# Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods

## TL;DR

This paper introduces nearly-linear and near-quadratic time algorithms for matrix scaling and balancing, utilizing a new second-order optimization framework and interior point methods, advancing computational efficiency in scientific computing.

## Contribution

The paper develops a unified second-order optimization framework and algorithms that improve the efficiency of matrix scaling and balancing, especially for matrices with quasi-polynomial condition ratios.

## Key findings

- Algorithms run in nearly-linear time for quasi-polynomial condition ratios.
- An interior point method achieves near-quadratic time complexity.
- The framework generalizes Laplacian system solving for broader function classes.

## Abstract

In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time $\widetilde{O}\left(m\log \kappa \log^2 (1/\epsilon)\right)$ where $\epsilon$ is the amount of error we are willing to tolerate. Here, $\kappa$ represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever $\kappa$ is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time $\widetilde{O}(m^{3/2} \log (1/\epsilon))$. In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.