arXiv research

We find lower and upper bounds for the risk of estimating a manifold in Hausdorff distance under several models. We also show that there are close connections between manifold estimation and the problem of deconvolving a singular measure.

The Hausdorff Distance (HD) is widely used in evaluating medical image segmentation methods. However, existing segmentation methods do not attempt to reduce HD directly. In this paper, we present novel loss functions for training convolutional neural network (CNN)-based segmentation methods with the goal of reducing HD…

A method refines weights to estimate smooth manifolds from noisy data.

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

The paper extends vector bundle theory to non-Hausdorff manifolds.

We construct a sequence of smooth Ricci flows on $T^2$, with standard uniform $C/t$ curvature decay, and with initial metrics converging to the standard flat unit-area square torus $g_0$ in the Gromov-Hausdorff sense, with the property that the flows themselves converge not to the static Ricci flow $g(t)\equiv g_0$, bu…

This paper introduces Hausdorff measure and its applications in fractal geometry.

We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.

Study of one-dimensional non-Hausdorff manifolds and their quotient to CW complexes.

Hausdorff reflection keeps space shape intact.

Study of de Rham cohomology on non-Hausdorff manifolds.

A novel UNet detector detects sheep in UAV imagery.

Geodesics with bounded angles have zero Hausdorff dimension.

The paper proves stability in compact finite dimensional Alexandrov spaces using equivariant Gromov--Hausdorff convergence.

New examples show non-integer Hausdorff dimensions in collapsing spaces.

Paper studies Hausdorff dimension of limit sets for Kleinian groups.

Compact Kahler-Einstein manifolds converge to semi-log canonical models.

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

Study of Calabi-Yau metrics on converging manifolds, resolving conjectures.

The paper classifies Kleinian groups with Hausdorff dimension less than 1.

Upper bound for Hausdorff distance between hyperbolic space and its medianization.

Study Gromov-Hausdorff convergence of metric pairs and tuples.

New Hausdorff integrations for Lie algebroids and symplectic groupoids.

Research shows surfaces close to planes in Hausdorff distance.

Study describes limits of surfaces in a mathematical space.

The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.

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

Compactness theorem for timed-metric spaces established.

Study on Hausdorff dimension of Anosov subgroup limit sets under specific affine complexity.

Polynomial algorithm estimates mGH distance between unweighted graphs.

Study extends Hausdorff dimension Hessian results to new hyperconvex representations.

Paper calculates Gromov-Hausdorff distance between simplexes and 2-distance spaces.

Investigates Künneth formula for foliated de Rham cohomology, overcoming non-Hausdorff issues.

Paper studies Hausdorff dimensions of specific limit sets for groups on curved spaces.

Stability of Wasserstein spaces under various convergence types.

This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i.e., a smooth submanifold N for which the growth vector of the distributi…

We study the relation between critical exponents and Hausdorff dimensions of limit sets for projective Anosov representations. We prove that the Hausdorff dimension of the symmetric limit set in $\mathbf{P}(\mathbb{R}^{n}) \times \mathbf{P}({\mathbb{R}^{n}}^*)$ is bounded between two critical exponents associated respe…

We prove here that the Poincaré exponent of a geometrically finite group od isometries of the 3-dimensionnal hyperbolic space coincides with the Hausdorff dimension of its limit set. We also compare the natural measures supported by this set: the Patterson measure and the Hausdorff and packing measures corresponding to…

We continue our investigation of the space of geodesic laminations on a surface, endowed with the Hausdorff topology. We determine the topology of this space for the once-punctured torus and the 4-times-punctured sphere. For these two surfaces, we also compute the Hausdorff dimension of the space of geodesic lamination…

It is known that shape injectivity implies homotopical Hausdorff and that the converse does not hold, even if the space is required to be a Peano continuum. This paper gives an alternative definition of homotopical Hausdorff inspired by a new topology on the set of fixed endpoint homotopy classes of paths. This version…

Let $π: \mathcal{X}^* \rightarrow B^*$ be an algebraic family of compact Kähler manifolds of complex dimension $n$ with negative first Chern class over a punctured disc $B^*\in \mathbb{C}$. Let $g_t$ be the unique Kähler-Einstein metric on $\mathcal{X}_t= π^{-1}(t)$. We show that as $t\rightarrow 0$, $(\mathcal{X}_t, g…

We study topological properties of the Gromov-Hausdorff metric on the set of isometry classes of nonnegatively curved $2$-spheres.

A clustering algorithm based on the Hausdorff distance is introduced and compared to the single and complete linkage. The three clustering procedures are applied to a toy example and to the time series of financial data. The dendrograms are scrutinized and their features confronted. The Hausdorff linkage relies of firm…

New geometric measure simplifies complex analysis.

New theorem on flat tori stability using harmonic maps and Ricci flow.

The paper connects geometric and topological concepts to bound distances between metric spaces.

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal posi…

A clustering procedure, based on the Hausdorff distance, is introduced and tested on the financial time series of the Dow Jones Industrial Average (DJIA) index.