Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding

TL;DR

This paper introduces Semi-Simplicial Neural Networks (SSNs), a novel topological deep learning model that captures directed higher-order relationships, significantly improving brain activity decoding and outperforming existing models in accuracy.

Contribution

The paper presents SSNs, a new class of TDL models operating on semi-simplicial sets to encode directed higher-order motifs, with a scalable routing extension and proven greater expressiveness.

Findings

SSNs outperform state-of-the-art models in brain activity classification by up to 27%.

Routing-SSNs improve relation selection, enhancing scalability.

SSNs show competitive results on standard node and edge tasks.

Abstract

Graph Neural Networks (GNNs) excel at learning from pairwise interactions but often overlook multi-way and hierarchical relationships. Topological Deep Learning (TDL) addresses this limitation by leveraging combinatorial topological spaces. However, existing TDL models are restricted to undirected settings and fail to capture the higher-order directed patterns prevalent in many complex systems, e.g., brain networks, where such interactions are both abundant and functionally significant. To fill this gap, we introduce Semi-Simplicial Neural Networks (SSNs), a principled class of TDL models that operate on semi-simplicial sets -- combinatorial structures that encode directed higher-order motifs and their directional relationships. To enhance scalability, we propose Routing-SSNs, which dynamically select the most informative relations in a learnable manner. We prove that SSNs are strictly more expressive than standard graph and TDL models. We then introduce a new principled framework for brain dynamics representation learning, grounded in the ability of SSNs to provably recover topological descriptors shown to successfully characterize brain activity. Empirically, SSNs achieve state-of-the-art performance on brain dynamics classification tasks, outperforming the second-best model by up to 27%, and message passing GNNs by up to 50% in accuracy. Our results highlight the potential of principled topological models for learning from structured brain data, establishing a unique real-world case study for TDL. We also test SSNs on standard node classification and edge regression tasks, showing competitive performance. We will make the code and data publicly available.