Learning Without Training
type: CreativeWork
description: 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 liftin…
abstract: 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 liftin…
author: O'Dowd, Ryan
keywords: Active Learning, Approximation Theory, Classification, Harmonic Analysis, Machine Learning, Manifold Learning, Mathematics
date published: 2026-01-01T08:00:00Z
date modified: 2026-07-24T01:41:20Z
provider: Claremont Graduate University
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: Claremont Graduate University