Non-Local Variational Problems with Applications to Data Science

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Machine learning algorithms have grown increasingly powerful and integral to our society in recent years. However, our understanding of how and why these algorithms work remains incomplete, especially in terms of their robustness and reliability. On the other hand, there has been a significant effort in the past century to mathematically understand theoretical properties of variational problems stemming from physics and engineering, and many related data science tasks that utilize machine learning algorithms can be cast in terms of optimization or variational problems. This project will study theoretical properties of machine learning algorithms by understanding their variational structure, and by doing so will develop novel mathematical techniques. Graduate and undergraduate students will also be trained as part of this project. This award focuses on two classes of anisotropic, non-local variational problems that have arisen in data science tasks. The first is a data-adapted, non-local perimeter which appears naturally in the context of adversarial learning, and the second is a non-local Dirichlet energy used in graph-based learning methods. This work will deepen our understanding of the regularity associated with these energies and corresponding evolution equations and will provide a new avenue for understanding classical elliptic regularity theory. Doing so will improve upon the current work in the machine learning community on the theory of these algorithms, helping to aid in rigorous choices for interpretable and computationally efficient algorithm design. Finally, the expected results will expand the types of tasks and questions that are computationally tractable, for example by permitting efficient exploration of the effect of adversarial power and its effect on topology and accuracy, or by permitting the accurate inclusion of boundary constraints in non-local Laplacian problems.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.