Thread by @tolga_birdal

Graphs & geometric deep learning are cool, but what's next? Our latest work leverages topology to model higher order relationships beyond graphs, covering domains like meshes, sets & their mixtures naturally. Learn more: arxiv.org/abs/2206.00606 #TopologicalDeepLearning #TDL

So what's a #HigherOrderRelationship? Essentially, it's a relation between relations. Graphs only consider node-to-node relations (via so called edges). Yet, relations between edges or edges & faces, encode interesting patterns, leading to a full hierarchy of relationships.

At the core of our work, lies a new, unifying data structure, we call 'combinatorial complex' (CC), equipped with set-type relations and their hierarchies. Graphs, simplicial complexes and cell complexes are all special cases of CCs, making CCs useful in summarizing complex data.

We then introduce some of the necessary and elementary building blocks leading to novel Combinatorial Complex Neural Networks (CCNNs) natively operating on CCs. CCNNs utilize higher order message passing protocols to communicate cells of potentially different order (rank). #CCNN

Our promising results are just the beginning of our exploration into high order understanding. CCNNs can be applied to many common forms of 3D representation in computer vision and graphics, and have achieved accurate predictions on par with carefully designed architectures.

Along with our paper, we release three software packages:
NetworkX for topological domains: github.com/pyt-team/TopoN…
Node2Vec type algorithms for topological domains:
github.com/pyt-team/TopoE…
Deep learning on topological domains:
github.com/pyt-team/TopoM…

Our work suggests a new direction asking for all ML frameworks to consider higher order relations as a central component. Happy to be part of this gigantic effort led by @HajijMustafa, and an amazing team including @ghadazamzmi, @theopapamarkou, @ninamiolane and @PaulRosenPHD.