Scaling positive random matrices: concentration and asymptotic convergence

Understand

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 $A = \mathbb{E}[\widetilde{A}]$.