# Spectral analysis of matrix scaling and operator scaling

## TL;DR

This paper provides a spectral analysis of matrix and operator scaling, showing linear convergence of gradient flows under spectral gap conditions, with implications for various applications in mathematics and quantum information.

## Contribution

It introduces a spectral gap condition that guarantees linear convergence of gradient methods for matrix and operator scaling, and derives bounds relevant for multiple applications.

## Key findings

- Gradient flow converges linearly with spectral gap
- Bounds on condition number and capacity derived
- Applications include expander graphs and quantum information

## Abstract

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the capacity of the matrix and operator. These bounds can be used in various applications of scaling problems, including matrix scaling on expander graphs, permanent lower bounds on random matrices, the Paulsen problem on random frames, and Brascamp-Lieb constants on random operators. In some applications, the inputs of interest satisfy the spectral condition and we prove significantly stronger bounds than the worst case bounds.