Variational and Partial Differential Equation Methods for Geometrical Machine Learning

Abstract

This thesis is concerned with the study of geometric machine learning methods through vari- ational and PDE methods. First, through an analysis of asymptotic consistency relying on op- timal transport and Γ-convergence, we characterize the well-posedness and precise behaviour of various (hyper)graph algorithms. In particular, we show that the semi-supervised problem is only effectively solved through fractional Laplacian regularization on graphs or p-Laplacian learning on hypergraphs if the underlying geometrical data structures are correctly designed as the number of data samples increases. We also develop numerical schemes to efficiently solve supervised problems with p-Laplacian regularization on graphs. Lastly, we present a compar- ative study of various machine learning methodologies for a hyperspectral imaging problem.

Date of Award	1 Aug 2024
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Kody Law (Supervisor) & Matthew Thorpe (Supervisor)