Much Faster Algorithms for Matrix Scaling

Abstract

We develop several efficient algorithms for the classical Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input n× n matrix A, this problem asks to find diagonal (scaling) matrices X and Y (if they exist), so that X A Y ε-approximates a doubly stochastic matrix, 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}/ε).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.