Variational Problems and Partial Differential Equations on Discrete Random Structures: Analysis and Applications to Data Science

研究人员研究数据科学问题的变分和偏微分方程(PDE)方法。现代技术使我们能够获得关于我们生活的世界几乎任何方面的大量数据。数据科学的目标是提取和解释数据包含的信息。实现这一点将导致机器学习任务，如回归、聚类、分类、降维、半监督学习和学习数据表示(例如，深度学习)。这些任务通常被归结为优化问题，其中最小化目标泛函，该目标泛函对所寻找的对象的所需属性进行建模。目标泛函和目标泛函的最小化通常是在可用数据样本上提出的，这导致了图和表示数据的相关结构上的离散变分问题。这个项目的目标是开发一个数学框架来研究随机数据样本上的变分和基于偏微分方程的问题。研究人员利用基于连续统的变分问题和偏微分方程组的见解来改进离散环境中的现有方法，并为数据科学的相关问题引入新的模型和算法。研究生从事这一项目的研究，研究人员将分析工具应用于离散随机环境，以显示变分问题和此类结构上的偏微分方程组的基本性质。他致力于建立和使用随机离散结构上的问题与大样本限制中出现的连续统问题之间的联系。特别是，他研究了基于拉普拉斯的正则化和基于p-拉普拉斯的正则化在半监督学习中的行为；研究了检测数据集的边界和施加期望的边界条件的稳定方法；并为数据集在极限内逼近的连续统问题开发了精确的基于图的离散化。该项目的第二部分致力于研究随机离散结构上的梯度流。在这里，研究者研究了这类问题的稳定性和渐近性质，以及它们所代表的非局部连续统问题的性质。研究生从事该项目的研究。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。

The investigator studies variational and partial differential equation (PDE) approaches to problems of data science. Modern technology enables us to obtain large amounts of data about virtually any aspect of the world we live in. The goal of data science is to extract and interpret the information the data contain. Achieving this leads to machine learning tasks such as regression, clustering, classification, dimensionality reduction, semi-supervised learning, and learning data representation (e.g. deep learning). These tasks are regularly cast as optimization problems where one minimizes an objective functional that models the desired properties of the object sought. The objective functionals and the resulting minimization are often posed on the available data sample, which leads to discrete variational problems on graphs and related structures representing the data. The goal of this project is to develop a mathematical framework to study variational and PDE-based problems on random data samples. The investigator uses insights from the continuum-based variational problems and PDEs to improve existing approaches in the discrete setting and introduce new models and algorithms for pertinent problems of data science. Graduate students are engaged in the research of the project.The investigator adapts tools of analysis to the discrete random setting in order to show the fundamental properties of the variational problems and PDEs on such structures. He works on establishing and using the connection between problems on random discrete structures and the continuum problems that arise in the large-sample limit. In particular, he investigates the behavior of Laplacian-based and p-Laplacian-based regularizations in semi-supervised learning; studies stable ways to detect the boundaries of the data sets and impose the desired boundary conditions; and develops accurate graph-based discretizations for the continuum problems that the data sets approximate in the limit. The second part of the project is devoted to gradient flows on random discrete structures. Here the investigator studies stability and asymptotic properties of such problems, as well as the properties of the nonlocal continuum problems that they represent. Graduate students are engaged in the research of the project.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.