Entrywise Approximate Solutions for SDDM Systems in Almost-Linear Time

TL;DR

This paper introduces an efficient algorithm for approximately solving SDDM linear systems in nearly linear time, significantly improving computational speed for large sparse matrices common in graph-based applications.

Contribution

The paper presents a novel algorithm that computes entrywise approximate solutions for SDDM systems in almost-linear time, a substantial advancement over previous methods.

Findings

Achieves solution in ilde{O}(m n^{o(1)}) time

Works for any invertible SDDM matrix

High-probability approximation guarantees

Abstract

We present an algorithm that given any invertible symmetric diagonally dominant M-matrix (SDDM), i.e., a principal submatrix of a graph Laplacian, $\boldsymbol{\mathit{L}}$ and a nonnegative vector $\boldsymbol{\mathit{b}}$, computes an entrywise approximation to the solution of $\boldsymbol{\mathit{L}} \boldsymbol{\mathit{x}} = \boldsymbol{\mathit{b}}$ in $\tilde{O}(m n^{o(1)})$ time with high probability, where $m$ is the number of nonzero entries and $n$ is the dimension of the system.