Partitioning signal classes using transport transforms for data analysis and machine learning

TL;DR

This paper explores the mathematical properties of transport-based transforms like CDT, R-CDT, and LOT, demonstrating how they convexify signal classes generated by algebraic models to aid data classification and analysis.

Contribution

It provides conditions under which these transforms convexify algebraic signal classes, enhancing understanding of their theoretical foundations and potential applications.

Findings

Transport transforms convexify algebraic signal classes

Conditions for convexification are established

Transform domain simplifies classification tasks

Abstract

A relatively new set of transport-based transforms (CDT, R-CDT, LOT) have shown their strength and great potential in various image and data processing tasks such as parametric signal estimation, classification, cancer detection among many others. It is hence worthwhile to elucidate some of the mathematical properties that explain the successes of these transforms when they are used as tools in data analysis, signal processing or data classification. In particular, we give conditions under which classes of signals that are created by algebraic generative models are transformed into convex sets by the transport transforms. Such convexification of the classes simplify the classification and other data analysis and processing problems when viewed in the transform domain. More specifically, we study the extent and limitation of the convexification ability of these transforms under an algebraic generative modeling framework. We hope that this paper will serve as an introduction to these transforms and will encourage mathematicians and other researchers to further explore the theoretical underpinnings and algorithmic tools that will help understand the successes of these transforms and lay the groundwork for further successful applications.