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

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.

ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain

This paper describes the 2nd edition of the ICML Topological Deep Learning Challenge that was hosted within the ICML 2024 ELLIS Workshop on Geometry-grounded Representation Learning and Generative Modeling (GRaM). The challenge focused on the problem of representing data in different discrete topological domains in order to bridge the gap between Topological Deep Learning (TDL) and other types of structured datasets (e.g. point clouds, graphs). Specifically, participants were asked to design and implement topological liftings, i.e. mappings between different data structures and topological domains --like hypergraphs, or simplicial/cell/combinatorial complexes. The challenge received 52 submissions satisfying all the requirements. This paper introduces the main scope of the challenge, and summarizes the main results and findings.

TopoBenchmarkX: A Framework for Benchmarking Topological Deep Learning

This work introduces TopoBenchmarkX, a modular open-source library designed to standardize benchmarking and accelerate research in Topological Deep Learning (TDL). TopoBenchmarkX maps the TDL pipeline into a sequence of independent and modular components for data loading and processing, as well as model training, optimization, and evaluation. This modular organization provides flexibility for modifications and facilitates the adaptation and optimization of various TDL pipelines. A key feature of TopoBenchmarkX is that it allows for the transformation and lifting between topological domains. This enables, for example, to obtain richer data representations and more fine-grained analyses by mapping the topology and features of a graph to higher-order topological domains such as simplicial and cell complexes. The range of applicability of TopoBenchmarkX is demonstrated by benchmarking several TDL architectures for various tasks and datasets.