Hodge-Aware Contrastive Learning
Alexander Möllers Alexander Immer Vincent Fortuin Elvin Isufi
Simplicial complexes prove effective in modeling data with multiway
dependencies, such as data defined along the edges of networks or within other
higher-order structures. Their spectrum can be decomposed into three
interpretable subspaces via the Hodge decomposition, resulting foundational in
numerous applications. We leverage this decomposition to develop a contrastive
self-supervised learning approach for processing simplicial data and generating
embeddings that encapsulate specific spectral information.Specifically, we
encode the pertinent data invariances through simplicial neural networks and
devise augmentations that yield positive contrastive examples with suitable
spectral properties for downstream tasks. Additionally, we reweight the
significance of negative examples in the contrastive loss, considering the
similarity of their Hodge components to the anchor. By encouraging a stronger
separation among less similar instances, we obtain an embedding space that
reflects the spectral properties of the data. The numerical results on two
standard edge flow classification tasks show a superior performance even when
compared to supervised learning techniques. Our findings underscore the
importance of adopting a spectral perspective for contrastive learning with
higher-order data.