DIVOL , Vincent

We focus on the problem of manifold estimation: given a set of observations sampled close to some unknown submanifold M , one wants to recover information about the geometry of M . Minimax estimators which have been proposed so far all depend crucially on the a priori knowledge of parameters quantifying the underlying distribution generating...

We provide a short proof that the Wasserstein distance between the empirical measure of a n-sample and the estimated measure is of order n^−1/d , if the measure has a lower and upper bounded density on the d-dimensional flat torus.

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...

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 R 2 that can equivalently be seen as discrete measures in R 2. When the data come as a random point cloud, these discrete measures...

Persistence diagrams are efficient descriptors of the topology of a point cloud. As they do not naturally belong to a Hilbert space, standard statistical methods cannot be directly applied to them. Instead, feature maps (or representations) are commonly used for the analysis. A large class of feature maps, which we call linear, depends on some...

Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of discrete measures, and by...

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...

Persistence diagrams (PDs) are the most common descriptors used to encode the topology of structured data appearing in challenging learning tasks; think e.g. of graphs, time series or point clouds sampled close to a manifold. Given random objects and the corresponding distribution of PDs, one may want to build a statistical summary-such as a...

In general, the critical points of the distance function d_M to a compact submanifold M ⊂ R^D can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and µ-critical points of a submanifold and by proving that they are generically satisfied and stable with respect to...