Related papers: Monge-Kantorovich Depth, Quantiles, Ranks, and Sig…
The quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories. Motivated by the categorical interpretation of partial metric spaces, we are led to investigate the…
Universality of local eigenvalue statistics is one of the most striking phenomena of Random Matrix Theory, that also accounts for a lot of the attention that the field has attracted over the past 15 years. In this paper we focus on the…
Statistical depths provide a fundamental generalization of quantiles and medians to data in higher dimensions. This paper proposes a new type of globally defined statistical depth, based upon control theory and eikonal equations, which…
The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space $\mathbb{R}^d$, we introduce…
In this note, following \cite{Chitescuetal2014}, we show that the Monge-Kantorovich norm on the vector space of countably additive measures on a compact metric space has a primal representation analogous to the Hanin norm, meaning that…
The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category…
Inspired by the recent theory of Entropy-Transport problems and by the $\mathbf{D}$-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly…
Optimization problems with stochastic dominance constraints provide a possibility to shape risk by selecting a benchmark random outcome with a desired distribution. The comparison of the relevant random outcomes to the respective benchmarks…
In 2010, Vershik proposed a new combinatorial invariant of metric spaces given by a class of polytopes that arise in the theory of optimal transport and are called ``Wasserstein polytopes'' or ``Kantorovich-Rubinstein polytopes'' in the…
The Monge-Kantorovich mass-transportation problem has been shown to be fundamental for various basic problems in analysis and geometry in recent years. Shen and Zheng (2010) proposed a probability method to transform the celebrated…
In this paper we introduce a nonextensive quantum information theoretic measure which may be defined between any arbitrary number of density matrices, and we analyze its fundamental properties in the spectral graph-theoretic framework.…
These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures…
We define a new two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family contains the classical Monge-Kantorovich…
The concept of depth has proved very important for multivariate and functional data analysis, as it essentially acts as a surrogate for the notion a ranking of observations which is absent in more than one dimension. Motivated by the rapid…
We study the log-concave measures, their characterization via the Pr\'ekopa-Leindler property and also define a subset of it whose elements are called super log-concave measures which have the property of satisfying a logarithmic Sobolev…
In this paper, we extend the upper and lower bounds for the "pseudo-distance" on quantum densities analogous to the quadratic Monge-Kantorovich(-Vasershtein) distance introduced in [F. Golse, C. Mouhot, T. Paul, Commun. Math. Phys. 343…
Discrepancy measures between probability distributions are at the core of statistical inference and machine learning. In many applications, distributions of interest are supported on different spaces, and yet a meaningful correspondence…
We investigate the properties of convex functions in the plane that satisfy a local inequality which generalizes the notion of sub-solution of Monge-Ampere equation for a Monge-Kantorovich problem with quadratic cost between non-absolutely…
We provide a Central Limit Theorem for the Monge-Kantorovich distance between two empirical distributions with size $n$ and $m$, $W_p(P_n,Q_m)$ for $p>1$ for observations on the real line, using a minimal amount of assumptions. We provide…
A measure theoretical approach is presented to study the Monge-Kantorovich optimal mass transport problem. This approach together with Kantorovich duality provide an effective tool to answer a long standing question about the support of…