CDS&E: Robust Symmetry-Preserving Machine Learning: Theory and Application

CDS

• 批准号: 2244976
• 负责人: Wei Zhu
• 金额: $ 16万
• 依托单位: University of Massachusetts Amherst
• 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: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2244976&HistoricalAwards=false
• 关键词: CDS& Robust; Symmetry; Preserving; Machine

Deep neural networks (DNNs) have been a major driving force behind recent advances in data science and engineering. An emerging theme in DNN research is to exploit the intrinsic structure of the learning problems, such as symmetry, to improve the data-efficiency of DNNs in the small-data regime. Recent work on symmetry-preserving machine learning typically studies it in the ideal setting where the symmetry transformations are perfect, whereas in reality, however, they are usually “contaminated” by various sources of signal deformation. The aim of this project is to rigorously measure and guarantee the deformation robustness of general symmetry-preserving DNNs, as well as quantifying their resulting performance gain. Results of the research are expected to advance understanding of robust geometric deep learning, with a diverse range of applications from computer vision to scientific computing with limited data. The project will provide interdisciplinary training in applied mathematics, engineering, and data science to undergraduate and graduate students. The overarching theme of the project is to leverage mathematical tools from differential geometry, applied harmonic analysis, and applied probability to improve the statistical-efficiency of machine learning models. Special emphasis has been placed on the rigorous analysis and promotion of robust symmetry-preservation that is broadly applicable to arbitrary Lie group representations on general feature fields. In addition, the project aims to extend the idea of symmetry-preservation to deep distribution learning, and proposes a unified framework for data-efficient generation of distributions with intrinsic structures including—but not limited to—group symmetry; the improved statistical efficiency will be rigorously quantified through sample complexity analysis. The techniques to be developed in this project will be widely applicable across different disciplines, providing fundamental building blocks for the next generation of mathematical tools for the computational and geometric modeling of Big Data.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.

深度神经网络（DNN）一直是数据科学和工程最新进展的主要驱动力。DNN研究中的一个新兴主题是利用学习问题的内在结构（如对称性）来提高DNN在小数据范围内的数据效率。最近关于保留对称性的机器学习的工作通常在对称变换完美的理想环境中研究它，然而，在现实中，它们通常被各种信号变形源“污染”。该项目的目的是严格测量和保证一般保密DNN的变形鲁棒性，并量化其产生的性能增益。预计研究结果将促进对强大的几何深度学习的理解，从计算机视觉到有限数据的科学计算等各种应用。该项目将为本科生和研究生提供应用数学，工程和数据科学的跨学科培训。该项目的首要主题是利用微分几何，应用调和分析和应用概率的数学工具来提高机器学习模型的学习效率。特别强调了严格的分析和推广鲁棒的保序性，广泛适用于一般特征场的任意李群表示。此外，该项目旨在将数据保留的思想扩展到深度分布学习，并提出一个统一的框架，用于数据高效生成具有内在结构的分布，包括但不限于群对称性;改进的统计效率将通过样本复杂性分析进行严格量化。该项目开发的技术将广泛应用于不同学科，为下一代大数据计算和几何建模的数学工具提供基础构建模块。该奖项反映了NSF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。