Open Access
2022 Scaling positive random matrices: concentration and asymptotic convergence
Boris Landa
Author Affiliations +
Electron. Commun. Probab. 27: 1-13 (2022). DOI: 10.1214/22-ECP502


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 A˜RM×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 A˜ around those of A=E[A˜]. This result is employed to study the convergence rate of the scaling factors of A˜ to those of A, as well as the concentration of the scaled version of A˜ around the scaled version of A in operator norm, as M,N. We demonstrate our results in several simulations.

Funding Statement

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.


Download Citation

Boris Landa. "Scaling positive random matrices: concentration and asymptotic convergence." Electron. Commun. Probab. 27 1 - 13, 2022.


Received: 22 November 2021; Accepted: 20 November 2022; Published: 2022
First available in Project Euclid: 15 December 2022

MathSciNet: MR4529629
zbMATH: 1506.60014
Digital Object Identifier: 10.1214/22-ECP502

Primary: 60B20 , 60F10

Keywords: concentration inequality , doubly stochastic matrix , matrix balancing , matrix scaling

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