  Obtaining continuous representations of structural data such as directed
acyclic graphs (DAGs) has gained attention in machine learning and artificial
intelligence. However, embedding complex DAGs in which both ancestors and
descendants of nodes are exponentially increasing is difficult. Tackling in
this problem, we develop Disk Embeddings, which is a framework for embedding
DAGs into quasi-metric spaces. Existing state-of-the-art methods, Order
Embeddings and Hyperbolic Entailment Cones, are instances of Disk Embedding in
Euclidean space and spheres respectively. Furthermore, we propose a novel
method Hyperbolic Disk Embeddings to handle exponential growth of relations.
The results of our experiments show that our Disk Embedding models outperform
existing methods especially in complex DAGs other than trees.
