Variational Problems on Random Structures: Analysis and Applications to Data Science

该研究项目涉及将数学分析的现代工具应用于数据科学中当前重要主题的研究，包括海量数据分析（数据云）和机器学习。 现代数据采集技术产生了大量关于我们生活的世界的数据。 从数据中提取信息会导致机器学习/统计任务，例如聚类，分类，低维嵌入等。 该项目介绍了新的数学工具，用于理解一些重要数据分析任务的最先进方法。 所获得的结论将提高其可靠性，鲁棒性，速度和可扩展性。 分析的见解有望为创建数据分析的新模型和相关任务的新方法奠定基础。本项目将研究数据分析和机器学习中出现的变分问题。 它将这样做，通过考虑变分描述这些问题的答案是通过最小化的目标功能。 该项目将开发一个数学框架，适用于研究随机样本和相关随机几何的变分问题的渐近性质。 特别是，它将研究随机离散结构（如数据云的邻域图）上的变分问题与连续变分问题之间的关系。 它结合了变分法，应用分析，最优运输，概率和统计的技术，以获得有关随机设置中的离散变分问题的见解。

This research project is concerned with applying modern tools from mathematical analysis to the study of currently important topics in data science, including massive data analysis (data clouds) and machine learning. Modern data-acquisition techniques produce a wealth of data about the world we live in. Extracting the information from the data leads to machine learning/statistics tasks such as clustering, classification, low-dimensional embedding, and others. This project introduces new mathematical tools for understanding of some of the state-of-art approaches to important data analysis tasks. The conclusions gained will improve their reliability, robustness, speed, and scalability. The insights of the analysis are expected to lay the foundation to create new models for data analysis and new approaches to the pertinent tasks.This project will investigate variational problems that arise in data analysis and machine learning. It will do so by considering variational descriptions of these problems in which the answer is obtained by minimizing an objective functional. The project will develop a mathematical framework suitable for studies of asymptotic properties of variational problems posed on random samples and related random geometries. In particular, it will investigate the relationship between variational problems on random discrete structures, such as a neighborhood graph of a data cloud, and continuum variational problems. It combines the techniques of calculus of variations, applied analysis, optimal transportation, probability, and statistics to gain insights about discrete variational problems in a random setting.