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

离散随机结构的变分问题和偏微分方程：分析及其在数据科学中的应用

基本信息

批准号：
1814991
负责人：
Dejan Slepcev
金额：
$ 24.56万
依托单位：
Carnegie-Mellon University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1814991&HistoricalAwards=false
关键词：
Variational Problems Partial Differential Equations

项目摘要

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.

研究者研究了数据科学问题的变分和部分微分方程（PDE）。现代技术使我们能够获得有关我们所生活的世界上任何方面的大量数据。数据科学的目的是提取和解释数据所包含的信息。实现这一目标会导致机器学习任务，例如回归，聚类，分类，减少维度，半监督学习和学习数据表示（例如，深度学习）。这些任务经常被视为优化问题，其中人们最大程度地降低了对物体所需属性建模的目标功能。目标函数和最小化通常在可用的数据样本上构成，这会导致图形和代表数据的相关结构上的离散变化问题。该项目的目的是开发一个数学框架，以研究随机数据样本上的基于变异的问题和基于PDE的问题。研究人员使用基于连续的变分问题和PDE的见解来改善离散设置中的现有方法，并引入有关数据科学问题的新模型和算法。研究生从事该项目的研究。研究人员将分析工具适应离散的随机设置，以显示变异问题和PDE的基本属性，而PDE则在此类结构上。他致力于在随机离散结构上建立和使用问题之间的联系，以及在大样本限制中出现的连续性问题。特别是，他研究了半监督学习中基于拉普拉斯和基于p拉普拉斯的正规化的行为；研究稳定的方法来检测数据集的边界并施加所需的边界条件；并为数据设置在限制中近似的连续性问题开发基于图的准确离散。该项目的第二部分致力于在随机离散结构上进行梯度流。在这里，研究者研究了此类问题的稳定性和渐近特性，以及它们所代表的非局部连续性问题的特性。研究生从事该项目的研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点评估和更广泛影响的评论标准来支持的。

项目成果

期刊论文数量（9）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates

DOI：
发表时间：
2020-06
期刊：
影响因子：
0
作者：
J. Calder;Brendan Cook;Matthew Thorpe;D. Slepčev
通讯作者：
J. Calder;Brendan Cook;Matthew Thorpe;D. Slepčev

Accurate Quantization of Measures via Interacting Particle-based Optimization

DOI：
发表时间：
2022
期刊：
影响因子：
0
作者：
Lantian Xu;Anna Korba;D. Slepčev
通讯作者：
Lantian Xu;Anna Korba;D. Slepčev

Properly-Weighted Graph Laplacian for Semi-supervised Learning

DOI：
10.1007/s00245-019-09637-3
发表时间：
2018-10
期刊：
Applied Mathematics & Optimization
影响因子：
1.8
作者：
J. Calder;D. Slepčev
通讯作者：
J. Calder;D. Slepčev

Nonlocal-Interaction Equation on Graphs: Gradient Flow Structure and Continuum Limit

图上的非局部相互作用方程：梯度流结构和连续极限

DOI：
10.1007/s00205-021-01631-w
发表时间：
2021
期刊：
Archive for Rational Mechanics and Analysis
影响因子：
2.5
作者：
Esposito, Antonio;Patacchini, Francesco S.;Schlichting, André;Slepčev, Dejan
通讯作者：
Slepčev, Dejan

Large data and zero noise limits of graph-based semi-supervised learning algorithms