Much Faster Algorithms for Matrix Scaling

TL;DR

This paper introduces significantly faster algorithms for the matrix scaling problem, improving efficiency from previous methods and applicable to various special cases, with potential broader impact in optimization.

Contribution

The authors develop new algorithms for matrix scaling that achieve faster total complexity, especially when the matrix has few nonzero entries, advancing the state-of-the-art.

Findings

Achieve total complexity of O(m + n^{4/3}) for certain matrix scaling cases.

Improve upon previous algorithms with complexity O(n^4) or O(m n^{1/2}/).

Utilize tailored first and second order optimization techniques.

Abstract

We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this problem asks to find diagonal (scaling) matrices $X$ and $Y$ (if they exist), so that $X A Y$ $\varepsilon$-approximates a doubly stochastic, or more generally a matrix with prescribed row and column sums. We address the general scaling problem as well as some important special cases. In particular, if $A$ has $m$ nonzero entries, and if there exist $X$ and $Y$ with polynomially large entries such that $X A Y$ is doubly stochastic, then we can solve the problem in total complexity $\tilde{O}(m + n^{4/3})$. This greatly improves on the best known previous results, which were either $\tilde{O}(n^4)$ or $O(m n^{1/2}/\varepsilon)$. Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems.

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