| dc.description.abstract | In this thesis, we delve into the application of optimal transport theory in tackling classification and estimation problems within the realm of Machine Learning and Data Science. Our primary focus is the introduction of novel mathematical signal transforms known as the Signed Cumulative Distribution Transform (SCDT) and Knothe Rosenblatt Cumulative Distribution Transform (KR-CDT) which derive their roots from the theory of optimal transport. Initially, we examine the Cumulative Distribution Transform (CDT) for positive 1D signals and extend it to a more comprehensive measure-theoretic framework that includes positive as well as signed measures to establish the SCDT. We furnish both the forward and inverse formulas for the SCDT and expound upon its essential properties. These properties enable effective signal manipulation and analysis, as exemplified through practical applications of the SCDT in classification and estimation applications. Expanding our investigation to encompass 2D signals, we extend the concept of the CDT to two dimensions by introducing the Knothe-Rosenblatt Cumulative Distribution Transform (KR-CDT). Employing the framework of optimal transport, we define the KR-CDT and furnish a comprehensive understanding of its formulation and properties. By extending the transform to 2D signals, we unlock new possibilities for signal analysis and processing across a broader range of applications. Throughout this thesis, we underscore the theoretical foundations of these transforms, their mathematical properties, and their practical applications in solving classification and estimation problems. By leveraging the theory of optimal transport and pioneering innovative signal transforms such as the SCDT and KR-CDT, this research makes a contribution to the field of data science, offering fresh perspectives and novel approaches to address signal analysis challenges. | |
