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.

翻译：单纯复形在建模具有多路依赖的数据（如网络边缘或高阶结构上定义的数据）方面已被证明有效。通过霍奇分解，其谱可被分解为三个可解释的子空间，这为众多应用奠定了基础。我们利用该分解开发了一种对比自监督学习方法，用于处理单纯复形数据并生成包含特定谱信息的嵌入。具体而言，我们通过单纯复形神经网络对相关数据不变性进行编码，并设计数据增强方法，为下游任务生成具有合适谱属性的正例。此外，我们在对比损失中重新加权负例的重要性，考虑其霍奇分量与锚点的相似性。通过增强不相似实例之间的区分度，我们获得了反映数据谱属性的嵌入空间。在两个标准边流分类任务上的数值结果表明，即使与监督学习方法相比，我们的方法也展现出更优性能。我们的发现强调了在高阶数据对比学习中采用谱视角的重要性。