Topological Deep Learning with State-Space Models: A Mamba Approach for Simplicial Complexes (2409.12033v1)

Abstract: Graph Neural Networks based on the message-passing (MP) mechanism are a dominant approach for handling graph-structured data. However, they are inherently limited to modeling only pairwise interactions, making it difficult to explicitly capture the complexity of systems with $n$-body relations. To address this, topological deep learning has emerged as a promising field for studying and modeling higher-order interactions using various topological domains, such as simplicial and cellular complexes. While these new domains provide powerful representations, they introduce new challenges, such as effectively modeling the interactions among higher-order structures through higher-order MP. Meanwhile, structured state-space sequence models have proven to be effective for sequence modeling and have recently been adapted for graph data by encoding the neighborhood of a node as a sequence, thereby avoiding the MP mechanism. In this work, we propose a novel architecture designed to operate with simplicial complexes, utilizing the Mamba state-space model as its backbone. Our approach generates sequences for the nodes based on the neighboring cells, enabling direct communication between all higher-order structures, regardless of their rank. We extensively validate our model, demonstrating that it achieves competitive performance compared to state-of-the-art models developed for simplicial complexes.

• The paper introduces a novel framework using a Mamba state-space model to incorporate simplicial complexes, capturing higher-order interactions beyond pairwise node relationships.
• The paper bypasses traditional GNN message-passing by applying implicit state-space modeling, enabling direct communication across different complex ranks.
• The paper demonstrates algorithmic efficiency and scalability with a new batching strategy, achieving improved accuracy and lower computational costs on large-scale datasets.

Insights into "Topological Deep Learning with State-Space Models: A Mamba Approach for Simplicial Complexes"

The paper presents a novel approach to incorporate topological features in deep learning models using state-space representations, specifically targeted at simplicial complexes. The authors, Montagna et al., address the inherent limitation within Graph Neural Networks (GNNs) which predominantly use message-passing mechanisms suited for pairwise interactions, thereby limiting their capacity to capture complex relationships that extend beyond graph node interactions in higher-dimensional topological structures.

Key Contributions

1. Combining State-Space Models with Topological Data: The paper introduces a new architectural framework utilizing simplicial complexes within a deep learning context, typically not well-represented by traditional GNNs. By leveraging the Mamba state-space model, the authors effectively create sequences based on node neighborhoods, achieving comprehensive higher-order interaction capturing.
2. Elimination of Pairwise Limitation: The unique aspect of this method lies in its ability to effectively transcend the inherent pairwise engagement limitation present in standard GNNs. By bypassing message-passing through implicit state-space modeling, it facilitates direct communication across different ranks of simplicial complexes.
3. Algorithmic Efficiency: Essential aspects of the methodology include the hardware-efficient execution of Mamba, both in the forward and backward propagation across sequences, demonstrated to handle large-scale networks without the prohibitive computational costs often associated with purely graph-based or transformer-based architectures in handling higher-order data.

Implications and Results

Empirical validation on multiple datasets, transformed to simplicial complexes, illustrates the robustness of the proposed architecture. Significant performance gains were noted when compared to benchmark simplicial complex models such as Simplicial Complex Neural (SCN) networks and Simplicial Complex Convolutional Neural Networks (SCCNNs). The Mamba-based architecture demonstrated competitive results, showcasing improved accuracy and computational efficiency in tasks involving nodes with multiplied ranks.

Observations on Batching and Scalability

A noteworthy engineering contribution is the introduction of novel batching strategy grounded on the simplicial node incidence relation, reducing memory overhead and enhancing training throughput compared to conventional batching techniques. This is particularly instrumental for processing larger datasets where standard full-batch training is infeasible due to resource constraints.

Future Directions

The framework proposed lays foundational pathways for broad adaptability across different topological domains, potentially integrating cellular and combinatorial complex processing within a cohesive modeling strategy. Furthermore, the exploration of graph-structured datasets complexified by simplicial expansions proposes a new frontier where state-space mechanisms make pertinent higher-order interactions tractable and efficient.

In advancing this research, efforts might focus on extending Mamba's applicability to typical challenges in topological deep learning, such as adapting to domains like hypergraphs and enhancing scalability to handle even more complex topological transformations.