Past Event: Babuška Forum

Using Continuum Limits To Understand Data Clustering And Classification

Franca Hoffmann, Bonn Junior Fellow, University of Bonn, Germany

Abstract

Graph Laplacians encode geometric information contained in data, via the eigenfunctions associated with their small eigenvalues. These spectral properties provide powerful tools in data clustering and data classification. When a large number of data points are available one may consider continuum limits of the graph Laplacian, both to give insight and, potentially, as the basis for numerical methods. We summarize recent insights into the properties of a family of weighted elliptic operators arising in the large data limit of different graph Laplacian normalizations, and propose an inverse problem formalism for continuous semi-supervised learning algorithms, making use of these differential operators. This is joint work with Bamdad Hosseini (Caltech), Assad A. Oberai (USC) and Andrew M. Stuart (Caltech). Bio Franca Hoffmann is a Bonn Junior Fellow at University of Bonn (Germany). After completing her PhD at the Cambridge Centre for Analysis at University of Cambridge (UK) in 2017, she held the position of von Karman instructor at California Institute of Technology (US) from 2017 to 2020. Her research is focused on the applied mathematics/data analysis interface, driven by the need to provide rigorous mathematical foundations for modeling tools used in applications. In particular, Franca is interested in the theory of nonlinear and nonlocal PDEs, as well as in developing novel tools for data analysis and mathematical approaches to machine learning. (The Babuška Forum series was started by Professor Ivo Babuška several years ago to expose students to interesting and curious topics relevant to computational engineering and science with technical content at the graduate student level (i.e. the focus of the lectures is on main ideas with some technical content). Seminar credit is given to those students who attend.)