一般分布间最优传输映射估计的统计收敛速率

Empirical Regularized Optimal Transport: Statistical Theory and Applications

We derive limit distributions for certain empirical regularized optimal transport distances between probability distributions supported on a finite metric space and show consistency of the (naive) bootstrap. In particular, we prove that the empirical regularized transport plan itself asymptotically follows a Gaussian law. The theory includes the Boltzmann-Shannon entropy regularization and hence a limit law for the widely applied Sinkhorn divergence. Our approach is based on an application of the implicit function theorem to necessary and sufficient optimality conditions for the regularized tr…

Distributed Statistical Estimation and Rates of Convergence in Normal Approximation

This paper presents a class of new algorithms for distributed statistical estimation that exploit divide-and-conquer approach. We show that one of the key benefits of the divide-and-conquer strategy is robustness, an important characteristic for large distributed systems. We establish connections between performance of these distributed algorithms and the rates of convergence in normal approximation, and prove non-asymptotic deviations guarantees, as well as limit theorems, for the resulting estimators. Our techniques are illustrated through several examples: in particular, we obtain new resul…

基于最优传输的对数凹测度坐标上升变分推断的收敛性

Convergence of coordinate ascent variational inference for log-concave measures via optimal transport

本文针对对数凹测度下坐标上升变分推断（CAVI）收敛性不明确的问题，借助最优传输中的位移凸性性质，证明了CAVI的收敛性，并分别给出了对数凹密度带Lipschitz梯度及强对数凹情形下的线性与指数收敛速率。

Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport

We derive nearly tight and non-asymptotic convergence bounds for solutions of entropic semi-discrete optimal transport. These bounds quantify the stability of the dual solutions of the regularized problem (sometimes called Sinkhorn potentials) w.r.t. the regularization parameter, for which we ensure a better than Lipschitz dependence. Such facts may be a first step towards a mathematical justification of annealing or $\varepsilon$-scaling heuristics for the numerical resolution of regularized semi-discrete optimal transport. Our results also entail a non-asymptotic and tight expansion of the d…

Posterior convergence for approximated unknowns in non-Gaussian statistical inverse problems

The statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation is studied in the Bayesian framework. In practice, one often considers only finite-dimensional unknowns and investigates numerically their probabilities. As many unknowns are function-valued, it is of interest to know whether the estimated probabilities converge when the finite-dimensional approximations of the unknown are refined. In this work, the generalized Bayes formula is shown to be a powerful tool in the convergence studies. With the help of…

Poisson点过程的收敛及最优传输正则化在PET重建变分分析中的应用

Convergence of Poisson point processes and of optimal transport regularization with application in variational analysis of PET reconstruction

本文针对PET等反问题，在信噪比趋于无穷大时，对使用最优传输正则化的动态贝叶斯重建泛函进行变分分析，研究其重建收敛性及与探测器尺寸等参数的相互作用。

Empirical optimal transport on countable metric spaces: Distributional limits and statistical applications

We derive distributional limits for empirical transport distances between probability measures supported on countable sets. Our approach is based on sensitivity analysis of optimal values of infinite dimensional mathematical programs and a delta method for non-linear derivatives. A careful calibration of the norm on the space of probability measures is needed in order to combine differentiability and weak convergence of the underlying empirical process. Based on this we provide a sufficient and necessary condition for the underlying distribution on the countable metric space for such a distrib…

Statistical Convergence of the EM Algorithm on Gaussian Mixture Models

We study the convergence behavior of the Expectation Maximization (EM) algorithm on Gaussian mixture models with an arbitrary number of mixture components and mixing weights. We show that as long as the means of the components are separated by at least $Ω(\sqrt{\min\{M,d\}})$, where $M$ is the number of components and $d$ is the dimension, the EM algorithm converges locally to the global optimum of the log-likelihood. Further, we show that the convergence rate is linear and characterize the size of the basin of attraction to the global optimum.

Stability and upper bounds for statistical estimation of unbalanced transport potentials

In this note, we derive upper-bounds on the statistical estimation rates of unbalanced optimal transport (UOT) maps for the quadratic cost. Our work relies on the stability of the semi-dual formulation of optimal transport (OT) extended to the unbalanced case. Depending on the considered variant of UOT, our stability result interpolates between the OT (balanced) case where the semi-dual is only locally strongly convex with respect the Sobolev semi-norm H1 dot and the case where it is locally strongly convex with respect to the H 1 norm. When the optimal potential belongs to a certain class C w…

概率测度间熵正则最优传输随机算法的渐近分布与收敛速率

Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures

针对熵正则最优传输的 Sinkhorn 散度估计，论文提出基于半对偶形式的 Robbins-Monro 递推随机算法，证明其在离散与半离散情形下的几乎必然收敛、渐近正态性及期望超额风险收敛速率。

Graphical and uniform consistency of estimated optimal transport plans

A general theory is provided delivering convergence of maximal cyclically monotone mappings containing the supports of coupling measures of sequences of pairs of possibly random probability measures on Euclidean space. The theory is based on the identification of such a mapping with a closed subset of a Cartesian product of Euclidean spaces and leveraging tools from random set theory. Weak convergence in the appropriate Fell space together with the maximal cyclical monotonicity then automatically yields local uniform convergence of the associated mappings. Viewing such mappings as optimal tran…

通过适应最优输运的依赖性度量：稳定性与收敛速率

Dependence Measures via Adapted Optimal Transport: Stability and Rates of Convergence

提出基于适应最优输运的收敛模式，恢复条件分布的弱连续性，并导出依赖度量插件估计器的O(N^{-1/3})收敛速率。