A Survey on Algorithmic Developments in Optimal Transport Problem with Applications

TL;DR

This survey comprehensively reviews the theoretical foundations, algorithms, emerging trends, and applications of Optimal Transport, highlighting its growing importance and challenges in machine learning and data analysis.

Contribution

It provides an extensive overview of recent algorithmic developments, applications, and future directions in Optimal Transport research.

Findings

Efficient algorithms like Sinkhorn iterations enable scalable OT computations.

OT's integration into machine learning enhances data analysis capabilities.

Emerging variants and applications expand OT's versatility across domains.

Abstract

Optimal Transport (OT) has established itself as a robust framework for quantifying differences between distributions, with applications that span fields such as machine learning, data science, and computer vision. This paper offers a detailed examination of the OT problem, beginning with its theoretical foundations, including the classical formulations of Monge and Kantorovich and their extensions to modern computational techniques. It explores cutting-edge algorithms, including Sinkhorn iterations, primal-dual strategies, and reduction-based approaches, emphasizing their efficiency and scalability in addressing high-dimensional problems. The paper also highlights emerging trends, such as integrating OT into machine learning frameworks, the development of novel problem variants, and ongoing theoretical advancements. Applications of OT are presented across a range of domains, with particular attention to its innovative application in time series data analysis via Optimal Transport Warping (OTW), a robust alternative to methods like Dynamic Time Warping. Despite the significant progress made, challenges related to scalability, robustness, and ethical considerations remain, necessitating further research. The paper underscores OT's potential to bridge theoretical depth and practical utility, fostering impactful advancements across diverse disciplines.