CDS&E MSS: Collaborative Research: Multiparameter Topological Data Analysis

Award Number and Duration

NSF DMS 2301361 (University of Utah)

NSF DMS 2301360 (Purdue University)

NSF DMS 2301359 (Ohio State University)

September 1, 2023 to August 31, 2026

PIs

Bei Wang (University of Utah, PI)
Associate Professor
School of Computing and Scientific Computing and Imaging Institute
University of Utah
beiwang AT sci.utah.edu
Homepage

Facundo Mémoli (OSU, PI)
Professor
Department of Mathematics
Department of Computer Science and Engineering
Ohio State University
memoli AT math.osu.edu
Homepage

Tamal Dey (Perdue University, PI)
Professor
Department of Computer Science
Purdue University
tamaldey AT purdue.edu
Homepage

Collaborators

Kelin Xia, Associate Professor, School of Physical & Mathematical Sciences, Nanyang Technological University.

Overview

Complex datasets arise in many disciplines of science and engineering and require multiparameter data analysis, which broadly speaking, studies the dependency of a phenomenon or a space on multiple parameters. In recent years, topological data analysis (TDA) has evolved as an emerging area in data science. So far, most of its applications have been limited to the single parameter case, that is, to data expressing the behavior of a single variable. As its reach to applications expands, the task of extracting intelligent summaries out of diverse, complex data demands the study of multiparameter dependencies. This project will help address this demand.

Although TDA involving a single parameter has been well researched and developed, the same is not true for the multiparameter case. At its current nascent stage, multiparameter TDA is yet to develop tools to practically handle complex, diverse, and high-dimensional data. To meet this challenge, this project will make both mathematical and algorithmic advances for multiparameter TDA. To scope effectively, focus will be mainly on three research thrusts to: (I) explore multipa- rameter persistence for generalized features and develop algorithms to compute them; (II) exploit the connections of zigzag persistence to multiparameter settings to support dynamic data analysis, and (III) generalize topological descriptors such as merge trees, Reeb spaces, and mapper. The overarching goal lacing all three thrust areas remains that of developing actionable and practicable tools in applications including Cytometry, Materials Science, Climate Simulations and Ecology. From a methodology point of view, the geometric and topological ideas behind the proposed work represent novel directions and inject new ideas and perspectives to the important field of computational data analysis. In particular, the project team will investigate several novel mathematical concepts in conjunction with algorithms to address various challenges appearing in the aforementioned topics. The resulting TDA methodologies can complement and augment traditional data analysis approaches in fields such as machine learning and statistical data analysis. The PIs bring together expertise in theoretical computer science, algorithms design, mathematics, and in particular topological data analysis to conduct this research.

Broader Impacts

This project will develop a sound mathematical theory supported by efficient algorithmic tools for the aforementioned thrusts and, thus, will provide a powerful platform for data exploration and analysis in a range of applications in science and engineering. The educational impact will be accelerated by the synergy between mathematics and computer science and integrated applications. Graduate students supported by the project will be trained to: develop skills in mathematics and theoretical computer science, most notably in algorithms and topology, write efficient and usable software, and analyze real-world data sets. The PIs will provide best practice recruiting and mentoring of students from underrepresented groups. The PIs plan to broaden research engagement via workshops or tutorials at computational topology and TDA venues. A nationally focused hackathon will introduce methodologies and software tools for multiparameter TDA to the research community as where participants work on data analysis projects while being mentored by disciplinary experts. The outreach activities including a K-12 summer camp and a data science day will further converge disciplinary communities and introduce high school students to novel modern data analysis.

Publications and Manuscripts

Presentations, Educational Development and Broader Impacts

1. Bei Wang, Invited Talk: Capturing Robust Topology in Data.
   IMS-NTU joint workshop on Biomolecular Topology: Modeling and Data Analysis, June 24, 2024.
2. Bei Wang, Conference Talk: Measure-Theoretic Reeb Graphs and Reeb Spaces.
   International Symposium on Computational Geometry (SoCG), Greece, June 14, 2024.
3. Bei Wang, Invited Talk: Reeb Graphs and Their Variants: Theory and Application.
   Dagstuhl Seminar 24092: Applied and Combinatorial Topology, Dagstuhl, Germany, Feb 26, 2024.
4. Bei Wang, Invited Talk: Reeb Graphs and Measure Theoretic Variants: Theory and Applications.
   MPI Geometry Seminar, Max Planck Institute for Mathematics in the Sciences, Berlin, Germany, Jan 23, 2024.
5. Bei Wang, Invited Talk: Reeb Graphs and Measure Theoretic Variants: Theory and Applications.
   MATH+ Workshop on Small Data Analysis, Zuse Institute Berlin (ZIB), Leipzig, Germany, Jan 17, 2024.

Postdocs and Students

Qingsong Wang (Mathematics Postdoc)
Weiran (Nancy) Lyu (CS PhD student)
Nathaniel Gorski (CS PhD student)
Guanquan Ma (CS PhD student)
Dhruv Meduri (CS PhD student)

Acknowledgement

This material is based upon work supported or partially supported by the National Science Foundation under Grant No. 2301361.

Any opinions, findings, and conclusions or recommendations expressed in this project are those of author(s) and do not necessarily reflect the views of the National Science Foundation.

Web page last update: September 7, 2024.