CHAZAL , Frédéric

Modern data often come as point clouds embedded in high dimensional Euclidean spaces, or possibly more general metric spaces. They are usually not distributed uniformly, but lie around some highly nonlinear geometric structures with nontrivial topology. Topological data analysis (TDA) is an emerging field whose goal is to provide mathematical...

In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary structures from data sampled around them. We prove that they can be approximated, with respect to the...

Given a probability measure with density, Fermat distances and density-driven metrics are conformal transformation of the Euclidean metric that shrink distances in high density areas and enlarge distances in low density areas.Although they have been widely studied and have shown to be useful in various machine learning tasks, they are limited...

Persistence diagrams play a fundamental role in Topological Data Analysis where they are used as topological descriptors of filtrations built on top of data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$ that can equivalently be seen as discrete measures in $\mathbb{R}^2$. When the data is assumed to be random, these...

Given a smooth compact codimension one submanifold S of Rk and a compact approximation K of S, we prove that it is possible to reconstruct S and to approximate the medial axis of S with topological guarantees using unions of balls centered on K. We consider two notions of noisy-approximation that generalize sampling conditions introduced by...

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space $\R^d$. However, the complexities of these methods scale up...

In many real-world applications data appear to be sampled around1-dimensional filamentary structures that can be seen as topologicalmetric graphs. In this paper we address the metric reconstructionproblem of such filamentary structures from data sampled aroundthem. We prove that they can be approximated, with respect tothe Gromov-Hausdorff...