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

研究人员研究用变分和偏微分方程（PDE）方法解决数据科学问题。 现代技术使我们能够获得关于我们生活的世界的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中包含的信息。 实现这一点会导致机器学习任务，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）。 这些任务通常被视为优化问题，其中最小化目标泛函，该目标泛函对所寻求的对象的期望属性进行建模。 目标泛函和由此产生的最小化往往是对现有的数据样本，这导致离散变分问题的图和相关结构表示的数据。 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的变分和偏微分方程为基础的问题。 研究人员使用基于连续变分问题和偏微分方程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。研究生参与了该项目的研究。研究者将分析工具应用于离散随机设置，以显示此类结构上的变分问题和偏微分方程的基本性质。 他致力于建立和使用随机离散结构问题与大样本极限中出现的连续问题之间的联系。 特别是，他研究了半监督学习中基于拉普拉斯和基于p-Laplacian的正则化的行为;研究了检测数据集边界并施加所需边界条件的稳定方法;并为数据集近似于极限的连续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随机离散结构上的梯度流。 在这里，研究人员研究这些问题的稳定性和渐近性质，以及它们所代表的非局部连续问题的性质。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。

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.