TY - JOUR

T1 - Scaling positive random matrices

T2 - concentration and asymptotic convergence

AU - Landa, Boris

N1 - Publisher Copyright:
© 2022, Institute of Mathematical Statistics. All rights reserved.

PY - 2022

Y1 - 2022

N2 - 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 Ã ϵ 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 Ã around those of A = E[Ã]. This result is employed to study the convergence rate of the scaling factors of Ã to those of A, as well as the concentration of the scaled version of Ã around the scaled version of A in operator norm, as M, N → ∞. We demonstrate our results in several simulations.

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

KW - concentration inequality

KW - doubly stochastic matrix

KW - matrix balancing

KW - matrix scaling

UR - http://www.scopus.com/inward/record.url?scp=85143894854&partnerID=8YFLogxK

U2 - 10.1214/22-ECP502

DO - 10.1214/22-ECP502

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AN - SCOPUS:85143894854

SN - 1083-589X

VL - 27

JO - Electronic Communications in Probability

JF - Electronic Communications in Probability

M1 - 61

ER -