Scaling positive random matrices: concentration and asymptotic convergence

TL;DR

This paper analyzes the behavior of random positive matrices under scaling, providing concentration inequalities and convergence rates for the scaling factors and scaled matrices as dimensions grow large.

Contribution

It introduces concentration bounds and asymptotic convergence results for the scaling factors and scaled matrices of random positive matrices, extending matrix scaling theory to stochastic settings.

Findings

Concentration inequality for scaling factors around their expectations.

Bounded convergence rate of scaling factors as matrix dimensions increase.

High-probability bounds for the scaled matrix's operator norm in large dimensions.

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 (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 $\widetilde{A}\in\mathbb{R}^{M\times N}$ 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 $\widetilde{A}$ 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 $M,N\rightarrow\infty$. 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.