arXiv research

A new method using energy distance for ensemble and scenario reduction.

Energy distance measures feature heterogeneity in federated learning.

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, maximum mean discrepancies (MMD), that is, distances between embeddings of distributions to reproduc…

We provide a unifying framework linking two classes of statistics used in two-sample and independence testing: on the one hand, the energy distances and distance covariances from the statistics literature; on the other, distances between embeddings of distributions to reproducing kernel Hilbert spaces (RKHS), as establ…

Paper extends multivariate rank tests for robust subspace detection.

This work examines the sensitivity of energy distance to mean differences compared to covariance differences.

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…

Proposes an energy-based sliced Wasserstein distance for improved probability measure comparison.

Study spider mechanism configuration spaces using squared distance function.

Energy functional for Legendrian knots in Heisenberg group, invariant under PU(2,1).

We show that balls, circles and 2-spheres can be identified by generalized Riesz energy among compact submanifolds of the Euclidean space that are either closed or with codimension 0, where the Riesz energy is defined as the double integral of some power of the distance between pairs of points. As a consequence, we obt…

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…

Study geodesic distances and convexity in contact sets.

We present Optimal Transport GAN (OT-GAN), a variant of generative adversarial nets minimizing a new metric measuring the distance between the generator distribution and the data distribution. This metric, which we call mini-batch energy distance, combines optimal transport in primal form with an energy distance define…

Lowered regularity assumption for a phase-dependent Helfrich energy equation.

Paper proves stability of positive mass theorem for specific types of manifolds.

New distances for comparing heterogeneous probability measures efficiently.

In this paper, we first prove a folklore conjecture on a greatest lower bound of the Calabi energy in all Kähler manifold. Similar result in algebriac setting was obtained by S. K. Donaldson. Secondly, we give an upper/lower bound estimate of the K energy in terms of the geodesic distance and the Calabi energy. This is…

Study the hanging chain shape around a circle.

New method improves speech synthesis quality.

Paper proposes a new method for estimating treatment effects using interpretable deep learning models.

In this paper we study 1/k-geodesics, those closed geodesics that minimize on any subinterval of length $l(γ)/k$. We employ energy methods to provide a relationship between the 1/k-geodesics and what we define as the balanced points of the uniform energy. We show that classes of balanced points of the uniform energy pe…

We establish a fundamental connection between smooth and polygonal knot energies, showing that the Minimum Distance Energy for polygons inscribed in a smooth knot converges to the Moebius Energy of the smooth knot as the polygons converge to the smooth knot. However, the polygons must converge in a ``nice'' way, and th…

We propose a geometric method for quantifying the difference between parametrized curves in Euclidean space by introducing a distance function on the space of parametrized curves up to rigid transformations (rotations and translations). Given two curves, the distance between them is defined as the infimum of an energy …

New neural method calculates EMD for particle physics data.

Paper uses news data to model asset correlations without market data.

Study finite-energy metrics over complex manifold degenerations.

We show that K-energy minimizing movements agree with smooth solutions to Calabi flow as long as the latter exist. As corollaries we conclude that in a general Kahler class long time solutions of Calabi flow minimize both K-energy and Calabi energy. Lastly, by applying convergence results from the theory of minimizing …

Sharp stability result for maps near infinitely concentrated minimisers.

Given a closed submanifold, or a compact regular domain, in euclidean space, we consider the Riesz energy defined as the double integral of some power of the distance between pairs of points. When this integral diverges, we compare two different regularization techniques (Hadamard's finite part and analytic continuatio…

Study distances between special functions on Kähler manifolds.

Algorithm finds best Dirac mass approximation of target measure.

Learning algorithms for implicit generative models can optimize a variety of criteria that measure how the data distribution differs from the implicit model distribution, including the Wasserstein distance, the Energy distance, and the Maximum Mean Discrepancy criterion. A careful look at the geometries induced by thes…

In a vacuum spacetime equipped with the Bondi's radiating metric which is asymptotically flat at spatial infinity including gravitational radiation ({\bf Condition D}), we establish the relation between the ADM total energy-momentum and the Bondi energy-momentum for perturbed radiative spatial infinity. The perturbatio…

Quantitative stability for nearly minimizing Yamabe metrics.

Meta learns low-rank covariance factors for better uncertainty estimation.

Study exact Lagrangian cobordisms in cotangent bundles, proving bounds on sheaf interleaving distance and shadow distance.

DCMA uses generative models to analyze complex treatment effects on outcome distributions.

Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.

We consider least energy solutions to the nonlinear equation $-Δ_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is …

Classifies soap film surfaces with vertical potentials.

The paper studies properties of Sliced Wasserstein energy for discrete measures.

In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us…

The study improves PAC-Bayesian bounds for adversarial generative models.

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they…

We study the non-asymptotic behavior of a Coulomb gas on a compact Riemannian manifold. This gas is a symmetric n-particle Gibbs measure associated to the two-body interaction energy given by the Green function. We encode such a particle system by using an empirical measure. Our main result is a concentration inequalit…

Study spherical curves with curvature dependent on distance to a great circle.

The paper presents new metrics to quantify and test for (i) the equality of distributions and (ii) the independence between two high-dimensional random vectors. We show that the energy distance based on the usual Euclidean distance cannot completely characterize the homogeneity of two high-dimensional distributions in …