arXiv research

Study on Teichmüller rays' asymptotic behavior and distances.

The paper studies the past inextendibility of FLRW spacetimes using the VDR asymptote.

Boundary distances determine conformal metrics

Asymptotic geodesics in convex polygons are convex for large distances.

In the spirit of Otal and Croke, we prove that a negatively-curved asymptotically hyperbolic surface is boundary distance rigid, where the distance between two points on the boundary at infinity is defined by a renormalized quantity.

This paper provides performance guarantees for neural estimation of statistical distances.

The paper is devoted to the large scale geometry of the Heisenberg group $\mathbb H$ equipped with left-invariant Riemannian distances. We prove that two such distances have bounded difference if and only if they are asymptotic, i.e., their ratio goes to one, at infinity. Moreover, we show that for every left-invariant…

The study examines the asymptotic behavior of extremal length in Teichmüller space.

The paper proposes methods to estimate MCMC quality with couplings, bounding Wasserstein distance.

The paper establishes criteria for spacetime inextendibility using asymptotic volume-distance-ratio analysis.

Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.

New smoothing technique improves Wasserstein distance estimation in high dimensions.

Exact inference method for Wasserstein distance with finite-sample coverage.

New online method estimates OT distances from sample streams.

In this paper, we consider the asymptotic behavior of two Teichmüller geodesic rays determined by Jenkins-Strebel differentials, and we obtain a generalization of a theorem in \cite{Amano14}. We also consider the infimum of the asymptotic distance in shifting base points of the rays along the geodesics. We show that th…

Corrects local error estimates for UBU integrator in SDEs, improving complexity guarantees.

Stability of positive mass theorem for hyperbolic manifolds studied.

Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.

In high dimensions, the mean and geometric median are nearly identical.

This article explains how to construct immersed Lagrangian submanifolds in C^2 that are asymptotic at large distance from the origin to a given braid in the 3-sphere. The self-intersections of the Lagrangians are related to the crossings of the braid. These Lagrangians are then used to construct immersed Lagrangians in…

Minimum expected distance estimation (MEDE) algorithms have been widely used for probabilistic models with intractable likelihood functions and they have become increasingly popular due to their use in implicit generative modeling (e.g. Wasserstein generative adversarial networks, Wasserstein autoencoders). Emerging fr…

We study the topological types of pants decompositions of a surface by associating to any pants decomposition $P,$ in a natural way its pants decomposition graph, $Γ(P).$ This perspective provides a convenient way to analyze the maximum distance in the pants complex of any pants decomposition to a pants decomposition c…

Research shows surfaces close to planes in Hausdorff distance.

We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert space techniques on the space of persistence diagrams. We also prove that when …

New distance comparison principle for curve shortening flow in higher dimensions.

We provide a proof and analyze the asymptotic behavior of a formula for the linking number of line segments.

The mass of asymptotically hyperbolic ends and manifolds is analyzed.

Derives new orthogonal coordinates for evolving surfaces and curves.

Study non-asymptotic Langevin Monte Carlo for Gibbs distributions.

Solves asymptotic $A_n$-realization problem for curves.

We study the asymptotic behaviour of 1-parameter subgroups with respect to Hofer's metric when the underlying symplectic manifold is an open surface of infinite area. We prove that, depending on the topology of the level sets of the Hamiltonian H, the distance either is bounded or behaves asymptotically linear. Moreove…

Paper analyzes SGLD for nonconvex optimization with local conditions.

Researchers derived heat kernel expansions for non-compact spaces using Witten deformation.

We use the notion of intrinsic flat distance to address the almost rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. In particular, we prove that a sequence of spherically symmetric asymptotically hyperbolic manifolds satisfying the conditions of the positive mass theorem converges to hyper…

We produce new non-Kähler complete steady gradient Ricci solitons whose asymptotics combine those of the Bryant solitons and the Hamilton cigar. We also obtain a family of complete Ricci-flat metrics with asymptotically locally conical asymptotics. Finally, we obtain numerical evidence for complete steady soliton struc…

Bayesian hierarchical clustering (BHC) is an agglomerative clustering method, where a probabilistic model is defined and its marginal likelihoods are evaluated to decide which clusters to merge. While BHC provides a few advantages over traditional distance-based agglomerative clustering algorithms, successive evaluatio…

The paper proves a quantitative positive mass theorem for spin manifolds with distance estimates.

New method calculates Ricci curvature from distances between weighted volumes.

This paper analyzes the bias of inexact MCMC methods in high dimensions.

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as converge…

New distances defined between space-times, proving some definite.

Comparing counterfactual distributions can provide more nuanced and valuable measures for causal effects, going beyond typical summary statistics such as averages. In this work, we consider characterizing causal effects via distributional distances, focusing on two kinds of target parameters. The first is the counterfa…

Identifying statistical dependence between the features and the label is a fundamental problem in supervised learning. This paper presents a framework for estimating dependence between numerical features and a categorical label using generalized Gini distance, an energy distance in reproducing kernel Hilbert spaces (RK…

Quantum codes with optimal distance and dimension for n-dimensional space.

In this article we study point configurations minimizing the discrete energy on a compact Riemannian manifold, where the energy kernel is taken to be the Green's function for the Laplacian. We show that every point in a minimizing configuration lies inside an open set called harmonic ball where no other point can enter…

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} dis…

Exact 1-Wasserstein distance between location-scale distributions derived, with privacy effects studied.

In this letter, we derive the optimal discriminant functions for modulation classification based on the sampled distribution distance. The proposed method classifies various candidate constellations using a low complexity approach based on the distribution distance at specific testpoints along the cumulative distributi…