analyzing the behavior of distance function under Ricci flows with bounded scalar curvature.
method Analyzing the distance function's Hölder continuity and extension up to singular time.
result The distance function is shown to be 1/2-Hölder continuous on small time-intervals.
Study curve flows with global forcing terms using a distance comparison principle.
problem Analyse the behavior of curves under curve flows with global forcing terms.
method Prove a distance comparison principle for curve shortening flow with arbitrary global forcing terms.
result Established a distance comparison principle for curve flows with global forcing terms.
Paper generalizes Ricci flow for inversive distance circle packings.
problem Deforming circle packings with prescribed cone angles.
method Generalized combinatorial Ricci flow for inversive distance.
result Generalized flow deforms any inversive distance circle packing to a unique packing with prescribed cone angles.
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.
Distance between evolving hypersurfaces is a PDE solution.
problem Tracking the distance between evolving hypersurfaces.
method Elliptic and parabolic PDEs, mean curvature flow.
result Local Harnack inequalities for the distance between evolving hypersurfaces.
A model of multi-layer flow network explores international trade dynamics.
problem Analyzing multi-layer flow networks of international trade.
method Flow distances and symmetric minimum flow distance introduced; multi-layer flow network model proposed.
result Clustering patterns in trading 'trophic levels' are similar for different commodities; centrality stratification observed.
Uniform distance distortion estimate for Ricci flows with bounded scalar curvature.
problem Analyzing Ricci flows with collapsing initial data.
method Uniform distance distortion estimate through renormalized metric-measure quantities.
result Uniform lower bounds of the renormalized heat kernel match with the lower bound of the renormalized volume ratio, proving distance distortion estimate.
Proposes a new method for posterior sampling using MMD with negative distance kernel.
problem Posterior sampling and conditional generative modeling.
method Approximates joint distribution using discrete Wasserstein gradient flows of MMD with negative distance kernel.
result Establishes an error bound for posterior distributions and proves the method is a Wasserstein gradient flow.
Bayesian neural flows improve Gaia distance estimates and dust modeling.
problem Improving precision of distance estimates from Gaia DR2 data.
method Normalizing flow for learning flexible color-magnitude diagrams.
result Distance posteriors improved by more than 48% over raw Gaia data.
New distance comparison principle for curve shortening flow in higher dimensions.
problem Understanding curve shortening flow in higher dimensions.
method Established a variant of Huisken's distance comparison principle.
result Symmetric curve shortening flow with one-to-one convex projection develops Type I singularities and becomes asymptotically circular.
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.
Proposes a Coulomb-like model for international trade flows, fitting real-world data.
problem Describing and predicting international trade flows between countries.
method Formulated a coulomb force model where GDP represents charge and distance is influenced by various factors.
result Developed a trade strength distribution equation that fits real-world data well.
We show that the distance function under the Ricci flow is uniformly continuous in the time direction, assuming only the scalar curvature is bounded.
Method approximates Wasserstein distance between 2D histograms using min cost flow.
problem Computing Wasserstein distance between 2D histograms efficiently.
method Transforms the problem into an uncapacitated min cost flow problem.
result Approximates optimal solution with reduced network size O(n). The paper studies a flow related to Ricci flow on manifolds with geometric singularities.
problem Analyzing geometric flows on manifolds with singularities.
method Introduced geometric flow on smooth compact manifolds with rough metrics, providing a regularity theory and demonstrating distance consistency.
result The flow on spaces with singularities preserves the distance metric as in smooth cases, maintaining smoothness away from singular points.
Two singular mean curvature flows converge if Hausdorff distance decreases faster than inverse time.
problem Understanding convergence of singular mean curvature flows.
method Analyzing Hausdorff distance between flows approaching a singularity.
result Two flows become identical if Hausdorff distance decreases faster than inverse time.
Article proves Liouville theorem for heat equation in super Ricci flow.
problem Proving Liouville theorem for heat equation in super Ricci flow.
method Formulated under a growth condition concerning Perelman's reduced distance.
result Established Liouville theorem for heat equation in ancient super Ricci flow.
Study curve flows with a global forcing term, proving distance comparison and convexity.
problem Analyzing the behavior of curves under curve shortening flow with a global forcing term.
method Distance comparison principle, finite time exclusion of singularities, convexity and convergence analysis.
result Convexity and smooth exponential convergence to a circle for closed curves.
Derives heat equation estimates linked to Ricci flow on compact and noncompact manifolds.
problem Estimating heat equation coupled to Ricci flow on noncompact manifolds.
method Local derivative estimates for the heat equation coupled to the Ricci flow.
result Extends results on distance distortion and backward pseudolocality to noncompact manifolds.
Paper analyzes convergence of ODE samplers in Wasserstein distances.
problem Limited theoretical understanding of convergence properties of probability flow ODEs.
method Convergence analysis for general probability flow ODEs in 2-Wasserstein distance.
result First non-asymptotic convergence analysis for probability flow ODE samplers.
The abstract discusses smoothing and non-smoothing effects using a metric transformation.
problem Analyzing smoothing and non-smoothing effects in metric measure spaces.
method A transformation of metric measure spaces using the length distance induced by heat kernel measures.
result The transformation smooths some Euclidean cones but not the Heisenberg group.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
In this paper, we define a reduced distance function based at a point at the singular time T<∞ of a Ricci flow. We also show the monotonicity of the corresponding reduced volume based at time T, with equality iff the Ricci flow is a gradient shrinking soliton. Our curvature bound assumption is more general than …
Establishes exponential contraction in Wasserstein distance on manifolds and flows.
problem Analyzing contraction rates in Wasserstein distance on manifolds and their evolution.
method Explicit estimates and extension to evolving manifolds under geometric flow.
result Gradient estimates with exponential contraction rate under weak curvature conditions.
