Topological Deep Learning: Going Beyond Graph Data (2206.00606v3)

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.

• The paper introduces Combinatorial Complexes (CCs) as a generalized topological structure to enable deep learning on non-Euclidean data with higher-order interactions, extending beyond graph-based models.
• It proposes Combinatorial Complex Neural Networks (CCNNs) built on CCs, detailing their architecture and operators for processing complex relational data while maintaining desirable equivariances.
• Empirical results show CCNNs achieve competitive performance on tasks like mesh and graph classification, and the authors provide Python packages (TopoNetX, TopoModelX) to facilitate their implementation.

An Examination of Topological Deep Learning and Combinatorial Complex Neural Networks

This paper, titled "Topological Deep Learning," provides a comprehensive framework for developing and evaluating deep learning models applied to data residing on non-Euclidean domains. This contribution is formulated through the introduction of Combinatorial Complexes (CCs), which serve as generalized topological structures. The focus extends beyond conventional graph-based learning to effectively harness higher-order interactions inherent in complex domains such as hypergraphs, simplicial complexes, and cell complexes.

Conceptual Framework and Motivations

The authors justify the advancement toward topological deep learning by elucidating the limitations of current models restricted to graph structures. Graph-based models, though powerful, are constrained in their ability to encapsulate non-local dependencies and higher-order interactions. By leveraging CCs, researchers can imbue models with hierarchical and set-type relations, crucial for interpreting data with intricate dependencies.

CCs amalgamate the key aspects of existing higher-order structures like simplicial and cell complexes, thus accommodating arbitrary set-type relations while maintaining hierarchical architectures. This blending supports the robust abstraction essential for synthesizing deep learning and topological data analysis, enabling sophisticated representations that are invariant to certain geometric transformations.

Combinatorial Complexes and Associated Neural Networks

The paper introduces Combinatorial Complex Neural Networks (CCNNs), a versatile class of neural networks constructed on CCs, which are equipped to handle complex relational data. Attention-based CCNNs receive particular emphasis, detailing their permutation and orientation equivariances. These properties refer to the network's ability to maintain consistent performance regardless of the transformations applied to the underlying data structure, a desirable feature in many computational learning scenarios.

CCNNs operate through a combination of novel operators, such as the push-forward operation, which cohesively integrates the computational strategies for message-passing and (un)pooling. In particular, the attention mechanisms integrated within CCNNs enable nuanced signal processing by adjusting weights assigned to adjacent cells based on their task-relevant information, enhancing the network's expressive capacity.

Implementation Results

The research work rigorously evaluates the proposed framework across a series of empirical tasks—mesh segmentation, point cloud classification, and graph classification—on both simulated and real-world datasets. Results demonstrate that CCNNs achieve competitive, if not superior, predictive performance compared to current state-of-the-art models, underscoring their operational efficacy in capturing higher-order relationships prevalent in these datasets.

The authors provide robust Python packages: TopoNetX, TopoEmbedX, and TopoModelX, which facilitate the construction and implementation of topological models. These tools support diverse topological spaces, promoting further exploration and application in varied machine learning challenges.

Implications and Future Directions

The implications of this work are manifold, extending the reach of deep learning into domains previously less explored due to structural complexity constraints. By incorporating topological principles within the learning framework, there is an anticipated improvement in model generalization and the ability to conduct more profound analyses on complex, multi-faceted datasets.

Future research could delve into optimizing these computational models for scalability and efficiency, particularly as CCNNs involve intricacies beyond traditional network architectures. Exploring CCNNs in dynamic and evolving datasets, as well as potential integrations with quantum computing paradigms, presents an intriguing direction that could further bolster the framework’s applicability and performance in intensive data environments.

In conclusion, this paper presents significant advancements in topological deep learning, providing a foundational approach for leveraging higher-order structures to improve learning machinery efficiency and flexibility. The development and implementation of CCNNs are seminal, setting the stage for ongoing innovations and applications across scientific and engineering domains.