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 than standard GNNs on a variety of node classification tasks, particularly in graphs with high heterophily.

翻译：欧拉特征变换（ECT）是一种可高效计算的几何拓扑不变量，用于刻画数据的全局形状。本文提出局部欧拉特征变换（$\ell$-ECT），作为ECT的一种新颖扩展，专门用于增强图表示学习中的表达能力和可解释性。与可能通过聚合丢失关键局部细节的传统图神经网络（GNNs）不同，$\ell$-ECT能够无损地表示局部邻域。该方法通过保留细微的局部结构同时维持全局可解释性，解决了GNNs的关键局限。此外，我们基于$\ell$-ECT构建了一种旋转不变度量，用于数据空间的空间对齐。在各种节点分类任务上，尤其在异配性较高的图中，我们的方法展现出优于标准GNNs的性能。