A Modified Newton Method for Multilinear PageRank

Abstract

When studying the multilinear PageRank problem, a system of polynomial equations needs to be solved. In this paper, we propose a modified Newton method and develop a monotone convergence theory for a third-order tensor when $\alpha \lt 1/2$. In this parameter regime, the sequence of vectors produced by the Newton-like method is monotonically increasing and converges to the solution. When $\alpha \gt 1/2$ we present an always-stochastic modified Newton iteration. Numerical results illustrate the effectiveness of this method.