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

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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.