Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms

TL;DR

This paper introduces the Local Euler Characteristic Transform ($ ext{l}$-ECT), a new graph representation method that preserves local details and enhances interpretability, outperforming traditional GNNs in node classification tasks.

Contribution

The paper presents $ ext{l}$-ECT, a novel extension of ECT for graphs that captures local structures losslessly and provides a rotation-invariant metric for data alignment.

Findings

$ ext{l}$-ECT improves node classification accuracy.

The method offers theoretical guarantees of effectiveness.

It maintains local detail while ensuring global interpretability.

Abstract

The Euler Characteristic Transform (ECT) is an efficiently-computable geometrical-topological invariant that characterizes the global shape of data. In this paper, we introduce the Local Euler Characteristic Transform ($\ell$-ECT), a novel extension of the ECT particularly designed to enhance expressivity and interpretability in graph representation learning. Unlike traditional Graph Neural Networks (GNNs), which may lose critical local details through aggregation, the $\ell$-ECT provides a lossless representation of local neighborhoods. This approach addresses key limitations in GNNs by preserving nuanced local structures while maintaining global interpretability. Moreover, we construct a rotation-invariant metric based on $\ell$-ECTs for spatial alignment of data spaces. Our method exhibits superior performance compared to standard GNNs on a variety of node-classification tasks, while also offering theoretical guarantees that demonstrate its effectiveness.