Related papers: Quantization and martingale couplings
The Wasserstein metric has become increasingly important in many machine learning applications such as generative modeling, image retrieval and domain adaptation. Despite its appeal, it is often too costly to compute. This has motivated…
We investigate here the optimal transportation problem on configuration space for the quadratic cost. It is shown that, as usual, provided that the corresponding Wasserstein is finite, there exists one unique optimal measure and that this…
We propose a new method to estimate Wasserstein distances and optimal transport plans between two probability distributions from samples in high dimension. Unlike plug-in rules that simply replace the true distributions by their empirical…
We show that combining randomized measurement protocols with importance sampling allows for characterizing entanglement in significantly larger quantum systems and in a more efficient way than in previous work. A drastic reduction of…
The objective of this paper is to develop a duality between a novel Entropy Martingale Optimal Transport problem (A) and an associated optimization problem (B). In (A) we follow the approach taken in the Entropy Optimal Transport (EOT)…
The optimal transport problem with quadratic regularization is useful when sparse couplings are desired. The density of the optimal coupling is described by two functions called potentials; equivalently, potentials can be defined as a…
Given a determinate (multivariate) probability measure $\mu$, we characterize Gaussian mixtures $\nu\_\phi$ which minimize the Wasserstein distance $W\_2(\mu,\nu\_\phi)$ to $\mu$ when the mixing probability measure $\phi$ on the parameters…
In this thesis, we consider the Wasserstein barycenter problem of discrete probability measures from computational and statistical sides. The statistical focus is estimating the sample size of measures necessary to calculate an…
Consider the empirical measure, $\hat{\mathbb{P}}_N$, associated to $N$ i.i.d. samples of a given probability distribution $\mathbb{P}$ on the unit interval. For fixed $\mathbb{P}$ the Wasserstein distance between $\hat{\mathbb{P}}_N$ and…
Given the univariate marginals of a real-valued, continuous-time martingale, (respectively, a family of measures parameterised by $t \in [0,T]$ which is increasing in convex order, or a double continuum of call prices) we construct a family…
We consider the Monge-Kantorovich problem between two random measuress. More precisely, given probability measures $\mathbb{P}_1,\mathbb{P}_2\in\mathcal{P}(\mathcal{P}(M))$ on the space $\mathcal{P}(M)$ of probability measures on a smooth…
Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ is also optimal (between its…
This article establishes cutoff convergence or abrupt convergence of three statistical quantities for multivariate (Hurwitz) stable geometric Brownian motion: the autocorrelation function, the Wasserstein distance between the current state…
This article presents a general approximation-theoretic framework to analyze measure transport algorithms for probabilistic modeling. A primary motivating application for such algorithms is sampling -- a central task in statistical…
Distributionally robust chance constrained programs minimize a deterministic cost function subject to the satisfaction of one or more safety conditions with high probability, given that the probability distribution of the uncertain problem…
Inverse problems in physical or biological sciences often involve recovering an unknown parameter that is random. The sought-after quantity is a probability distribution of the unknown parameter, that produces data that aligns with…
We study optimization problems in which a linear functional is maximized over probability measures that are dominated by a given measure according to an integral stochastic order in an arbitrary dimension. We show that the following four…
Randomized measurements constitute a simple measurement primitive that exploits the information encoded in the outcome statistics of samples of local quantum measurements defined through randomly selected bases. In this work we exploit the…
The general notion of a stochastic ordering is that one probability distribution is smaller than a second one if the second attaches more probability to higher values than the first. Motivated by recent work on barycentric maps on spaces of…
We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. We formulate this first-order optimality condition using…