Topologically Attributed Graphs for Shape Discrimination

TL;DR

This paper introduces a new family of attributed graphs derived from Mapper and Reeb graph approximations, enriched with persistent homology to improve shape discrimination and classification.

Contribution

It presents a novel topologically attributed graph construction that incorporates stable homological invariants for enhanced shape analysis.

Findings

Achieves competitive shape classification results

Enriches graph representations with stable topological invariants

Demonstrates effectiveness in shape discrimination tasks

Abstract

In this paper we introduce a novel family of attributed graphs for the purpose of shape discrimination. Our graphs typically arise from variations on the Mapper graph construction, which is an approximation of the Reeb graph for point cloud data. Our attributions enrich these constructions with (persistent) homology in ways that are provably stable, thereby recording extra topological information that is typically lost in these graph constructions. We provide experiments which illustrate the use of these invariants for shape representation and classification. In particular, we obtain competitive shape classification results when using our topologically attributed graphs as inputs to a simple graph neural network classifier.