论文标题
缩放阳性随机矩阵：浓度和渐近收敛性
Scaling positive random matrices: concentration and asymptotic convergence
论文作者
论文摘要
众所周知，任何正矩阵都可以缩放以通过将行和色谱柱乘以某些正缩放因子（唯一到正量表）来缩放行和列总和。该过程称为矩阵缩放，并在操作研究，经济学，图像处理和机器学习中发现了许多应用。在这项工作中，我们研究了缩放因子的行为以及当要缩放的矩阵是随机的。具体而言，让$ \ widetilde {a} \ in \ mathbb {r}^{m \ times n} $是一个正面的随机矩阵，其条目假设某种类型的独立性，我们为$ \ didetilde {a} $ a = a = a = a = \ mathbb {e} [\ widetilde {a}] $。该结果被用来将$ \ widetilde {a} $的比例因子的收敛率与$ a $的收敛速度联系起来，以及$ \ widetilde {a} $的缩放版本的浓度，围绕缩放版本的运营商Norm的$ a $ a $ a a $ m，n \ rightarrow \ rightarrow \ rightarrow \ iffty $。当$ \ widetilde {a} $的条目是独立的，$ m = n $，并且所有处方的行和列总和为$ 1 $（即，偶然到达的矩阵缩放），这两个先前提到的界限均为$ \ nathcal {O}（O}（O}（O}）我们在几个模拟中演示了结果。
It is well known that any positive matrix can be scaled to have prescribed row and column sums by multiplying its rows and columns by certain positive scaling factors (which are unique up to a positive scalar). This procedure is known as matrix scaling, and has found numerous applications in operations research, economics, image processing, and machine learning. In this work, we investigate the behavior of the scaling factors and the resulting scaled matrix when the matrix to be scaled is random. Specifically, letting $\widetilde{A}\in\mathbb{R}^{M\times N}$ be a positive and bounded random matrix whose entries assume a certain type of independence, we provide a concentration inequality for the scaling factors of $\widetilde{A}$ around those of $A = \mathbb{E}[\widetilde{A}]$. This result is employed to bound the convergence rate of the scaling factors of $\widetilde{A}$ to those of $A$, as well as the concentration of the scaled version of $\widetilde{A}$ around the scaled version of $A$ in operator norm, as $M,N\rightarrow\infty$. When the entries of $\widetilde{A}$ are independent, $M=N$, and all prescribed row and column sums are $1$ (i.e., doubly-stochastic matrix scaling), both of the previously-mentioned bounds are $\mathcal{O}(\sqrt{\log N / N})$ with high probability. We demonstrate our results in several simulations.
