Geometry-Aware Simplicial Message Passing

TL;DR

This paper introduces GSWL, a geometry-aware extension of the WL test for simplicial complexes, enhancing the ability of message passing networks to distinguish geometric structures.

Contribution

It develops GSWL, bounding the expressivity of geometry-aware message passing schemes and linking them with the Euler Characteristic Transform for comprehensive geometric analysis.

Findings

GSWL's discriminating power matches certain geometry-aware message passing schemes.

Experiments validate the hierarchy from combinatorial to geometry-aware models.

The framework provides a complete invariant for geometric simplicial complexes.

Abstract

The Weisfeiler--Lehman (WL) test and its simplicial extension (SWL) characterize the combinatorial expressivity of message passing networks, but they are blind to geometry, i.e., meshes with identical connectivity but different embeddings are indistinguishable. We introduce the Geometric Simplicial Weisfeiler--Lehman (GSWL) test, which incorporates vertex coordinates into color refinement for geometric simplicial complexes. In addition, we show that (i) the expressivity of geometry-aware simplicial message passing schemes is bounded above by GSWL, and (ii) that there exist parameters such that the discriminating power of GSWL is matched by these schemes on any fixed finite family of geometric simplicial complexes. Combined with the Euler Characteristic Transform (ECT), a complete invariant for geometric simplicial complexes, this yields a geometric expressivity characterization together with an approximation framework. Experiments on synthetic and mesh datasets serve to validate our theory, showing a clear hierarchy from combinatorial to geometry-aware models.