Given a non-negative \(n \times m\) real matrix A, the matrix scaling problem is to determine if it is possible to scale the rows and columns so that each row and each column sums to a specified positive target values. The Sinkhorn–Knopp algorithm is a simple and classic procedure which alternately scales all rows and all columns to meet these targets. The focus of this paper is the worst-case theoretical analysis of this algorithm. We present an elementary convergence analysis for this algorithm that improves upon the previous best bound. In a nutshell, our approach is to show (i) a simple bound on the number of iterations needed so that the KL-divergence between the current row-sums and the target row-sums drops below a specified threshold \(\delta \), and (ii) then show that for a suitable choice of \(\delta \), whenever KL-divergence is below \(\delta \), then the \(\ell _1\)-error or the \(\ell _2\)-error is below \(\varepsilon \). The well-known Pinsker’s inequality immediately allows us to translate a bound on the KL divergence to a bound on \(\ell _1\)-error. To bound the \(\ell _2\)-error in terms of the KL-divergence, we establish a new inequality, referred to as (KL vs \(\ell _1/\ell _2\)). This inequality is a strengthening of Pinsker’s inequality and may be of independent interest.

给定非负\（n×m）实矩阵A，矩阵缩放问题是确定是否可以缩放行和列，以使每一行和每一列的总和为指定的正目标值。Sinkhorn–Knopp算法是一种简单而经典的过程，可以交替缩放所有行和所有列以满足这些目标。本文的重点是该算法的最坏情况理论分析。我们提出了对该算法的基本收敛性分析，该分析对先前的最佳范围进行了改进。简而言之，我们的方法是显示（i）所需迭代次数的简单界限，以使当前行总和与目标行总和之间的KL散度降至指定阈值以下\（\三角洲\） ，和（ii）然后表明，适当选择\（\三角洲\） ，每当KL散度低于\（\三角洲\） ，则\（\ ELL _1 \） -error或\（\ ell _2 \） -错误低于\（\ varepsilon \）。著名的Pinsker不等式立即使我们能够将KL散度的边界转换为\（\ ell _1 \） -error的边界。为了根据KL散度限制\（\ ell _2 \） -误差，我们建立了一个新的不等式，称为（KL vs \（\ ell _1 / \ ell _2 \））。这种不平等加剧了Pinsker的不平等，并且可能具有独立利益。