Limit Distributions and Sensitivity Analysis for Empirical Entropic Optimal Transport on Countable Spaces

For probability measures on countable spaces we derive distributional limits for empirical entropic optimal transport quantities. More precisely, we show that the empirical optimal transport plan weakly converges to a centered Gaussian process and that the empirical entropic optimal transport value is asymptotically normal. The results are valid for a large class of cost functions and generalize distributional limits for empirical entropic optimal transport quantities on finite spaces. Our proofs are based on a sensitivity analysis with respect to norms induced by suitable function classes, which arise from novel quantitative bounds for primal and dual optimizers, that are related to the exponential penalty term in the dual formulation. The distributional limits then follow from the functional delta method together with weak convergence of the empirical process in that respective norm, for which we provide sharp conditions on the underlying measures. As a byproduct of our proof technique, consistency of the bootstrap for statistical applications is shown.Comment: 68 page

Regularized Optimal Transport and the Rot Mover's Distance

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification

Quantum Computing and Hidden Variables I: Mapping Unitary to Stochastic Matrices

This paper initiates the study of hidden variables from the discrete, abstract perspective of quantum computing. For us, a hidden-variable theory is simply a way to convert a unitary matrix that maps one quantum state to another, into a stochastic matrix that maps the initial probability distribution to the final one in some fixed basis. We list seven axioms that we might want such a theory to satisfy, and then investigate which of the axioms can be satisfied simultaneously. Toward this end, we construct a new hidden-variable theory that is both robust to small perturbations and indifferent to the identity operation, by exploiting an unexpected connection between unitary matrices and network flows. We also analyze previous hidden-variable theories of Dieks and Schrodinger in terms of our axioms. In a companion paper, we will show that actually sampling the history of a hidden variable under reasonable axioms is at least as hard as solving the Graph Isomorphism problem; and indeed is probably intractable even for quantum computers.Comment: 19 pages, 1 figure. Together with a companion paper to appear, subsumes the earlier paper "Quantum Computing and Dynamical Quantum Models" (quant-ph/0205059

Computing Wasserstein Barycenter via operator splitting: the method of averaged marginals

The Wasserstein barycenter (WB) is an important tool for summarizing sets of probabilities. It finds applications in applied probability, clustering, image processing, etc. When the probability supports are finite and fixed, the problem of computing a WB is formulated as a linear optimization problem whose dimensions generally exceed standard solvers' capabilities. For this reason, the WB problem is often replaced with a simpler nonlinear optimization model constructed via an entropic regularization function so that specialized algorithms can be employed to compute an approximate WB efficiently. Contrary to such a widespread inexact scheme, we propose an exact approach based on the Douglas-Rachford splitting method applied directly to the WB linear optimization problem for applications requiring accurate WB. Our algorithm, which has the interesting interpretation of being built upon averaging marginals, operates series of simple (and exact) projections that can be parallelized and even randomized, making it suitable for large-scale datasets. As a result, our method achieves good performance in terms of speed while still attaining accuracy. Furthermore, the same algorithm can be applied to compute generalized barycenters of sets of measures with different total masses by allowing for mass creation and destruction upon setting an additional parameter. Our contribution to the field lies in the development of an exact and efficient algorithm for computing barycenters, enabling its wider use in practical applications. The approach's mathematical properties are examined, and the method is benchmarked against the state-of-the-art methods on several data sets from the literature

Weak limits of entropy regularized Optimal Transport; potentials, plans and divergences

This work deals with the asymptotic distribution of both potentials and couplings of entropic regularized optimal transport for compactly supported probabilities in $\R^d$. We first provide the central limit theorem of the Sinkhorn potentials -- the solutions of the dual problem -- as a Gaussian process in \Cs. Then we obtain the weak limits of the couplings -- the solutions of the primal problem -- evaluated on integrable functions, proving a conjecture of \cite{ChaosDecom}. In both cases, their limit is a real Gaussian random variable. Finally we consider the weak limit of the entropic Sinkhorn divergence under both assumptions $H_0:\ {\rm P}={\rm Q}$ or $H_1:\ {\rm P}\neq{\rm Q}$. Under $H_0$ the limit is a quadratic form applied to a Gaussian process in a Sobolev space, while under $H_1$, the limit is Gaussian. We provide also a different characterisation of the limit under $H_0$ in terms of an infinite sum of an i.i.d. sequence of standard Gaussian random variables. Such results enable statistical inference based on entropic regularized optimal transport