Entropic Regularization of Optimal Transport

最优输运（OT）是将一个质量分布尽可能有效地移动到另一个质量分布的一般问题，是离散匹配问题的连续延伸。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）中的恰好一个数据点关联起来。这种匹配的发生是有成本的（比如病人需要走到医院的距离）。最优匹配是使平均成本最小化的匹配。Monge Kantorovich OT的数学已经发展成为许多科学学科的统一主题，从纯数学领域，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，两项菲尔兹奖（Villani 2010, Figalli 2018）强调了最优运输在黎曼几何不等式和里奇曲率研究中的影响。另一方面，在分析来自流形的高维数据时，OT已经成为经典的基于最大似然的方法的可行替代方案。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分析，干细胞生物学的新应用，以及诺贝尔奖得主Gale-Shapely算法的推广，在线拍卖和期权定价等经济应用。其中大部分是由于OT中基于熵的正则化的计算方法的令人印象深刻的飞跃。本项目主要研究最优运输问题熵正则化领域的新概率问题及其应用。该项目还为研究生提供了培训机会。PI在概率论和Monge-Kantorovich OT的交叉点上提出了广泛的问题，跨越了几个数学和应用学科。它为统计学家、计算机科学家、分析学家和几何学者感兴趣的质量传输问题带来了新的概率工具和视角（如互换性、高斯混沌展开、Metropolis算法、平均场粒子相互作用）。主要关注的是离散和连续熵正则化OT的渐近分析，也称为薛定谔桥，无论是当温度趋近于零，还是当数据大小趋近于无穷大，或两者兼而有之。提出的问题之一涉及基于马尔可夫链的OT计算方法，如果解决，将立即应用于依赖于OT计算的各种应用学科。其他应用包括数学金融和机器人中的交易成本分析，其中大量相互作用的无人机的动力学由一种新颖的McKean-Vlasov型极限描述。大量的数学读者会发现这些材料接近他们的兴趣，该项目有望促进在OT中工作的概率学家和非概率学家之间的进一步跨学科合作。pi领导的PIMS Kantorovich倡议也促进了这一点，该倡议为位于太平洋西北地区的OT跨学科工作创建了基础设施。该奖项反映了美国国家科学基金会的法定使命，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.