This paper proves variants of the triangle inequality for the quantum analogues of the Wasserstein metric of exponent 2 introduced in Golse et al. (2016) to compare two density operators, and in Golse and Paul (2017) to compare a phase space probability measure and a density operator. The argument differs noticeably from the classical proof of the ...
In this thesis we apply the optimal transport (OT) theory to various disciplines of applied and computational mathematics such as scientific computing, numerical analysis, and dynamical systems. The research consists of three aspects: (1) We focus on solving OT problems from different perspectives including (a) direct approximation of the OT map in...
Based on the novel concept of multivariate center-outward quantiles introduced recently in Chernozhukov et al. (2017) and Hallin et al. (2021), we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional pr...
Enthusiasm for Machine Learning is spreading to nearly all fields such as transportation, energy, medicine, banking or insurance as the ubiquity of sensors through IoT makes more and more data at disposal with an ever finer granularity. The abundance of new applications for monitoring of complex infrastructures (e.g. aircrafts, energy networks) tog...
Graph management systems are emerging as an efficient solution to store and query graph-oriented data. To assess the performance and compare such systems, practitioners often design benchmarks in which they use large scale graphs. However, such graphs either do not fit the scale requirements or are not publicly available. This has been the incentiv...
Published in Journal of Statistical Physics
Landauer’s Principle states that the energy cost of information processing must exceed the product of the temperature, Boltzmann’s constant, and the change in Shannon entropy of the information-bearing degrees of freedom. However, this lower bound is achievable only for quasistatic, near-equilibrium computations—that is, only over infinite time. In...
Using optimal transport in image processing tasks has become very popular. However, it still faces difficult computational issues when dealing with high dimensional distributions. We propose here to use the recently introduced GMM-OT formulation, which consists in restricting the optimal transport problem to the set of Gaussian mixture models. As a...
The labour market is closely linked to the rental markets for professionals and for individuals. The purpose of this thesis is to study the interactions of these markets.In a first step, we develop and study a mean field game model that links the labour market with the rental market for professionals. In a specific framework where firms' production...
We address the problem of proving a Central Limit Theorem for the empirical optimal transport cost, $\sqrt{n}\{\mathcal{T}_c(P_n,Q)-\mathcal{T}_c(P,Q)\}$, in the semi discrete case, i.e when the distribution $P$ is finitely supported. We show that the asymptotic distribution is the supremun of a centered Gaussian process which is Gaussian under som...
Published in Alzheimer's Research & Therapy
BackgroundTo help clinicians provide timely treatment and delay disease progression, it is crucial to identify dementia patients during the mild cognitive impairment (MCI) stage and stratify these MCI patients into early and late MCI stages before they progress to Alzheimer’s disease (AD). In the process of diagnosing MCI and AD in living patients,...
