Variational Problems and Partial Differential Equations on Discrete Random Structures: Analysis and Applications to Data Science

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The investigator studies variational and partial differential equation (PDE) approaches to problems of data science. Modern technology enables us to obtain large amounts of data about virtually any aspect of the world we live in. The goal of data science is to extract and interpret the information the data contain. Achieving this leads to machine learning tasks such as regression, clustering, classification, dimensionality reduction, semi-supervised learning, and learning data representation (e.g. deep learning). These tasks are regularly cast as optimization problems where one minimizes an objective functional that models the desired properties of the object sought. The objective functionals and the resulting minimization are often posed on the available data sample, which leads to discrete variational problems on graphs and related structures representing the data. The goal of this project is to develop a mathematical framework to study variational and PDE-based problems on random data samples. The investigator uses insights from the continuum-based variational problems and PDEs to improve existing approaches in the discrete setting and introduce new models and algorithms for pertinent problems of data science. Graduate students are engaged in the research of the project.The investigator adapts tools of analysis to the discrete random setting in order to show the fundamental properties of the variational problems and PDEs on such structures. He works on establishing and using the connection between problems on random discrete structures and the continuum problems that arise in the large-sample limit. In particular, he investigates the behavior of Laplacian-based and p-Laplacian-based regularizations in semi-supervised learning; studies stable ways to detect the boundaries of the data sets and impose the desired boundary conditions; and develops accurate graph-based discretizations for the continuum problems that the data sets approximate in the limit. The second part of the project is devoted to gradient flows on random discrete structures. Here the investigator studies stability and asymptotic properties of such problems, as well as the properties of the nonlocal continuum problems that they represent. Graduate students are engaged in the research of the project.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.