This paper presents a theoretical analysis of the convergence rate of the Sinkhorn-Knopp algorithm when the cost matrix is sparse. We derive bounds on the convergence rate that depend on the sparsity pattern and the degree of nonsparsity of the cost matrix. We also explore connections to existing convergence results for dense cost matrices. Our analysis provides new insights into the behavior of the Sinkhorn-Knopp algorithm in the presence of sparsity and highlights potential avenues for algorithmic improvements.

翻译：本文对代价矩阵稀疏时Sinkhorn-Knopp算法的收敛速率进行了理论分析。我们推导了收敛速率的界，该界依赖于代价矩阵的稀疏模式与非稀疏程度。同时探讨了该结果与稠密代价矩阵现有收敛理论的联系。我们的分析为理解稀疏条件下Sinkhorn-Knopp算法的行为提供了新见解，并指出了算法改进的潜在方向。