The Sinkhorn algorithm has emerged as a powerful tool for solving optimal transport problems, finding applications in various domains such as machine learning, image processing, and computational biology. Despite its widespread use, the intricate structure and scaling properties of the coupling matrices generated by the Sinkhorn algorithm remain largely unexplored. In this paper, we delve into the multifractal properties of these coupling matrices, aiming to unravel their complex behavior and shed light on the underlying dynamics of the Sinkhorn algorithm. We prove the existence of the multifractal spectrum and the singularity spectrum for the Sinkhorn coupling matrices. Furthermore, we derive bounds on the generalized dimensions, providing a comprehensive characterization of their scaling properties. Our findings not only deepen our understanding of the Sinkhorn algorithm but also pave the way for novel applications and algorithmic improvements in the realm of optimal transport.

翻译：Sinkhorn算法已成为解决最优传输问题的有力工具，在机器学习、图像处理和计算生物学等多个领域得到广泛应用。尽管该算法已被广泛使用，但Sinkhorn算法生成的耦合矩阵的复杂结构和标度特性在很大程度上仍未得到探索。本文深入研究了这些耦合矩阵的多重分形特性，旨在揭示其复杂行为并阐明Sinkhorn算法的底层动力学机制。我们证明了Sinkhorn耦合矩阵的多重分形谱和奇异性谱的存在性。此外，我们推导了广义维数的界限，从而全面刻画了其标度特性。我们的发现不仅加深了对Sinkhorn算法的理解，还为最优传输领域的新应用和算法改进铺平了道路。