Scaling positive random matrices: concentration and asymptotic convergence

Abstract

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. 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 establish the stability of matrix scaling to random bounded perturbations. Specifically, letting $\tilde{A}\in {\mathbb{R}}^{M\times N}$ 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 $\tilde{A}$ around those of $A=\mathbb{E}[\tilde{A}]$. This result is employed to study the convergence rate of the scaling factors of $\tilde{A}$ to those of A, as well as the concentration of the scaled version of $\tilde{A}$ around the scaled version of A in operator norm, as $M,N\to \mathrm{\infty }$. We demonstrate our results in several simulations.