arXiv subjects

Jérôme Taupin

Publications and source records attributed to Jérôme Taupin.

3 recordsLinked to original sources

arXiv2026

Estimation of Conformal Metrics

We study deformations of the geodesic distances on a domain of R N induced by a function called conformal factor. We show that under a positive reach assumption on the domain (not necessarily a submanifold) and mild assumptions on the conformal factor, geodesics for the conformal metric have good regularity properties in the form of a lower bounded reach. This regularity allows for efficient estimation of the conformal metric from a random point cloud with a relative error proportional to the Hausdorff distance between the point cloud and the original domain. We then establish convergence rates of order n^(-1/d) that are close to sharp when the intrinsic dimension d of the domain is large, for an estimator that can be computed in O(n^2 ) time. Finally, this paper includes a useful equivalence result between ball graphs and nearest-neighbors graphs when assuming Ahlfors regularity of the sampling measure, allowing to transpose results from one setting to another.

math.ST↗

arXiv2026

Fermat Distance-to-Measure: a robust Fermat-like metric

Given a probability measure with density, Fermat distances and density-driven metrics are conformal transformations 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 to measures with density (with respect to Lebesgue measure, or volume form on manifold). In this paper, by replacing the density with the Distance-to-Measure, we introduce a new metric, the Fermat Distance-to-Measure, defined for any probability measure in R^d. We derive strong stability properties for the Fermat Distance-to-Measure with respect to the measure and propose an estimator from random sampling of the same measure, featuring an explicit bound on its convergence rate.

math.ST↗

arXiv2025

Improved Trial and Error Learning for Random Games

When a game involves many agents or when communication between agents is not possible, it is useful to resort to distributed learning where each agent acts in complete autonomy without any information on the other agents' situations. Perturbation-based algorithms have already been used for such tasks. We propose some improvements based on practical observations to improve the performance of these algorithms. We show that the introduction of these changes preserves their theoretical convergence properties towards states that maximize the average reward and improve them in the case where optimal states exist. Moreover, we show that these algorithms can be made robust to the addition of randomness to the rewards, achieving similar convergence guarantees. Finally, we discuss the possibility for the perturbation factor of the algorithm to decrease during the learning process, akin to simulated annealing processes.

math.OC↗