Distributed algorithms for the least square solution of linear equations

TL;DR

This paper introduces novel distributed algorithms enabling multi-agent networks to collaboratively find least square solutions to linear equations, even with incomplete data and heterogeneous agent capabilities, ensuring exponential convergence and solution verification.

Contribution

It presents new distributed algorithms for least squares solutions that handle incomplete and heterogeneous data partitions with proven convergence and verification properties.

Findings

Algorithms achieve exponential convergence.

Effective in heterogeneous and incomplete data scenarios.

Simulation confirms practical effectiveness.

Abstract

This paper proposes distributed algorithms for solving linear equations to seek a least square solution via multi-agent networks. We consider that each agent has only access to a small and imcomplete block of linear equations rather than the complete row or column in the existing literatures. Firstly, we focus on the case of a homogeneous partition of linear equations. A distributed algorithm is proposed via a single-layered grid network, in which each agent only needs to control three scalar states. Secondly, we consider the case of heterogeneous partitions of linear equations. Two distributed algorithms with doubled-layered network are developed, which allows each agent's states to have different dimensions and can be applied to heterogeneous agents with different storage and computation capability. Rigorous proofs show that the proposed distributed algorithms collaboratively obtain a least square solution with exponential convergence, and also own a solvability verification property, i.e., a criterion to verify whether the obtained solution is an exact solution. Finally, some simulation examples are provided to demonstrate the effectiveness of the proposed algorithms.