Non-Local Variational Problems with Applications to Data Science

非局部变分问题及其在数据科学中的应用

• 批准号: 2307971
• 负责人: Ryan Murray
• 金额: $ 14.58万
• 依托单位: North Carolina State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307971&HistoricalAwards=false
• 关键词: Non; Local; Variational; Problems; Applications

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.

近年来，机器学习算法变得越来越强大，成为我们社会不可或缺的一部分。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，特别是在它们的鲁棒性和可靠性方面。另一方面，在过去的一个世纪里，人们在数学上理解源自物理和工程的变分问题的理论性质方面做出了重大努力，许多利用机器学习算法的相关数据科学任务可以用优化或变分问题的形式来表达。该项目将通过理解其变分结构来研究机器学习算法的理论性质，并通过这样做将开发新的数学技术。作为这个项目的一部分，研究生和本科生也将接受培训。该奖项主要关注数据科学任务中出现的两类各向异性、非局部变分问题。第一个是数据适应的，非局部周长，在对抗性学习的背景下自然出现，第二个是基于图的学习方法中使用的非局部狄利克雷能量。这项工作将加深我们对与这些能量和相应的演化方程相关的规律性的理解，并将为理解经典椭圆规律性理论提供新的途径。这样做将改善当前机器学习社区在这些算法理论方面的工作，有助于为可解释和计算效率高的算法设计提供严格的选择。最后，预期的结果将扩展可计算处理的任务和问题的类型，例如，允许有效地探索对抗力的影响及其对拓扑和精度的影响，或者允许在非局部拉普拉斯问题中精确地包含边界约束。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。