Collaborative Research: AF: Medium: A Unified Framework for Geometric and Topological Signature-Based Shape Comparison

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A fundamental aspect for data analysis is the ability to compare data sets, in order to measure (dis)similarity and quantify patterns present in the data. However, data is often too large and complex to analyze in its entirety, and therefore different techniques are used to summarize the data in order to work with smaller, more manageable representations of it. This project studies the data-comparison problem through the lens of mathematics, using geometric and topological signatures to represent these shapes concisely. This project will consider a variety of different kinds of shape data which live in some larger geometric or topological space (e.g., GIS trajectories, point sets, meshes, 3d scans, or graphs), and consider classes of algebraic, geometric, and graphical signatures which can be used to represent these shapes concisely. The project draws primarily upon the nascent yet rapidly developing area of topological data analysis, where tools from topology like homology or homotopy are combined with geometric measures to create robust analysis tools for analyzing the shape of data. Graduate and undergraduate students will be tightly integrated into the project, and special efforts will be made to involve students from underrepresented groups. Additional efforts by the research team include planning a workshop focused on women in this field, as well as broadening diversity and inclusion efforts in their own universities.The project focuses on shapes that have some common underlying annotation framework on top of the signature, which is usually additional structural or geometric information from the original embedding. The research consists of two major components. In the first, the investigators are initiating a principled study of algorithms and approaches to develop a unified framework which leverages multiple signatures for shape comparison. The goal of this phase is to provide theoretical results as well as empirical evaluations on a variety of data sets and signatures. The second major component of the project studies inverse problems, which aim to reconstruct shapes from a combination of signatures. Such problems are notoriously difficult for geometric or topological signatures, as they are necessarily lossy and remove certain types of information. During the course of the project, the investigators are also developing a shape signatures toolkit that enables computation of a range of signatures and distances, adding to the software both existing notions of distance and new ones developed over the course of the project.This project is jointly funded by the Algorithmic Foundations Core Program and by the Established Program to Stimulate Competitive Research (EPSCoR).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.