NervePool: A Simplicial Pooling Layer

TL;DR

NervePool introduces a learned pooling layer for data on simplicial complexes, enabling hierarchical coarsening by partitioning the vertex set and extending to higher-dimensional simplices via unions of stars and a nerve construction. The authors provide a dual formulation: a topological view based on nerve complexes and a parallel matrix implementation using block mappings that reproduce the same coarsening, with a rigorous equivalence shown under hard vertex partitions and permutation invariance. The method integrates with existing graph pooling approaches (e.g., DiffPool) and is demonstrated on graph classification benchmarks, where NervePool achieves competitive performance while generalizing to higher-order structures. This work enables scalable, topology-aware pooling in simplicial neural networks and opens avenues for incorporating higher-order topological information into learning pipelines.

Abstract

For deep learning problems on graph-structured data, pooling layers are important for down sampling, reducing computational cost, and to minimize overfitting. We define a pooling layer, nervePool, for data structured as simplicial complexes, which are generalizations of graphs that include higher-dimensional simplices beyond vertices and edges; this structure allows for greater flexibility in modeling higher-order relationships. The proposed simplicial coarsening scheme is built upon partitions of vertices, which allow us to generate hierarchical representations of simplicial complexes, collapsing information in a learned fashion. NervePool builds on the learned vertex cluster assignments and extends to coarsening of higher dimensional simplices in a deterministic fashion. While in practice the pooling operations are computed via a series of matrix operations, the topological motivation is a set-theoretic construction based on unions of stars of simplices and the nerve complex.

Figures (9)

• Figure 1: Non-oriented boundary matrices $\mathbf{B}_1$ and $\mathbf{B}_2$ for an example simplicial complex.
• Figure 2: For simplex $\sigma$, geometrically realized as an edge, its four different types of adjacent simplices are: boundary adjacent, coboundary adjacent, lower adjacent, and upper adjacent.
• Figure 3: A visual representation of NervePool for an example simplicial complex and choice of soft partition of vertices. Shaded-in matrix entries indicate non-zero values, and elsewhere are zero-valued entries. Darker grey entry shading within the $\mathbf{S}^{(\ell)}$ matrix indicate the diagonal sub-blocks which are used for subsequent pooling operations. The output simplicial complex given by the matrix implementation is equivalent to the nerve complex output, up to weighting of $p$-simplices and the addition of "self-loops", as indicated by non-zero entries on the diagonal of adjacency matrices.
• Figure 4: The three green highlighted vertices come from a single cover element $U_i$ of a given example cover; the full cover example will be continued in Fig. \ref{['fig:sx_pool_diagram']}. The extended cover $\widetilde{U_i}$ for this collection of vertices, defined by the union of the star of every vertex in the cluster, is shown by the highlighted green simplices. These simplices in $K^{(\ell)}$ all contribute information to the meta vertex $\sigma \in K^{(\ell+1)}$.
• Figure 5: Illustration of NervePool on an example $3$-dim simplicial complex, applied to coarsen the complex into a $2$-dimensional simplicial complex. The left-most complex is the input simplicial complex, with vertex cluster membership indicated by color. The center complex depicts the extended clusters $\mathcal{U} = \{\widetilde{U}_i\}$ and the right-most complex is the pooled simplicial complex, determined by $\mathrm{Nrv}(\mathcal{U})$.

Theorems & Definitions (6)

• Theorem 1: Equivalence of Topological and Matrix Formulations
• proof
• Theorem 2: NervePool identity function
• proof
• Theorem 3: NervePool Permutation Invariance
• proof