  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.
