Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks (2304.10031v3)

Abstract: The natural world is full of complex systems characterized by intricate relations between their components: from social interactions between individuals in a social network to electrostatic interactions between atoms in a protein. Topological Deep Learning (TDL) provides a comprehensive framework to process and extract knowledge from data associated with these systems, such as predicting the social community to which an individual belongs or predicting whether a protein can be a reasonable target for drug development. TDL has demonstrated theoretical and practical advantages that hold the promise of breaking ground in the applied sciences and beyond. However, the rapid growth of the TDL literature for relational systems has also led to a lack of unification in notation and language across message-passing Topological Neural Network (TNN) architectures. This presents a real obstacle for building upon existing works and for deploying message-passing TNNs to new real-world problems. To address this issue, we provide an accessible introduction to TDL for relational systems, and compare the recently published message-passing TNNs using a unified mathematical and graphical notation. Through an intuitive and critical review of the emerging field of TDL, we extract valuable insights into current challenges and exciting opportunities for future development.

• The paper presents a unified framework that decomposes message-passing operations in topological neural networks.
• It details various topological structures such as hypergraphs and simplicial complexes to capture higher-order relationships in data.
• The survey highlights benchmarking challenges and advocates for standardized metrics to advance practical TDL applications.

Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks

The paper "Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks," authored by Mathilde Papillon, Sophia Sanborn, Mustafa Hajij, and Nina Miolane, provides an exhaustive synthesis of the field of Topological Deep Learning (TDL) with a specific focus on Message-Passing Topological Neural Networks (TNNs). The researchers tackle the burgeoning complexity and fragmentation in the TDL literature, an issue that has been exacerbated by the various notations and terminologies employed across numerous studies. The analysis in this survey lays a comprehensive groundwork that unifies existing approaches through consistent mathematical nomenclature and graphical representations.

The paper sets forth a detailed examination of the use of topological frameworks, moving beyond mere graph structures to capture higher-order relational data, such as those found in social networks and biochemical interactions. Topologies such as hypergraphs, simplicial complexes, cellular complexes, and combinatorial complexes come into focus, each offering unique capabilities to encode complex relationships and hierarchy in data.

Core Concepts and Framework

The authors present a structured overview of TNNs by decomposing the message-passing operations into four essential steps: message computation, within-neighborhood aggregation, between-neighborhood aggregation, and feature update. This framework provides a standard against which various TNN architectures can be compared.

Domains and Structures: The paper elaborates on different domains used in TDL, ranging from graphs, extending to more complex structures like simplicial complexes and combinatorial complexes, explaining the unique attributes and merits of each in dealing with higher-order interactions and abstractions.

Neighborhood and Message Passing: By surveying the types of neighborhood structures and message-passing functions leveraged across TNN architectures, the authors identify several innovative strategies employed to exploit the topological and geometric properties of the data domain, ensuring robustness and expressivity in the learned representations.

Critical Review and Observations

The authors critically evaluate existing TNN architectures in terms of the tasks they address, the depth of benchmarking against both established and cutting-edge models, and their adaptability across different topological domains. Structures such as hypergraphs and simplicial complexes have seen the most exploration, often focused on node and edge-level tasks like classification and prediction.

Despite the promising developments, several areas necessitate further inquiry:

1. Benchmarking and Comparison: The lack of standardized benchmarks across domains hinders seamless model evaluations, underlining a need for comprehensive datasets and uniform comparison metrics.
2. Generalization and Adaptability: While hypergraph models have successfully extended concepts from graph neural networks (GNNs), similar adaptations for more general topological domains like cellular and combinatorial complexes remain underexplored.
3. Dynamic Models and Deeper Architectures: The paper highlights potential advancements by integrating dynamic domains and addressing the challenge of oversmoothing, known in GNNs but less so in TNNs. Methods to counteract oversmoothing could allow TNNs to scale in depth and complexity.

Practical and Theoretical Implications

The insights from this survey have practical implications, potentially impacting domains such as drug discovery, social network analysis, and complex system simulations. Theoretical advancements in topological expressivity and computational efficiency could significantly enhance TNNs' capabilities, positioning them as essential tools for capturing and processing multifaceted systems.

The authors suggest that future research should also aim to bridge the conceptual and practical gaps between various topological methods and foster innovations that could lead to novel architectures. These developments would further solidify TDL's standing within AI research and its application to real-world challenges.

In conclusion, this survey of Message-Passing Topological Neural Networks sets a new standard for understanding and advancing TDL methodologies. By unifying diverse strands of research, the authors provide a critical resource for researchers aiming to harness topology's full potential in deep learning.