Implementing distributed graph filters by elementary matrix decomposition

TL;DR

This paper presents a method to implement distributed graph filters using elementary matrix decomposition, enabling nodes to compute filters locally based on neighboring signals, with a proof of concept and example.

Contribution

It introduces a novel approach using Gaussian elimination to decompose graph filters into locally implementable components for connected graphs.

Findings

Any connected graph filter can be implemented through elementary matrix decomposition.

The method allows distributed implementation using only local neighbor signals.

A concrete example demonstrates the practical application of the approach.

Abstract

In this letter, we consider the implementation problem of distributed graph filters, where each node only has access to the signals of the current and its neighboring nodes. By using Gaussian elimination, we show that as long as the graph is connected, we can implement any graph filter by decomposing the filter into a product of directly implementable filters, filters that only use the signals at the current and neighboring nodes as inputs. We have also included a concrete example as an illustration.