genealogical networks • Asymmetric / transitive relation of nodes → Partially ordered set (poset) DAG Embedding • Embedding nodes into continuous poset so that transitive relation is preserved. DAG Embedding Models
Embedding: General framework for embedding DAGs Our Contributions • Novel Hyperbolic Disk Embedding • Experiments • Theorems: Existing methods are special cases of DE (+ extra restrictions)
extensible to general metric spaces Further generalizations • Metric spaces → Quasi-metric spaces • (Closed) Disks → Formal disks Existing methods can be understood as special cases of DEs X y x
x y x Closed disk Open disk Dx ⊆ Dy D∘ x ⊆ D∘ y (x, r) (x, rx ) ⊑ (y, ry ) d(x, y) + rx − ry ≤ 0 Formal Disk Radii can be negative → Reversibility of Disk Embeddings
radius: reversibility & translational symmetry • Avoid gradient vanishing on loss functions Applicable for various (quasi-)metric spaces. Positive sample Negative sample Case of Negative Radius Loss Functions Ours Existing methods
etc. (Minkolov 2013) Order Embeddings (Vendrov 2016) Sphere Several approaches Hyperbolic Entailment Cones (Ganea 2018) Hyperbolic space Poincaré Embedding (Nickel 2017) Hyperbolic Disk Embedding