Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity (2408.05486v2)

Abstract: Topological deep learning (TDL) is a rapidly growing field that seeks to leverage topological structure in data and facilitate learning from data supported on topological objects, ranging from molecules to 3D shapes. Most TDL architectures can be unified under the framework of higher-order message-passing (HOMP), which generalizes graph message-passing to higher-order domains. In the first part of the paper, we explore HOMP's expressive power from a topological perspective, demonstrating the framework's inability to capture fundamental topological and metric invariants such as diameter, orientability, planarity, and homology. In addition, we demonstrate HOMP's limitations in fully leveraging lifting and pooling methods on graphs. To the best of our knowledge, this is the first work to study the expressivity of TDL from a \emph{topological} perspective. In the second part of the paper, we develop two new classes of architectures -- multi-cellular networks (MCN) and scalable MCN (SMCN) -- which draw inspiration from expressive GNNs. MCN can reach full expressivity, but scaling it to large data objects can be computationally expansive. Designed as a more scalable alternative, SMCN still mitigates many of HOMP's expressivity limitations. Finally, we create new benchmarks for evaluating models based on their ability to learn topological properties of complexes. We then evaluate SMCN on these benchmarks and on real-world graph datasets, demonstrating improvements over both HOMP baselines and expressive graph methods, highlighting the value of expressively leveraging topological information. Code and data are available at https://github.com/yoavgelberg/SMCN.

• The paper reveals that HOMP models struggle to capture fundamental topological features such as homology, diameter, and orientability.
• It introduces two novel architectures—Multi-Cellular Networks (MCN) for full expressivity and Scalable MCN (SMCN) for balanced computational efficiency.
• Empirical validation on a synthetic Torus dataset demonstrates that SMCN successfully distinguishes all topological object pairs, outperforming traditional HOMP.

Topological Blind Spots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity

"Topological Blind Spots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity" by Eitan et al. provides a comprehensive examination of the expressivity limitations in Higher-Order Message-Passing (HOMP) models used in Topological Deep Learning (TDL). The paper introduces novel architectures aimed at addressing these limitations and demonstrates their effectiveness both theoretically and empirically.

Landscape of Topological Deep Learning and HOMP

Topological Deep Learning is a subfield focused on learning from data represented by topological structures like hypergraphs, simplicial complexes, and combinatorial complexes (CCs). HOMP extends traditional Message-Passing Neural Networks (MPNNs) to higher-dimensional topological spaces, making it a powerful tool for a wide range of applications. However, similar to MPNNs, HOMP faces significant expressivity limitations.

Limitations of HOMP

The authors investigate the expressivity of HOMP from multiple perspectives:

1. Topological and Metric Invariants: HOMP's inability to differentiate between fundamental topological and metric properties such as diameter, orientability, planarity, and homology is highlighted. Despite these properties being critical in many applications, HOMP struggles to encapsulate them due to its inherent architectural constraints.
2. Lifting and Pooling Operators: HOMP's inadequacy in leveraging lifting and pooling operators to fully exploit topological structures is explored. The authors provide instances where HOMP fails to distinguish between topological structures generated using common graph lifting and pooling operations.
3. Comparison with Hypergraph Architectures: HOMP is compared with hypergraph networks, specifically the Equivariant Hypergraph Neural Networks (EHNN). While HOMP shows some advantages in expressivity, it still falls short in several scenarios.

Introduction of Multi-Cellular Networks

To address HOMP's limitations, the authors propose two novel classes of TDL models: Multi-Cellular Networks (MCN) and Scalable Multi-Cellular Networks (SMCN). These models are inspired by architectures designed to overcome expressivity limitations in graph neural networks.

Multi-Cellular Networks (MCN) and Scalable Multi-Cellular Networks (SMCN)

MCN makes use of expressive graph architectures and introduces equivariant linear layers into the HOMP framework. MCN can achieve full expressivity but suffers from scalability issues.

SMCN, a scalable alternative to MCN, incorporates elements from Provably Powerful Graph Networks (PPGN) and Sub-Complex Networks (SCN). These adaptations significantly enhance expressivity while remaining computationally feasible. SMCN leverages sparse connectivity structures typical in higher-order cells, enabling it to balance expressivity and scalability effectively.

Empirical Validation and Contributions

The paper’s theoretical findings are empirically validated using a synthetic dataset, the Torus dataset, constructed to challenge the expressivity of TDL models. The results show that while HOMP is unable to distinguish between any pairs of topological objects in the dataset, the proposed SMCN model successfully distinguishes all pairs, thereby empirically validating the theoretical claims.

The key contributions of the paper are summarized as follows:

1. Comprehensive Analysis of HOMP’s Expressive Power: The paper provides a comparative analysis against hypergraph architectures and evaluates HOMP's ability to capture topological and metric invariants.
2. Novel Architectures: Introduction of Multi-Cellular Networks (MCN), which can achieve full expressivity, and Scalable Multi-Cellular Networks (SMCN), which mitigates many of HOMP's expressivity limitations.
3. Empirical Validation: Introduction and successful use of the Torus dataset to empirically validate the proposed models, demonstrating significant expressivity gains over HOMP.

Implications and Future Developments

The implications of this work are both practical and theoretical. Practically, the introduction of SMCN provides a more expressive and scalable alternative to HOMP, promising better performance on complex topological datasets. Theoretically, this work opens new avenues for exploring the expressivity of TDL models in more depth, potentially leading to the development of even more effective architectures.

Future research may delve into further scalability improvements for MCN and SMCN or explore new pooling operations that could enhance the practicality of these models. Moreover, more comprehensive synthetic datasets could be designed to better benchmark the expressivity and efficiency of future TDL models.

In conclusion, this paper advances the state of Topological Deep Learning by not only identifying critical expressivity limitations in HOMP but also proposing and validating innovative solutions that pave the way for more expressive and versatile topological models.