Marco Montagna

Guillermo Bernárdez, Lev Telyatnikov, Marco Montagna et al.

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.

Marco Montagna, Simone Scardapane, Lev Telyatnikov

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.

Lev Telyatnikov, Guillermo Bernardez, Marco Montagna et al.

This work introduces TopoBench, an open-source library designed to standardize benchmarking and accelerate research in topological deep learning (TDL). TopoBench decomposes TDL into a sequence of independent modules for data generation, loading, transforming 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 TopoBench is its support for transformations and lifting across topological domains. Mapping the topology and features of a graph to higher-order topological domains, such as simplicial and cell complexes, enables richer data representations and more fine-grained analyses. The applicability of TopoBench is demonstrated by benchmarking several TDL architectures across diverse tasks and datasets.

Martin Carrasco, Guillermo Bernardez, Marco Montagna et al.

While Graph Neural Networks (GNNs) have proven highly effective at modeling relational data, pairwise connections cannot fully capture multi-way relationships naturally present in complex real-world systems. In response to this, Topological Deep Learning (TDL) leverages more general combinatorial representations -- such as simplicial or cellular complexes -- to accommodate higher-order interactions. Existing TDL methods often extend GNNs through Higher-Order Message Passing (HOMP), but face critical \emph{scalability challenges} due to \textit{(i)} a combinatorial explosion of message-passing routes, and \textit{(ii)} significant complexity overhead from the propagation mechanism. This work presents HOPSE (Higher-Order Positional and Structural Encoder), an alternative method to solve tasks involving higher-order interactions \emph{without message passing}. Instead, HOPSE breaks \emph{arbitrary higher-order domains} into their neighborhood relationships using a Hasse graph decomposition. This method shows that decoupling the representation learning of neighborhood topology from that of attributes results in lower computational complexity, casting doubt on the need for HOMP. The experiments on molecular graph tasks and topological benchmarks show that HOPSE matches performance on traditional TDL datasets and outperforms HOMP methods on topological tasks, achieving up to $7\times$ speedups over HOMP-based models, opening a new path for scalable TDL.