New kernel improves MMDs with theoretical guarantees for gradient flows.
problem Non-smoothness of negative distance kernel in MMDs.
method Smoothed 1D absolute value function followed by fractional integral transform.
result Improved theoretical guarantees for Wasserstein gradient flows.
Continuous curve evolution depends on initial shape on sphere.
problem Evolution of a curve on a sphere by curvature flow.
method Study of curve evolution using curvature flow and level-set flow.
result Evolution depends continuously on initial curve in Fréchet distance.
New model improves data augmentation for causal tasks.
problem Optimizing causal models robustly under Wasserstein distances.
method Proposes a new G-Causal Normalizing Flow architecture.
result Empirically outperforms standard generative models.
New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.
problem Finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows with surgery for inversive distance circle packings.
result Longtime existence and global convergence of combinatorial curvature flows with surgery.
In this paper, we use the distance comparison principle, first been developed by G. Huisken, to study the spatial curve shortening flow. We have got the result that if the initial curve is the helix, then the local minimum of the ratio of the extrinsic and intrinsic distance is non-decreasing. And we have proved a Gray…
Paper introduces a differentially private generative model using gradient flow and sliced Wasserstein distance.
problem Protecting privacy in sensitive training data for generative models.
method Gradient flow in the space of probability measures, Gaussian-smoothed Sliced Wasserstein Distance, and numerical scheme for SDE.
result Demonstrates higher-fidelity data generation at low privacy budget compared to existing methods.
In this paper we consider compact, Riemannian manifolds M1,M2 each equipped with a one-parameter family of metrics g1(t),g2(t) satisfying the Ricci flow equation. Motivated by a characterization of the super Ricci flow developed by McCann-Topping, we introduce the notion of a super Ricci flow for a family of …
The paper deforms circle packings on surfaces with constant curvature.
problem Deforming circle packings on surfaces to constant curvature.
method Using Ge-Xu's α-flow to deform initial inversive distance circle packings.
result The inversive distance circle packing with constant α-curvature is unique under certain conditions.
Proves rigidity of maps between manifolds with scalar curvature constraints.
problem Lipschitz rigidity problem in scalar curvature geometry.
method Harmonic map heat flow coupled with Ricci flow.
result Continuous maps between manifolds with scalar curvature constraints are either isometries or have scalar curvature strictly less than the sphere.
Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
The paper describes flows of MMD functionals with distance kernel and quantile functions.
problem Wasserstein gradient flows of MMD functionals with negative distance kernel.
method Characterization via Cauchy problem on L2(0,1), solution via subdifferential construction. result Flow invariance and smoothing properties on subsets of C(0,1), absolute continuity of initial measures. In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to the contractivity of normalized L-Wasserstein distance under backward Ricci flow.
The study examines the limitations of bi-Lipschitz Normalizing Flows in approximating certain distributions.
problem The expressivity of bi-Lipschitz Normalizing Flows in approximating specific target distributions.
method Characterization of expressivity through lower bounds on Total Variation distance and discussion of potential remedies.
result Several target distributions are difficult to approximate using bi-Lipschitz Normalizing Flows, and lower bounds on their approximation are provided.
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if t…
New Sliced-Wasserstein distances for non-Euclidean data.
problem Computational burden of Wasserstein distance on non-Euclidean manifolds.
method Derive Sliced-Wasserstein distances and flows on Cartan-Hadamard manifolds.
result General constructions and non-parametric schemes for minimizing new distances.
Flow Matching improves Wasserstein 1 distance convergence in high dimensions.
problem Improving Wasserstein 1 distance estimation for unbounded distributions.
method Flow Matching approach based on ODEs, controlling Lipschitz constant.
result Derives a convergence rate for Wasserstein 1 distance, improving previous results.
It is a generally shared opinion that significant information about the topology of a bounded domain Ω of a riemannian manifold M is encoded into the properties of the distance, d∂Ω, %, d:Ω→[0,∞[, from the boundary of Ω. To confirm such an idea we propose an approach based on the in…
Two surfaces with similar boundary distances are isometric.
problem Determining when two Riemannian surfaces are isometric based on boundary distances.
method Analyzing Riemannian metrics with specific properties and using boundary rigidity results.
result Two surfaces with the same marked boundary distance are isometric.
Flow matching KL divergence bound derived for smooth distributions.
problem Estimating smooth distributions efficiently.
method Deterministic upper bound on KL divergence derived from flow-matching loss.
result Flow matching achieves nearly minimax-optimal efficiency under TV distance.
Estimates for solutions on manifolds under Ricci flow.
problem Gradient estimates for solutions on manifolds.
method Two-point function estimates for quasilinear parabolic equations under Ricci flow.
result Gradient estimates for solutions at any two points related to the distance between points.
Paper computes Kantorovich-Wasserstein distances on d-dimensional histograms efficiently.
problem Computing exact Kantorovich-Wasserstein distances between d-dimensional histograms. method Uses (d+1)-partite graph to solve as uncapacitated minimum cost flow problem. result Approach is competitive with state-of-the-art optimal transport algorithms.
The study extends Huisken's theorem to nonconvex surfaces that shrink to round points.
problem Extending Huisken's theorem to nonconvex surfaces.
method Constructing mean convex and non-mean convex hypersurfaces, using mean curvature flow.
result Found pathological examples of flows and sequences of flows that shrink to round points.