Clifford Group Equivariant Simplicial Message Passing Networks

Abstract

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

Figures (10)

• Figure 1: Illustration of our proposed architecture. Top left: a set of vertices (and edges) is lifted to a simplicial complex. We highlight three simplex types: vertices (0-simplices, ), edges (1-simplices, ), and triangles (2-simplices, ). In this case, the vertex feature is vector-valued and embedded as the grade $1$ part of a Clifford algebra element: a multivector. In three dimensions, a multivector has scalar ( ), vector ( ), bivector ( ) and trivector ( ) components. Higher-order simplices are initialized using the geometric product of their constituent vertices. As such, edges in the top left visualization are bivector-valued, and triangles are trivector-valued. The simplicial message-passing framework, denoted by $\phi$, refines the multivector-valued simplices, as portrayed in the bottom-left, by passing messages between simplices of different order. Crucially, $\phi$ maintains equivariance to the Clifford group's orthogonal action $\rho(w)$, representing a rotation here. In doing so, our method is ensured to respect the geometric symmetries of the input data.
• Figure 1: MSE ($\downarrow$) of the tested models on the convex hulls experiment.
• Figure 2: Left: we show how a simple graph (three fully-connected nodes) is lifted to a simplicial complex. Using simplicial message passing, we allow communication between objects of different dimensions. That is, between vertices ($0 \leftrightarrow 0$) , nodes and edges ($0 \rightarrow 1$ and $1 \rightarrow 0$) , edges ($1 \leftrightarrow 1$) , and between edges and triangles ($1 \rightarrow 2$ and $2 \rightarrow 1$) . Right: same as left, but a top-down view. It illustrates the hypergraph associated with the complex with several meta-vertices representing the simplices of various dimensionality. Instead of running message passing separately for all different communication types, we share the parameters of a single neural network operating on the extended graph. By conditioning on the message type, it is still able to leverage the simplicial complex.
• Figure 2: Left: MSE ($10^{-2}$) of the tested models on the CMU motion capture dataset. Right: Depiction (not cherry-picked) of an instance (the ground-truth target positions) vs. a csmpn prediction.
• Figure 3: In the convex hulls experiment, the task is to estimate the volume of the convex hull of eight five-dimensional random points. Here, we display a three-dimensional example, which is easier to visualize.

Theorems & Definitions (6)

• Theorem 2.1: All polynomials are Clifford group equivariant
• Theorem 2.2: All grade projections are Clifford group equivariant
• Definition 2.3: Simplicial Complex
• Definition 2.4: Simplicial Adjacencies
• Theorem 2.5: bodnar2021weisfeiler
• Theorem 2.6: bodnar2021weisfeiler