Title: Positional Encoding meets Persistent Homology on Graphs Show Chinese title

Title: 位置编码在图上的持久同调相遇

Abstract: The local inductive bias of message-passing graph neural networks (GNNs) hampers their ability to exploit key structural information (e.g., connectivity and cycles). Positional encoding (PE) and Persistent Homology (PH) have emerged as two promising approaches to mitigate this issue. PE schemes endow GNNs with location-aware features, while PH methods enhance GNNs with multiresolution topological features. However, a rigorous theoretical characterization of the relative merits and shortcomings of PE and PH has remained elusive. We bridge this gap by establishing that neither paradigm is more expressive than the other, providing novel constructions where one approach fails but the other succeeds. Our insights inform the design of a novel learnable method, PiPE (Persistence-informed Positional Encoding), which is provably more expressive than both PH and PE. PiPE demonstrates strong performance across a variety of tasks (e.g., molecule property prediction, graph classification, and out-of-distribution generalization), thereby advancing the frontiers of graph representation learning. Code is available at https://github.com/Aalto-QuML/PIPE. Show Chinese abstract

Abstract: 图神经网络（GNNs）的消息传递机制在局部归纳偏置方面存在局限性，这阻碍了它们利用关键结构信息（如连通性和循环）的能力。位置编码（PE）和持久同调（PH）作为两种有前景的方法，旨在缓解这一问题。PE 方法赋予 GNN 位置感知特征，而 PH 方法则通过多分辨率拓扑特征增强 GNN。然而，对 PE 和 PH 的相对优势与不足的严格理论刻画仍未明朗。 我们通过证明这两种范式在表达能力上并无高下之分来填补这一空白，并提供了新的构造实例，展示了一种方法失败但另一种成功的情况。我们的洞见指导了新型可学习方法 PiPE（Persistence-informed Positional Encoding）的设计，该方法在理论上被证明比 PH 和 PE 更具表达能力。PiPE 在多种任务（如分子属性预测、图分类和分布外泛化）中表现出色，从而推动了图表示学习的前沿发展。 代码可在 https://github.com/Aalto-QuML/PIPE 获取。