Topological filtering of a signal over a network

TL;DR

This paper introduces a novel topological filtering method for graph signals using persistent homology, bridging topological data analysis and signal processing, applicable to signals over general graphs and graphs with faces.

Contribution

The paper presents a new filtering approach based on Basin Hierarchy Trees that encode persistent homology, extending graph signal processing with topological data analysis techniques.

Findings

Effective filtering demonstrated on synthetic datasets

Successful application to real-world datasets

Bridges topological data analysis with signal processing

Abstract

Graph Signal Processing deals with the problem of analyzing and processing signals defined on graphs. In this paper, we introduce a novel filtering method for graph-based signals by employing ideas from topological data analysis. We begin by working with signals over general graphs and then extend our approach to what we term signals over graphs with faces. To construct the filter, we introduce a new structure called the Basin Hierarchy Tree, which encodes the persistent homology. We provide an efficient algorithm and demonstrate the effectiveness of our approach through examples with synthetic and real datasets. This work bridges topological data analysis and signal processing, presenting a new application of persistent homology as a topological data processing tool.