CDS&E-MSS: Recovery of High-Dimensional Structured Functions

CDS

• 批准号: 1622134
• 负责人: Simon Foucart
• 金额: $ 9.95万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1622134&HistoricalAwards=false
• 关键词: CDS& MSS; Recovery; Dimensional; Structured

Many scientific problems of crucial importance for the United States involve a very large number of parameters. This high dimensionality has long been a challenge for the numerical treatment of physical, chemical, and biological processes, and now it also represents an obstacle in data science, especially for the task of extracting useful information from only a limited amount of observations. Propitiously, the high-dimensional objects prevailing in real-life problems often possess some underlying structure simplifying their manipulation. The purpose of this project is to exploit this structure in order to develop a consistent theoretical framework and to conceive novel computational methods for the exact recovery (or good approximation) of the high-dimensional objects sketchily acquired.More specifically, the objects considered in this project are functions of very many variables. Handling them via traditional numerical methods is doomed by the so-called curse of dimensionality. But modern ideas such as sparsity and variable reduction make it possible to bypass the curse and thus are revolutionizing the approach to high-dimensional problems. Building upon the fundamentals from the theory of compressive sensing, the project will consider sparse recovery and simultaneously structured recovery as part of the more general recovery of high-dimensional functions depending on few reduced variables. The research strategy starts by investigating the theoretical limitations of any recovery method within a model, then continues by refining the model through confrontation with real-life problems, and finishes by implementing the algorithms proven to perform optimally. Since the project features interactions with several applied fields (in particular, Engineering and Bioinformatics), the novel numerical methods are to be tested in these areas, which will in turn provide fresh mathematical insight. A final part of the project is devoted to the integration of emerging concepts into the culture of the next scientific generation and in particular to the training of mathematicians in computational and data-related aspects.

许多对美国至关重要的科学问题涉及大量参数。长期以来，这种高维性一直是物理、化学和生物过程的数值处理的挑战，现在它也是数据科学的障碍，特别是对于从有限数量的观测中提取有用信息的任务。幸运的是，在现实生活中普遍存在的高维对象通常具有一些简化其操作的底层结构。这个项目的目的是利用这种结构，以发展一个一致的理论框架，并构思新的计算方法，精确恢复（或良好的近似）的高维对象粗略收购。更具体地说，在这个项目中考虑的对象是非常多的变量的函数。用传统的数值方法处理它们注定会受到所谓的维数灾难的影响。但是现代思想，如稀疏性和变量减少，使绕过诅咒成为可能，从而彻底改变了高维问题的方法。基于压缩感知理论的基本原理，该项目将考虑稀疏恢复和同时结构化恢复，作为依赖于少量减少变量的高维函数的更一般恢复的一部分。研究策略首先调查模型中任何恢复方法的理论局限性，然后通过与现实问题的对抗来改进模型，最后通过实施被证明具有最佳性能的算法来完成。由于该项目与多个应用领域（特别是工程和生物信息学）相互作用，因此将在这些领域测试新的数值方法，这反过来又将提供新的数学见解。该项目的最后一部分致力于将新出现的概念纳入下一代科学文化，特别是在计算和数据相关方面培训数学家。