Efficient Numerical Methods to Solve Sparse Linear Equations with Application to PageRank

TL;DR

This paper introduces three novel numerical algorithms tailored for efficiently solving sparse linear equations, specifically applied to the PageRank problem, by leveraging convex optimization techniques and supporting sparsity.

Contribution

It presents three new algorithms—L1 gradient descent, sparse Frank-Wolfe, and randomized mirror descent—for faster PageRank computation on sparse matrices.

Findings

Algorithms demonstrate faster convergence on sparse matrices.

Numerical experiments confirm effectiveness and efficiency.

Methods outperform traditional approaches in sparse settings.

Abstract

In this paper, we propose three methods to solve the PageRank problem for the transition matrices with both row and column sparsity. Our methods reduce the PageRank problem to the convex optimization problem over the simplex. The first algorithm is based on the gradient descent in L1 norm instead of the Euclidean one. The second algorithm extends the Frank-Wolfe to support sparse gradient updates. The third algorithm stands for the mirror descent algorithm with a randomized projection. We proof converges rates for these methods for sparse problems as well as numerical experiments support their effectiveness.