We live in an era of big data. Whether it be algorithms designed to help corporations efficiently allocate the use of their resources, systems to block or intercept transmissions in times of war, or the helpful pocket companion known as ChatGPT, machine learning is at the heart of managing the real-world problems associated with massive data. With the success of neural networks on such large-scale problems, more research in machine learning is being conducted now than ever before. This dissertation focuses on three different projects rooted in mathematical theory for machine learning applications. Common themes throughout involve the synthesis of mathematical ideas with problems in machine learning, yielding new theory, algorithms, and directions of study. The first project deals with supervised learning and manifold learning. In theory, one of the main problems in supervised learning is that of function approximation. At the surface level, classical approximation theory seems readily applicable to such a problem, but under the surface there are technical difficulties including unknown data domains, extremely high dimensional feature spaces, and noise. We introduce a method which aims to tackle these difficulties and remedies several of the theoretical shortcomings of the current paradigm. The second project deals with transfer learning, which is the study of how an approximation process or model learned on one domain can be leveraged to improve the approximation on another domain. This can be viewed as the lifting of a function from one manifold to another. This viewpoint enables us to connect some inverse problems in applied mathematics (such as the inverse Radon transform) with transfer learning. We study such liftings of functions when the data is assumed to be known only on a part of the whole domain. We are interested in determining subsets of the target data space on which the lifting can be defined, and how the local smoothness of the function and its lifting are related. The third project is concerned with the classification task in machine learning, particularly in the active learning paradigm. Classification has often been treated as an approximation problem as well, but we propose an alternative approach leveraging techniques originally introduced for signal separation problems. The analogue to point sources are the supports of distributions from which data belonging to each class is sampled from. We introduce theory to unify signal separation with classification and a new algorithm which yields competitive accuracy to other recent active learning algorithms while providing results much faster.
Active Learning, Approximation Theory, Classification, Harmonic Analysis, Machine Learning, Manifold Learning, Mathematics
Networked Digital Library of Theses and Dissertations