Entropic Regularization of Optimal Transport

最优运输（OT）是尽可能有效地将一种质量分布转移到另一种质量分布的一般问题，是离散匹配问题的连续扩展。匹配问题涉及将一个集合（比如病人）中的每个数据点与另一个集合（比如医院）中的一个数据点相关联。这种匹配需要花费一定的费用（比如患者需要前往医院的距离）。最佳匹配是使平均成本最小化的匹配。Monge Kantorovich OT的数学已经成为许多科学学科的统一主题，从纯粹的数学领域，如分析，几何和偏微分方程到经济学，统计学，机器学习和人工智能的革命性新方法。例如，在过去十年中，两个菲尔兹奖（Villani 2010，Figalli 2018）突出了最优运输在黎曼几何中的几何不等式和Ricci曲率研究中的影响。另一方面，OT已经确立了自己作为一个可行的替代经典的最大似然方法在分析高维数据来自一个流形。其他值得注意的应用包括机器学习和人工智能中的神经网络和对抗网络的分析，干细胞生物学的新应用，以及经济应用，如诺贝尔奖得主Gale-Shapely算法的推广，在线拍卖和期权定价。其中很大一部分是由于OT中的计算方法的令人印象深刻的飞跃，这些方法取决于基于熵的正则化。这个项目的重点是新的概率问题在这方面的熵正则化的最佳运输问题及其应用。该项目还为研究生提供培训机会。PI提出了概率论和蒙格-康托洛维奇OT交叉点的问题，这些问题范围广泛，跨越多个数学和应用学科。它为统计学家、计算机科学家、分析师和几何学家感兴趣的质量输运问题带来了新的概率工具和观点（如交换矩阵、高斯混沌展开、大都会算法、平均场粒子相互作用）。主要重点是离散和连续熵正则化OT（也称为薛定谔桥）的渐近分析，无论是温度趋于零还是数据大小趋于无穷大，或者两者兼而有之。提出的问题之一涉及基于马尔可夫链的计算方法OT，如果解决，将有直接的应用程序依赖于OT计算的各种应用学科。其他应用包括数学金融和机器人技术中的交易成本分析，其中大量相互作用的无人机的动态由新颖的McKean-Vlasov型极限描述。大量的数学观众会发现材料接近他们的兴趣，该项目预计将刺激概率学家和非概率学家在OT工作之间的进一步跨学科合作。由PI领导的PIMS Kantorovich倡议也促进了这一点，该倡议为基于位于太平洋西北部的OT的跨学科工作创建了基础设施。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Optimal transport (OT) is the general problem of moving one distribution of mass to another as efficiently as possible and is the continuum extension of the discrete problem of matching. A matching problem involves associating with each data point in one set (say patients), exactly one data point in another set (say hospitals). There is a cost incurred for this match to occur (say the distance the patient needs to travel to get to the hospital). An optimal matching is one that minimizes the average cost. The mathematics of Monge Kantorovich OT has grown to be a unifying theme in many scientific disciplines, from purely mathematical areas such as analysis, geometry, and partial differential equations to revolutionary new methods in economics, statistics, machine learning and artificial intelligence. For example, the impact of optimal transport in the study of geometric inequalities and Ricci curvature in Riemannian geometry was highlighted by two Fields medals (Villani 2010, Figalli 2018) in the last decade. On the other hand OT has established itself as a viable alternative to classical maximum likelihood based methods in analyzing high-dimensional data coming from a manifold. Other notable applications include analysis of neural networks and adversarial networks in machine learning and artificial intelligence, new applications to stem cell biology, and economic applications such as generalizations of the Nobel prize winning Gale-Shapely algorithm, online auctions, and option pricing. Much of these are due to impressive leaps in computational methods in OT that hinge on entropy based regularizations. This project focuses on new probability questions in this area of entropic regularization of optimal transport problems and their applications. The project also provides training opportunities for graduate students.The PI proposes problems at the intersection of probability and Monge-Kantorovich OT that are broad and cut across several mathematical and applied disciplines. It brings new probabilistic tools and perspectives (such as exchangeability, Gaussian chaos expansions, Metropolis algorithm, mean-field particle interactions) to problems of mass transport that are of interest to statisticians, computer scientists, analysts and geometers. The primary focus is on the asymptotic analysis of both discrete and continuous entropy-regularized OT, also called Schroedinger bridges, either as the temperature goes to zero or as the data size goes to infinity or both. One of the proposed problems involves Markov chain based computational method for OT, which, if resolved, will have immediate applications to various applied disciplines that depend on OT computations. Other applications involve analysis of transaction costs in mathematical finance and robotics where the dynamics of a large number of interacting drones is described by a novel McKean-Vlasov type limit. A large mathematical audience will find the material close to their interests and the project is expected to spur further cross-disciplinary collaborations between probabilists and non-probabilists working in OT. This is also facilitated by the PI-led PIMS Kantorovich Initiative that creates an infrastructure for interdisciplinary work based on OT located in the Pacific Northwest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.