Non-Local Variational Problems with Applications to Data Science

近年来，机器学习算法变得越来越强大，对我们的社会不可或缺。然而，我们对这些算法如何以及为什么工作的理解仍然不完整，特别是在它们的健壮性和可靠性方面。另一方面，在过去的一个世纪里，人们一直在努力从数学上理解源于物理和工程的变分问题的理论性质，许多利用机器学习算法的相关数据科学任务可以归结为优化或变分问题。这个项目将通过了解机器学习算法的变化结构来研究它们的理论性质，并通过这样做来开发新的数学技术。研究生和本科生也将作为该项目的一部分接受培训。该奖项集中于数据科学任务中出现的两类各向异性、非局部变分问题。第一种是在对抗性学习环境中自然出现的数据适应的非局部周界，第二种是基于图的学习方法中使用的非局部Dirichlet能量。这项工作将加深我们对与这些能量相关的正则性以及相应的演化方程的理解，并将为理解经典的椭圆正则性理论提供一条新的途径。这样做将改进机器学习社区目前在这些算法理论上的工作，有助于为可解释和计算高效的算法设计提供严格的选择。最后，预期结果将扩展在计算上容易处理的任务和问题的类型，例如，允许有效地探索对抗性力量的影响及其对拓扑和准确性的影响，或者允许在非局部拉普拉斯问题中准确地包括边界约束。该奖项反映了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。

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.