Towards Harmonic Analysis in Wasserstein Space: Low-Dimensional Structures, Learning, and Algorithms

开发有效的统计和计算数据分析方法是21世纪的一项重要任务。该项目将利用大型高维数据中的结构，这些结构有可能改变包括图像处理、自然语言处理和计算社会科学在内的一系列科学领域。数据将被建模为概率度量，以便开发易于处理和数据高效的机器学习方法，这些方法关键地利用数据的内在几何属性(例如，图像中的形状和文本文档中的语义)。重点放在与最先进的黑盒方法竞争的理论基础和可伸缩算法上，同时保持高度的可解释性。除了对数学、数据科学和机器学习的核心关注之外，这些框架和算法还立即应用于地球科学和地理。开发的新数据分析工具将使调查人员能够解决这些领域长期存在的未决问题，这将提供重要的验证数据来源。这个项目的一个主要组成部分是培养博士生和本科生在数学、数据科学和计算的交叉领域进行研究。具体地说，该项目的重点是以下基本问题：(1)用概率度量的本质低维集合进行建模和计算；(2)这些低维集合的可学习性和表现力。研究人员将数据视为Wasserstein空间中的度量，并建议通过允许相对于一组参考分布有效地合成和分析观测到的度量，来开发经典向量空间环境中的低维模型的类似物。重点讨论了与Wasserstein空间中的低维结构有关的两个主要问题。首先，对于沃瑟斯坦空间的本质低维的子集来说，什么是“正确的”模型？如何计算它们？研究人员建议通过提出三个模型来模拟低维子空间(精确和近似)的概念，这些模型利用了Wasserstein空间的几何特性以及最近在熵正则化方面的计算进展。这三个模型的目标是通过它们在Wasserstein空间的这些低维子集上的系数来有效地编码(分析)度量，并解码(合成)。重点是从统计(例如，从测量中估计给定样本)和计算(例如，设计具有亚立方复杂性的算法)两个角度的效率。第二，什么是沃瑟斯坦空间中有效的、可计算的和富有表现力的表象系统？研究人员建议解决数据驱动的问题，即在给定观察到的概率度量的情况下，如何识别和学习能够有效表示它们的少量参考度量。同时，研究人员将研究基本问题，即什么系统具有足够的表现力来表示沃瑟斯坦空间的典型元素。这提供了计算算法和沃瑟斯坦空间中调和分析的雏形理论之间的关键联系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Developing efficient methods for statistical and computational data analysis is an essential task in the 21st century. This project will leverage structures in large, high-dimensional data which has the potential to transform a range of scientific areas including image processing, natural language processing, and computational social sciences. Data will be modeled as probability measures in order to develop tractable and data-efficient machine learning methods that crucially leverage intrinsic geometric properties of the data (e.g., shapes in images and semantics in text documents). The focus is on theoretical foundations and scalable algorithms that compete with state-of-the-art black box methods while retaining a high degree of interpretability. Beyond the core focus on mathematics, data science, and machine learning, these frameworks and algorithms have immediate applications to geoscience and geography. The new data analysis tools developed will allow the investigators to address long-standing open problems in these fields, which will provide important sources of validation data. A major component of this project is training both PhD students and undergraduates in research at the intersection of mathematics, data science, and computing.Specifically, the focus of the project is on fundamental problems of (1) modeling with and computing in intrinsically low-dimensional sets of probability measures and (2) the learnability and expressivity of these low-dimensional sets. The investigators treat data as measures in Wasserstein space and propose to develop analogues of low-dimensional models in the classical vector space setting by allowing for efficient synthesis and analysis of observed measures with respect to a set of reference distributions. The focus is on two major problems pertaining to low-dimensional structures in Wasserstein space. First, what is the "correct" model for intrinsically low-dimensional subsets of Wasserstein space, and how can they be computed? The investigators propose to mimic notions of low-dimensional subspace (exact and approximate), by proposing three models that leverage the geometric properties of Wasserstein space, as well as recent computational advances in entropic regularization. The goal for these three models is to efficiently encode (analyze) measures by their coefficients in these low-dimensional subsets of Wasserstein space, and also decode (synthesize). The focus is on efficiency from both a statistical (e.g., estimating given samples from the measures) and computational (e.g., designing algorithms with sub-cubic complexity) perspective. Second, what are the efficient, computable, and expressive representational systems in Wasserstein space? The investigators propose to tackle the data-driven problem of, given observed probability measures, how to identify and learn a small number of reference measures that can represent them efficiently. In parallel, the investigators will study the fundamental problem of what systems are expressive enough to represent typical elements of Wasserstein space. This provides the crucial connection between the computational algorithms and an embryonic theory of harmonic analysis in Wasserstein space.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.