Topological Deep Learning: Going Beyond Graph Data

TL;DR

This work introduces combinatorial complexes (CCs) as a unifying topological domain for deep learning beyond graphs, and develops combinatorial complex neural networks (CCNNs) built around a push-forward operation, merge/split nodes, and pooling/unpooling mechanisms. It establishes a theory linking higher-order message passing, tensor-diagram representations, and attention (CCANNs) within CCs, and connects these ideas to a Hasse-graph perspective for equivariance and graph reductions. The authors provide a broad experimental validation across mesh shape analysis and graph classification, demonstrating competitive performance with state-of-the-art task-specific models and highlighting the advantages of incorporating higher-order relations. They also release software libraries (TopoNetX, TopoEmbedX, TopoModelX) and outline a rich set of future research directions, including scalable architectures, MOG-based pooling, and deeper theoretical understandings of CCNNs.

Abstract

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains. Specifically, we first introduce combinatorial complexes, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail. Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications.

Figures (36)

• Figure 1: A graphical abstract that visualizes our main contributions. (a): Different mathematical structures can be used to represent relations between abstract entities. Sets have entities with no connections, graphs encode binary relations between vertices, simplicial and cell complexes model hierarchical higher-order relations, and hypergraphs accommodate arbitrary set-type relations with no hierarchy. We introduce combinatorial complexes (CCs), which generalize graphs, simplicial and cell complexes, and hypergraphs. CCs are equipped with set-type relations as well as with a hierarchy of these relations. (b): By utilizing the hierarchical and topological structure of CCs, we introduce the push-forward operation, a fundamental building block for higher-order message-passing protocols and for (un)pooling operations on CCs. Our push-forward operations on CCs enable us to construct combinatorial complex neural networks (CCNNs), which provide a general conceptual framework for topological deep learning on higher-order domains.
• Figure 2: Data might be supported naturally on higher-order relations. (a): An edge-based vector field. (b): A face-based vector field. Both vector fields in (a) and (b) are defined on a cell complex torus. An interactive visualization of (a--b) is provided https://app.vectary.com/p/5uLAflZj6U2kvACv2kk2tN. (c): Class-labeled topological data might naturally be supported on higher-order relations. For instance, mesh segmentation labels for 2-faces are depicted by different colors (blue, green, turquoise, pink, brown) to represent different parts (head, neck, body, legs, tail) of the horse.
• Figure 3: Examples of processing data supported on graphs or on higher-order networks. (a): Graphs can be used to model particle interactions in fluid dynamics. Vertices represent particles, whereas particle-to-particle interactions are modeled via message passing among vertices sanchez2020learningshlomi2020graph. (b): When modeling springs and self-collision, it is natural to work with edges rather than vertices. This is because the behavior of the cloth is determined by the tension and compression forces acting along the edges, and not only by the position of individual particles. To model the interactions among multiple edges, polygonal faces can be used to represent the local geometry of the cloth. The polygonal faces provide a way to compute higher-order message passing among the edges.
• Figure 4: An illustration of representing hierarchical data via higher-order networks. Black arrows indicate graph augmentation by higher-order relations, whereas orange arrows indicate coarsened graph extraction. (a): A graph encodes binary relations (pink edges) among abstract entities (yellow vertices). (b): Higher-order relations, represented by blue cells, can be thought as relations among vertices or edges of the original graph. (c): Extraction of a coarsened version of the original graph. In the coarsened graph, vertices represent the higher-order relations (blue cells) of the original graph, and edges represent the intersections of these blue cells. (d--e): The same process repeats to obtain a more coarsened version of the original graph. The entire process corresponds to hierarchical higher-order relations, that is relations among relations, which extract meaning and content (including the 'shape of data'), a common task in topological data analysis carlsson2009topologyDW22.
• Figure 5: The graph in the middle can be augmented with higher-order relations to improve a learning task. In (a), cells have been added to the missing faces; a similar inductive bias has been considered in bodnar2021weisfeiler to improve classification on molecular data. In (b), one-hop neighborhoods have been added to some of the vertices in the graph; feng2019hypergraph use such an inductive bias, based on lifting a graph to its corresponding one-hop neighborhood hypergraph, to improve performance on vertex-based tasks.

Theorems & Definitions (92)

• Definition 1: Neighborhood function
• Definition 2: Neighborhood topology
• Definition 3: Topological space
• Definition 4: https://app.vectary.com/p/4HZRioKH7lZ2jWESIBrjhf
• Definition 5: https://app.vectary.com/p/3EBiRiJcYjFNvkbbWszQ0Z
• Definition 6: Hypergraph
• Definition 7: Rank function
• Definition 8: Set-type relations
• Definition 9: CC
• Remark 4.1