Matrix scaling problems with sparse cost matrices arise frequently in various domains, such as optimal transport, image processing, and machine learning. The Sinkhorn-Knopp algorithm is a popular iterative method for solving these problems, but its convergence properties in the presence of sparsity have not been thoroughly analyzed. This paper presents a theoretical analysis of the convergence rate of the Sinkhorn-Knopp algorithm specifically for sparse cost matrices. We derive novel bounds on the convergence rate that explicitly depend on the sparsity pattern and the degree of nonsparsity of the cost matrix. These bounds provide new insights into the behavior of the algorithm and highlight the potential for exploiting sparsity to develop more efficient solvers. We also explore connections between our sparse convergence results and existing convergence results for dense matrices, showing that our bounds generalize the dense case. Our analysis reveals that the convergence rate improves as the matrix becomes less sparse and as the minimum entry of the cost matrix increases relative to its maximum entry. These findings have important practical implications, suggesting that the Sinkhorn-Knopp algorithm may be particularly well-suited for large-scale matrix scaling problems with sparse cost matrices arising in real-world applications. Future research directions include investigating tighter bounds based on more sophisticated sparsity patterns, developing algorithm variants that actively exploit sparsity, and empirically validating the benefits of our theoretical results on real-world datasets. This work advances our understanding of the Sinkhorn-Knopp algorithm for an important class of matrix scaling problems and lays the foundation for designing more efficient and scalable solutions in practice.

翻译：具有稀疏成本矩阵的矩阵缩放问题常见于最优传输、图像处理和机器学习等多个领域。Sinkhorn-Knopp算法是求解此类问题的常用迭代方法，但其在稀疏性条件下的收敛性质尚未得到深入分析。本文针对稀疏成本矩阵，对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新颖上界，这些上界明确依赖于成本矩阵的稀疏模式与非稀疏程度。这些界限为理解算法行为提供了新视角，并揭示了利用稀疏性开发更高效求解器的潜力。我们还探讨了稀疏收敛结果与现有稠密矩阵收敛结果之间的联系，表明我们的界限推广了稠密情形。分析表明，当矩阵稀疏度降低且成本矩阵最小元素相对于最大元素的比值增大时，收敛速率会得到改善。这些发现具有重要的实际意义，表明Sinkhorn-Knopp算法可能特别适用于现实应用中出现的具有稀疏成本矩阵的大规模矩阵缩放问题。未来的研究方向包括：基于更复杂稀疏模式探究更紧的收敛界限、开发主动利用稀疏性的算法变体，以及在真实数据集上实证验证理论结果的有效性。本研究深化了我们对Sinkhorn-Knopp算法在重要矩阵缩放问题类别中的理解，并为在实践中设计更高效、可扩展的解决方案奠定了基础。