Groups with complex set intersection relations are a natural way to model a wide array of data, from the formation of social groups to the complex protein interactions which form the basis of biological life. One approach to representing such higher order relationships is as a hypergraph. However, efforts to apply machine learning techniques to hypergraph structured datasets have been limited thus far. In this paper, we address the problem of link prediction in knowledge hypergraphs as well as simple hypergraphs and develop a novel, simple, and effective optimization architecture that addresses both tasks. Additionally, we introduce a novel feature extraction technique using node level clustering and we show how integrating data from node-level labels can improve system performance. Our self-supervised approach achieves significant improvement over state of the art baselines on several hyperedge prediction and knowledge hypergraph completion benchmarks.