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

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提出了新的数学和计算挑战，这既是因为它们的公式复杂，也是因为它们的潜在规模非常大。在这种情况下，必须利用系统中可能存在的结构，其双重目标是简化分析和构建只需在每次迭代中执行基本任务的高效和灵活的优化算法。这项研究旨在通过开发新的数学工具和算法来解决这些问题，这些工具和算法围绕一类所谓的透视函数来构建，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有关的数学和计算问题，透视函数是一个强大的概念，它允许通过额外的尺度变量将一个凸函数扩展到一个联合凸函数。虽然透视函数隐含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性质，以及开发能够有效地求解它们的计算方法。因此，没有综合变分模型可用于统一涉及透视函数的一类优化问题。此外，在算法方面，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是已知的，这排除了使用强大的邻近分裂算法的可能性。填补这些空白是该项目的目标。该项目旨在为涉及透视函数及其推广的最小化问题的分析和数值解决奠定理论和计算基础，并将这些发现应用于数据分析中现有方法无法处理的问题。研究方法依赖于统一的结构化变分模型，这些模型在产品空间中重铸，并通过邻近分裂算法和对偶驱动策略求解。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。

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.