arXiv research

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…

We show that, on a smooth riemannian manifold, the laplacian of the distance function to a point $b$ is $-\infty$ in the sense of barriers, at every point of the cut locus with respect to $b$.

Discuss folklore statements about manifolds with curvature bounds.

We characterize the differentiable points of the distance function from a closed subset $N$ of an arbitrary dimensional Finsler manifold in terms of the number of $N$-segments. In the case of a 2-dimensional Finsler manifold, we prove the structure theorem of the cut locus of a closed subset $N$, namely that it is a lo…

New method calculates cut locus on Riemannian manifolds using optimal transport.

The paper proves the existence of a tubular neighborhood for Finsler submanifolds.

The paper connects cut locus, Thom space, and Morse-Bott functions in Riemannian geometry.

Nilpotent groups can't be biLipschitz embedded into $L^1$.

Maps on Sasakian manifolds limit to sub-Riemannian distance bounds.

Study exact formula and geodesics for Carnot-Carathéodory distance on 2-step groups.

New method calculates cut locus on surfaces without boundary.

We study the small time asymptotics of the gradient and Hessian of the logarithm of the heat kernel at the cut locus, giving, in principle, complete expansions for both quantities. We relate the leading terms of the expansions to the structure of the cut locus, especially to conjugacy, and we provide a probabilistic in…

Reduced sub-Riemannian time on a specific group structure.

We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove…

In this paper we study the invariant Carnot-Caratheodory metrics on $SU(2)\simeq S^3$, $SO(3)$ and $SL(2)$ induced by their Cartan decomposition and by the Killing form. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory dis…

We prove some sharp isoperimetric type inequalities for domains with smooth boundary on Riemannian manifolds. For example, using generalized convexity, we show that among all domains with a lower bound $l$ for the cut distance and Ricci curvature lower bound $(n-1)k$, the geodesic ball of radius $l$ in the space form o…

The paper studies the cut locus of submanifolds in Riemannian manifolds, providing geometric and topological insights.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Method learns graphons from graphs via Gromov-Wasserstein barycenters.

We study the problem of partitioning a small sample of $n$ individuals from a mixture of $k$ product distributions over a Boolean cube $\{0, 1\}^K$ according to their distributions. Each distribution is described by a vector of allele frequencies in $\R^K$. Given two distributions, we use $γ$ to denote the average $\el…

Online structure learning approaches, such as those stemming from Statistical Relational Learning, enable the discovery of complex relations in noisy data streams. However, these methods assume the existence of fully-labelled training data, which is unrealistic for most real-world applications. We present a novel appro…

Max-Cut decision tree improves classification accuracy and reduces computation time.

ExDAG solves DAG learning problems with low structural Hamming distance.

Transforms distance-based outlier scores into interpretable probabilistic estimates.

Study sub-Riemannian geodesics on a Heisenberg 3D nil-manifold.

A left-invariant sub-Riemannian metric $d$ on the shortened Lorentz group $SO_0(2,1)$ under the condition that $d$ is right-invariant relative to the orthogonal Lie subgroup $1\otimes SO(2)$ is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup $1\otimes SO(2)$ with the an…

In this paper we prove that if a point $p$ in a complete Riemannian manifold is not a cut point of any point whose distance to $p$ is $r$, then the injectivity radius of $p$ is strictly large than $r$. As a corollary we give a positive answer to a problem raised by Z. Sun and J. Wan.

The aim of this article is to present a comparative review of Riemannian and Finsler geometry. The structures of cut and conjugate loci on Riemannian manifolds have been discussed by many geometers including H. Busemann, M. Berger and W. Klingenberg. The key point in the study of Finsler manifolds is the non-symmetric …

Under the definition of Ricci curvature bounded below for Alexandrov spaces introduced by Zhang-Zhu, we generalize a result by Colding that an n dimentional manifold with Ricci curvature greater or equal to n minus 1 and volume close to that of the unit n sphere is close (in the Gromov-Hausdorff distance) to the sphere…

GCNs distinguish graph models based on embeddings, but depth matters.

Characterizes GM-groups via sub-Riemannian geometry properties.

Proposes a new method to improve target annotation in ATR.

Study shows effective resistance distance yields more accurate network barycenter than Hamming distance.

Let $V$ be a separable Hilbert space, possibly infinite dimensional. Let $\St(p,V)$ be the Stiefel manifold of orthonormal frames of $p$ vectors in $V$, and let $\Gr(p,V)$ be the Grassmann manifold of $p$ dimensional subspaces of $V$. We study the distance and the geodesics in these manifolds, by reducing the matter to…

Geodesics in Sol geometry described with invariant k and spiral properties.

NeuralCut learns to select cutting planes by looking ahead, outperforming traditional methods.

Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.

NeVI-Cut uses neural networks to efficiently propagate uncertainty without feedback.

Uniform heat kernel and diffusion bridge asymptotics for sub-Riemannian geometry.

Given a point (the "spider") on a rectangular box, we would like to find the minimal distance along the surface to its opposite point (the "fly" - the reflection of the spider across the center of the box). Without loss of generality, we can assume that the box has dimensions $1\times a\times b$ with the spider on one …

This paper improves non-asymptotic bounds for denoising diffusions, focusing on the Ornstein-Uhlenbeck process.

Paper connects probability density cuts to graph theory eigenfunctions.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

Study Riemannian metrics on lens spaces, find cut loci and diameters.

We consider a left invariant Riemannian metric on SO(3) with two equal eigenvalues. We find the cut locus and the equation for the cut time. We find the diameter of such metric and describe the set of all most distant points from the identity. Also we prove that the cut locus and the cut time converge to the cut locus …

Stability of cut locus under metric perturbations in compact Riemannian manifolds.

New equivalence relation for links using cut-diagrams.

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $\varphi(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut v…