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Computational Framework for Optimization with Perspective Functions and Applications to Data Analysis

透视函数优化的计算框架及其在数据分析中的应用

基本信息

批准号：
1818946
负责人：
Patrick Combettes
金额：
$ 37.26万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1818946&HistoricalAwards=false
关键词：
Computational Framework Optimization Perspective Functions

项目摘要

Data analysis is ubiquitous in modern science, essential in areas such as information technology, environmental sciences, medicine, biology, and homeland security. The mathematical formulations arising in modern data analysis pose new mathematical and computational challenges, both because of the sophistication of their formulations and their potentially very large size. In this context, it is essential to exploit structures that may be present in a system, with the dual objectives of simplifying the analysis and constructing efficient and flexible optimization algorithms that need only to perform basic tasks at each iteration. This research project aims to address these issues by developing new mathematical tools and algorithms structured around a class of so-called perspective functions that will facilitate handling of a broad range of data analysis questions.This research concerns mathematical and computational issues pertaining to perspective functions, a powerful concept that permits the extension of a convex function to a jointly convex one in terms of an additional scale variable. While perspective functions are implicitly or explicitly present in many variational formulations, especially in data analysis, few efforts have been devoted to the study of their mathematical properties and the development of computational methods that can solve them efficiently. Thus, no synthetic variational model is available to unify classes of optimization problems involving perspective functions. In addition, on the algorithmic side, there exists no principled strategy to solve such problems. In particular, the proximity operators of perspective functions are known only in limited cases, which precludes the use of powerful proximal splitting algorithms. It is the objective of this project to fill these gaps. The project aims to lay out theoretical and computational foundations for the analysis and the numerical solution of minimization problems involving perspective functions and generalizations thereof, and to apply these findings to problems in data analysis that are beyond the reach of current methods. The research methodology hinges on unifying structured variational models that are recast in product spaces and solved via proximal splitting algorithms as well as duality-driven strategies. Applications to several fields of data analysis are planned.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.

数据分析在现代科学中无处不在，在信息技术、环境科学、医学、生物学和国土安全等领域至关重要。现代数据分析中出现的数学公式带来了新的数学和计算挑战，这既是因为其公式的复杂性又是因为它们的规模可能非常大。在这种情况下，利用系统中可能存在的结构至关重要，其双重目标是简化分析和构建仅需要在每次迭代时执行基本任务的高效且灵活的优化算法。该研究项目旨在通过开发围绕一类所谓的透视函数构建的新数学工具和算法来解决这些问题，这将有助于处理广泛的数据分析问题。这项研究涉及与透视函数相关的数学和计算问题，透视函数是一个强大的概念，允许将凸函数扩展到附加尺度变量的联合凸函数。虽然透视函数隐式或显式地存在于许多变分公式中，特别是在数据分析中，但很少有人致力于研究它们的数学特性以及开发能够有效解决它们的计算方法。因此，没有可用的综合变分模型来统一涉及透视函数的优化问题类别。另外，在算法方面，也没有解决此类问题的原则性策略。特别是，透视函数的邻近算子仅在有限的情况下已知，这妨碍了强大的邻近分裂算法的使用。该项目的目标是填补这些空白。该项目旨在为涉及透视函数及其概括的最小化问题的分析和数值求解奠定理论和计算基础，并将这些发现应用于数据分析中当前方法无法解决的问题。研究方法取决于统一结构化变分模型，这些模型在产品空间中重新构建并通过近端分裂算法以及对偶驱动策略来解决。计划应用于多个数据分析领域。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。