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，生物学和国土安全性等领域至关重要。在现代数据分析中产生的数学表述构成了新的数学和计算挑战，这既是其配方的复杂性及其可能非常大的大小。在这种情况下，必须利用可能存在于系统中的结构，并具有简化分析并构建有效且灵活的优化算法的双重目标，这些算法仅需要在每次迭代中执行基本任务。 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.虽然透视功能在许多变化表述中隐含或明确地存在，尤其是在数据分析中，但很少有努力专门研究其数学属性以及可以有效解决它们的计算方法的发展。因此，没有任何合成变分模型可以统一涉及透视功能的优化问题类别。此外，在算法方面，没有解决此类问题的原则策略。特别是，仅在有限的情况下才知道透视函数的接近性运算符，这排除了强大的近端分裂算法的使用。填补这些空白是该项目的目的。该项目旨在为涉及透视功能及其概括的最小化问题分析的理论和计算基础，并将这些发现应用于数据分析中的问题，这些问题无法实现当前方法的影响。该研究方法取决于统一的结构化变分模型，这些模型是在产品空间中重铸并通过近端分裂算法和双重性驱动策略解决的。计划了对数据分析的几个领域的申请。该奖项反映了NSF的法定任务，并被认为是使用基金会的知识分子优点和更广泛的影响审查标准的评估值得支持的。