arXiv research

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

Identity testing for reversible Markov chains without symmetry assumption.

We define a new notion---the sub-index of a critical point of a distance function. We show how sub-index affects the homotopy type of sublevel sets of distance functions.

We study a notion of distance between knots, defined in terms of the number of saddles in ribbon concordances connecting the knots. We construct a lower bound on this distance using the X-action on Lee's perturbation of Khovanov homology.

Compactness theorem for timed-metric spaces established.

Proves compactness for timed-metric spaces using new distance and maps.

Study uses equivariant topology to measure distances between G metric spaces.

Study sequences of static spacetimes using null distance convergence.

Graph curvature measured by inverse resistance distance.

Theory of space-time currents for geometric evolutions.

A new dataset distance using optimal transport, agnostic of model and label sets.

New curvature concept preserves graph distances under operations.

Study shows intrinsic timed Hausdorff convergence leads to Gromov-Hausdorff and big bang convergence.

Paper proves circle packings converge to Riemann mapping for Jordan domains.

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian.

The aim of this article is to generalize the notion of the cut locus and to get the structure theorem for it. For this purpose, we first introduce a class of 1-Lipschitz functions, each member of which is called an {\it almost distance function}. Typical examples of an almost distance function are the distance function…

Extends Mahalanobis distance to Banach spaces for anomaly detection.

Extends manifold learning to non-Euclidean metrics.

Why does training deep neural networks using stochastic gradient descent (SGD) result in a generalization error that does not worsen with the number of parameters in the network? To answer this question, we advocate a notion of effective model capacity that is dependent on {\em a given random initialization of the netw…

We consider rotationally symmetric spaces with low regularity, which we regard as integral currents spaces or manifolds with Sobolev regularity and are assumed to have nonnegative scalar curvature. Relying on the flat distance and on Sobolev norms, we establish several nonlinear stability estimates about the ``distance…

Researchers modify $d_p$ distance to handle long, thin splines.

In infinite dimensional Heisenberg group, degenerate distances linked to unbounded curvature.

A slice distance for the class of weak abelian Lp-bundles in 3 dimensions was introduced in a previous article in collaboration with Tristan Rivière, where it was used to prove the closure of such class of bundles for the weak Lp-convergence. We further investigate this distance here, and we prove more properties of it…

If $Σ$ and $Σ'$ are homotopic embedded surfaces in a $4$-manifold then they may be related by a regular homotopy (at the expense of introducing double points) or by a sequence of stabilisations and destabilisations (at the expense of adding genus). This naturally gives rise to two integer-valued notions of distance bet…

This work proposes a model for geodesic distances and flows on manifolds.

Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…

Study stability of curvature-dimension condition for negative dimensions.

Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.

In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of …

We define a notion of Hempel distance for one-sided Heegaard splittings and show that the existence of alternate surfaces restricts distance for one-sided splittings in a manner similar to Hartshorn's and Scharlemann-Tomova's results for two-sided splittings. We also show that every geometrically compressible one-sided…

The paper shows how to recover true node positions from a graph or similarity matrix.

This paper presents a general notion of Mahalanobis distance for functional data that extends the classical multivariate concept to situations where the observed data are points belonging to curves generated by a stochastic process. More precisely, a new semi-distance for functional observations that generalize the usu…

Private density estimation in Wasserstein distance for geographic populations.

This note improves correlation stress tests using geodesic distance.

We consider various notions of strains; quantitative measures for the deviation of a linear transformation from an isometry. The main approach, which is motivated by physical applications and follows the work of Patrizio Neff and co-workers , is to select a Riemannian metric on $\text{GL}_n$, and use its induced geodes…

Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. T…

The paper tackles learning smooth distance functions using query-based methods.

Globalization is one of the central concepts of our age. The common perception of the process is that, due to declining communication and transport costs, distance becomes less and less important. However, the distance coefficient in the gravity model of trade, which grows in time, indicates that the role of distance i…

This guide explains statistical distances for evaluating generative models.

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

In case of a standard form vN-algebra, the Bures distance is the natural distance between the fibres of implementing vectors at normal positive linear forms. Thereby, it is well-known that to each two normal positive linear forms implementing vectors exist such that the Bures distance is attained by the metric distance…

New method improves robust point matching under probabilistic settings.

Study on volume of tubes and concentration in Riemannian geometry.

Many information retrieval algorithms rely on the notion of a good distance that allows to efficiently compare objects of different nature. Recently, a new promising metric called Word Mover's Distance was proposed to measure the divergence between text passages. In this paper, we demonstrate that this metric can be ex…

Inspired by the Gromov-Hausdorff distance, we define the intrinsic flat distance between oriented $m$ dimensional Riemannian manifolds with boundary by isometrically embedding the manifolds into a common metric space, measuring the flat distance between them and taking an infimum over all isometric embeddings and all c…

One of the most beautiful notions of metric geometry is the Gromov-Hausdorff distance which measures the difference between two metric spaces. To define the distance, let us isometrically embed these spaces into various metric spaces and measure the Hausdorff distance between their images. The best matching corresponds…

New Morse theory for shapes at distances.

Efficiently estimates binary product distributions with privacy.