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

Wasserstein 空间中的调和分析：低维结构、学习和算法

• 批准号: 2309519
• 负责人: James Murphy
• 金额: $ 37万
• 依托单位: Tufts University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309519&HistoricalAwards=false
• 关键词: Towards; Harmonic; Analysis; Wasserstein; Space

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.

发展有效的统计和计算数据分析方法是世纪的重要任务。 该项目将利用大型高维数据中的结构，这些数据有可能改变一系列科学领域，包括图像处理，自然语言处理和计算社会科学。 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，这些方法至关重要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）。 重点是理论基础和可扩展的算法，与最先进的黑盒方法竞争，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，这些框架和算法还可以直接应用于地球科学和地理学。 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问题，这将提供重要的验证数据来源。 该项目的主要内容是培养博士生和本科生在数学、数据科学和计算交叉领域的研究能力。具体来说，该项目的重点是（1）在固有低维概率测度集合中建模和计算的基本问题，以及（2）这些低维集合的可学习性和表达性。 研究人员将数据视为Wasserstein空间中的措施，并建议通过允许有效合成和分析一组参考分布的观测措施，在经典向量空间设置中开发低维模型的类似物。 重点是两个主要问题有关的低维结构在Wasserstein空间。 首先，什么是“正确的”模型，为内在低维子集的沃瑟斯坦空间，以及如何才能计算？ 研究人员建议通过提出三种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熵正则化的最新计算进展。 这三个模型的目标是通过Wasserstein空间的这些低维子集中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，从测量估计给定样本）和计算（例如，设计具有次立方复杂度的算法）的观点。 第二，什么是有效的，可计算的，并在沃瑟斯坦空间表现系统？ 研究人员建议解决数据驱动的问题，给定观察到的概率测度，如何识别和学习少量的参考测度，以有效地表示它们。 与此同时，研究人员将研究什么样的系统具有足够的表现力来表示Wasserstein空间的典型元素的基本问题。 这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。