Variational Problems on Random Structures: Analysis and Applications to Data Science

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

This research project is concerned with applying modern tools from mathematical analysis to the study of currently important topics in data science, including massive data analysis (data clouds) and machine learning. Modern data-acquisition techniques produce a wealth of data about the world we live in. Extracting the information from the data leads to machine learning/statistics tasks such as clustering, classification, low-dimensional embedding, and others. This project introduces new mathematical tools for understanding of some of the state-of-art approaches to important data analysis tasks. The conclusions gained will improve their reliability, robustness, speed, and scalability. The insights of the analysis are expected to lay the foundation to create new models for data analysis and new approaches to the pertinent tasks. This project will investigate variational problems that arise in data analysis and machine learning. It will do so by considering variational descriptions of these problems in which the answer is obtained by minimizing an objective functional. The project will develop a mathematical framework suitable for studies of asymptotic properties of variational problems posed on random samples and related random geometries. In particular, it will investigate the relationship between variational problems on random discrete structures, such as a neighborhood graph of a data cloud, and continuum variational problems. It combines the techniques of calculus of variations, applied analysis, optimal transportation, probability, and statistics to gain insights about discrete variational problems in a random setting.