Positional Encoding meets Persistent Homology on Graphs

TL;DR

This paper compares positional encoding and persistent homology for graph neural networks, establishing their relative expressiveness, and introduces PiPE, a new method that outperforms both in various graph tasks.

Contribution

It provides a theoretical comparison of PE and PH, and proposes PiPE, a learnable approach that surpasses both in expressiveness and performance.

Findings

Neither PE nor PH is strictly more expressive than the other.

PiPE is provably more expressive than both PE and PH.

PiPE achieves state-of-the-art results on multiple graph tasks.

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.