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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 $widetilde{A}inmathbb{R}^{Mtimes N}$ 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 $M,Nrightarrowinfty$. 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.
The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of r
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We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matr