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

TL;DR

This work tackles the loss of local information in traditional GNNs by introducing the Local Euler Characteristic Transform (\ell-ECT), a local, invertible descriptor that captures both topology and geometry of neighborhoods. It defines \ell-ECT_k(x;X) = \text{ECT}(N_k(x;X)) and develops a rotation-invariant metric d_{ECT} for aligning geometric graphs, with theoretical expressivity guarantees for featured graphs. Empirically, \ell-ECT-based representations achieve competitive or superior node classification performance across diverse datasets, including heterophilic graphs, while remaining model-agnostic and interpretable via downstream feature importances. The approach enables robust spatial alignment and offers a versatile framework for interpretable graph learning with potential extensions to higher-order domains and hybrid architectures.

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.

Figures (8)

• Figure 1: A comparison of the Hausdorff distances of aligned graphs. The black dots represents the Hausdorff distance between the original graph and a randomly-rotated version of itself. Our $\mathop{\mathrm{\ell -ECT}}\nolimits$-based alignment always results in substantially lower distances, with a median distance close to zero.
• Figure 2: Feature importance scores of an XGBoost model for the "Coauthor Physics" dataset (using $\mathop{\mathrm{\ell -ECT}}\nolimits_1$). Only a small number of features admit high importance scores.
• Figure 3: A comparison of the squared $L^2$ distances of $\mathop{\mathrm{\ell -ECT}}\nolimits$s of aligned and non-aligned MNIST digits of "1," respectively.
• Figure 4: Critical difference diagram showing the ranks of different models across all node-classification tasks from \ref{['sec:Experiments']}. Even the worst-performing $\mathop{\mathrm{\ell -ECT}}\nolimits$-based approach ($\mathop{\mathrm{\ell -ECT}}\nolimits_2$) exhibits superior performance to all other methods, when averaged across all tasks.
• Figure 5: A comparison of the squared $L^2$ distances of the $\mathop{\mathrm{ECT}}\nolimits$s of aligned and non-aligned wedged spheres, respectively. We see that alignment results in a median loss of zero, thus effectively showing that the two spaces are the same.

Theorems & Definitions (12)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Corollary 1
• Theorem 3
• Theorem 3
• proof
• Theorem 3
• proof