综述最优传输算法进展及其在时序分析等领域的应用
A Survey on Algorithmic Developments in Optimal Transport Problem with Applications
- 系统梳理了最优传输的理论基础与计算算法
- 提出基于Sinkhorn等方法的高效高维求解方案
- 适合机器学习与数据科学从业者了解前沿进展
最优传输（OT）已成为衡量分布差异的稳健框架，广泛应用于机器学习、数据科学和计算机视觉等领域。本文详细回顾了OT问题的理论基础，涵盖Monge与Kantorovich的经典形式及其向现代计算技术的拓展。重点分析了最新的算法，包括Sinkhorn迭代、原始-对偶策略及基于约简的方法，强调其在处理高维问题时的效率与可扩展性。文章还探讨了新兴趋势，如将OT融入机器学习框架、新型问题变体的发展以及持续的理论突破。应用场景覆盖多个领域，特别关注通过最优传输变形（OTW）在时间序列分析中的创新应用，作为动态时间规整（DTW）的鲁棒替代。尽管取得显著进展，可扩展性、鲁棒性及伦理考量仍存挑战，亟需进一步研究。本文强调了OT在连接理论深度与实际应用之间的潜力，推动多学科的实质性进步。
原文摘要 · Abstract (English)
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.
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