Cotsakis, Ryan Di Bernardino, Elena Opitz, Thomas

We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets.To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb R^2$ observed over regular square tilings. The proposed estimator acts on the empirical...

Divol, Vincent

Topological data analysis (or TDA for short) consists in a set of methods aiming to extract topological and geometric information from complex nonlinear datasets. This field is here tackled from two different perspectives.First, we consider techniques from geometric inference, whose goal is to reconstruct geometric invariants of a manifold thanks t...

Di Bernardino, Elena Duval, Céline

In the present article we study the average of Lipschitz-Killing (LK) curvatures of the excursion set of a stationary isotropic Gaussian field X on R 2. The novelty is that the field can be nonstandard, that is, with unknown mean and variance, which is more realistic from an applied viewpoint. To cope with the unknown location and scale parameters ...

Aaron, C Bodart, O

Consider a sample Xn = {X1 ,. .. , X n } of i.i.d variables drawn with a probability distribution P supported on a set M ⊂ R d. This article mainly deals with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M , a general convergence result is proved. Assuming M to be a man-ifold of know...

Cuel, Louis

Ces travaux s'inscrivent dans la thématique de l'inférence géométrique dont le but est de répondre au problème suivant : étant donné un objet géométrique dont on ne connaît qu'une approximation, peut-on estimer de manière robuste ses propriétés? On se place dans cette thèse dans le cas où l'approximation est un nuage de points ou un ensemble digita...

Guibas, Leonidas Morozov, Dmitriy Mérigot, Quentin
Published in
Discrete & Computational Geometry

Distance functions to compact sets play a central role in several areas of computational geometry. Methods that rely on them are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoni...

Chazal, Frédéric Cohen-Steiner, David Mérigot, Quentin
Published in
Foundations of Computational Mathematics

Data often comes in the form of a point cloud sampled from an unknown compact subset of Euclidean space. The general goal of geometric inference is then to recover geometric and topological features (e.g., Betti numbers, normals) of this subset from the approximating point cloud data. It appears that the study of distance functions allows one to ad...

Biau, Gérard Chazal, Frédéric Cohen-Steiner, David Devroye, Luc Rodriguez, Carlos

Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the k-nearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong appr...