Convergence of the D-iteration algorithm: convergence rate and asynchronous distributed scheme

TL;DR

This paper introduces a general framework for diffusion operators related to positive matrices, analyzing their properties and convergence behavior, with applications to distributed fixed point computations.

Contribution

It presents a novel framework for diffusion operators, providing new insights into convergence rates and enabling improved distributed matrix iteration schemes.

Findings

Established properties of diffusion operators and their state vectors.

Proved convergence and improved rates for fixed point problems.

Applied framework to distributed computation scenarios.

Abstract

In this paper, we define the general framework to describe the diffusion operators associated to a positive matrix. We define the equations associated to diffusion operators and present some general properties of their state vectors. We show how this can be applied to prove and improve the convergence of a fixed point problem associated to the matrix iteration scheme, including for distributed computation framework. The approach can be understood as a decomposition of the matrix-vector product operation in elementary operations at the vector entry level.