arXiv subjects

Thibaut Le Gouic

Publications and source records attributed to Thibaut Le Gouic.

24 records · Page 2Linked to original sources

arXiv2018

Unconstrained and Curvature-Constrained Shortest-Path Distances and their Approximation

We study shortest paths and their distances on a subset of a Euclidean space, and their approximation by their equivalents in a neighborhood graph defined on a sample from that subset. In particular, we recover and extend the results of Bernstein et al. (2000). We do the same with curvature-constrained shortest paths and their distances, establishing what we believe are the first approximation bounds for them.

cs.CG↗

arXiv2018

A notion of stability for k-means clustering

In this paper, we define and study a new notion of stability for the $k$-means clustering scheme building upon the notion of quantization of a probability measure. We connect this notion of stability to a geometric feature of the underlying distribution of the data, named absolute margin condition, inspired by recent works on the subject.

math.ST↗

arXiv2016

Existence and Consistency of Wasserstein Barycenters

In this paper, based on the Fr{é}chet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random distributions defined on a geodesic space (E, d). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the distributions or of a growing set of distributions.

math.ST↗

arXiv2015

Mass localization

For a given class $\mathcal{F}$ of closed sets of a measured metric space $(E,d,μ)$, we want to find the smallest element $B$ of the class $\mathcal{F}$ such that $μ(B)\geq 1-α$, for a given $0<α<1$. This set $B$ \textit{localizes the mass} of $μ$. Replacing the measure $μ$ by the empirical measure $μ_n$ gives an empirical smallest set $B_n$. The article introduces a formal definition of small sets (and their size) and study the convergence of the sets $B_n$ to $B$ and of their size.

math.ST↗

arXiv2013

Distribution's template estimate with Wasserstein metrics

In this paper we tackle the problem of comparing distributions of random variables and defining a mean pattern between a sample of random events. Using barycenters of measures in the Wasserstein space, we propose an iterative version as an estimation of the mean distribution. Moreover, when the distributions are a common measure warped by a centered random operator, then the barycenter enables to recover this distribution template.

math.ST↗

arXiv2011

On the mean speed of convergence of empirical and occupation measures in Wasserstein distance

In this work, we provide non-asymptotic bounds for the average speed of convergence of the empirical measure in the law of large numbers, in Wasserstein distance. We also consider occupation measures of ergodic Markov chains. One motivation is the approximation of a probability measure by finitely supported measures (the quantization problem). It is found that rates for empirical or occupation measures match or are close to previously known optimal quantization rates in several cases. This is notably highlighted in the example of infinite-dimensional Gaussian measures.

math.PR↗