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 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 around those of . This result is employed to study the convergence rate of the scaling factors of to those of A, as well as the concentration of the scaled version of around the scaled version of A in operator norm, as . We demonstrate our results in several simulations.
This research was supported by the National Institute of Health, grant numbers R01GM131642 and UM1DA051410.
The author would like to thank Thomas Zhang, Yuval Kluger, and Dan Kluger for their useful comments and suggestions.
"Scaling positive random matrices: concentration and asymptotic convergence." Electron. Commun. Probab. 27 1 - 13, 2022. https://doi.org/10.1214/22-ECP502