Shadow Cones: A Generalized Framework for Partial Order Embeddings

TL;DR

The paper tackles efficient encoding of partial orders on datasets with hierarchical structure by embedding entities in hyperbolic space. It introduces shadow cones, a physics-inspired, model-agnostic framework that represents entailment as shadow containment from a light source, yielding four embedding schemes across Poincaré ball and half-space models. The approach unifies umbral and penumbral constructions and defines a differentiable energy and a shadow loss, achieving state-of-the-art performance on several DAG benchmarks. It also explains and overcomes limitations of prior entailment-cone methods, such as the ε-hole, and offers avenues for multi-relational and downstream applications using multiple light sources.

Abstract

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincaré ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincaré ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincaré ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

Figures (14)

• Figure 1: Example of two sets of shadow cone embeddings in the Poincaré ball, and the partial relations it encodes. Marked in black are the encoded partial relations, while in blue are the embeddings, $u,v,w$ and $x,y,z$. Marked in red are the light source ($\mathcal{S}$), and the dotted geodesics representing light rays. Shaded areas represent shadows. The symbol "$\parallel$" denotes negative relations between unrelated, incomparable elements.
• Figure 2: Umbral-half-space embeddings of partial relation $u \preceq v$. Marked in red are light source at infinity ($\mathcal{S}$), directions of light ($e_n$), and geodesic shadow boundaries ($l$). Blue is the object, and green the shadow cones..
• Figure 3: Umbral-Poincaré-ball embeddings of relation $u \preceq v$.
• Figure 4: Penumbral-Poincaré-ball embeddings of relation $u \preceq v$.
• Figure 5: Penumbral-half-space embeddings of relation $u \preceq v$.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Theorem 3.1
• Theorem 3.2
• Remark 1: Hole around the light source
• Theorem 3.3
• Lemma 4.1: Umbral-half-space
• Theorem 4.2: Shortest Distance to Umbral Cones
• proof : Proof of transitivity
• proof : Proof of geodesic convexity for penumbral cones