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

翻译：本文对Sinkhorn算法在代价矩阵稀疏情况下的收敛速率进行了理论分析。我们推导了收敛速率的边界，该边界依赖于代价矩阵的稀疏模式与稀疏程度。同时，我们探讨了针对稠密代价矩阵的现有收敛结果是否能够适用于稀疏情形或得到改进。我们的分析为理解Sinkhorn算法在稀疏性条件下的行为提供了新的见解，并指出了算法改进的潜在方向。