Shadow Cones: A Generalized Framework for Modeling Partial Orders in Hyperbolic Space

แนวคิดหลัก

The shadow cones framework provides a novel, physically intuitive approach to model partial orders in hyperbolic space. It generalizes existing entailment cone constructions and offers clear advantages in terms of optimization properties and representation power.

บทคัดย่อ

The content introduces the "shadow cones" framework, a novel approach to model partial orders in hyperbolic space. It is inspired by the physics of light and shadows, and generalizes the existing "entailment cones" framework.
The key insights are:

Shadow cones model partial orders as subset relations between shadows cast by light sources and opaque objects in hyperbolic space. This results in a physically intuitive and transitive partial order representation.

The shadow cones framework encompasses a broad class of formulations, including the existing entailment cones as a special case. This generalization offers clear advantages, such as better optimization properties.

The authors propose four specific shadow cone constructions in the Poincaré ball and half-space models, and provide detailed mathematical characterizations of their properties.

Experiments on various graph datasets show that shadow cones consistently and significantly outperform existing entailment cone constructions, demonstrating their effectiveness in modeling partial orders.

Overall, the shadow cones framework provides a novel, powerful, and physically intuitive approach to embed partial orders in hyperbolic space, with clear theoretical and empirical advantages over prior work.

สถิติ

"Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs."
"Volumes in hyperbolic space grow exponentially for large radius, which matches the number of nodes in a tree; in contrast, this volume grows polynomially in Euclidean space."

คำพูด

"Shadow cones can be seen as a generalization of existing approaches that use hyperbolic cones to model partial orders ("entailment cones"), and are agnostic to choice of hyperbolic model."
"Shadow cones possess better optimization properties over constructions limited to the Poincaré ball."

ข้อมูลเชิงลึกที่สำคัญจาก

Shadow Cones

by Tao Yu,Toni ... ที่ arxiv.org 04-08-2024

https://arxiv.org/pdf/2305.15215.pdf

สอบถามเพิ่มเติม

How can the shadow cones framework be extended to model multiple relation types within a single embedding?

The shadow cones framework can be extended to model multiple relation types within a single embedding by incorporating multiple light sources, each casting shadows that represent different types of relations. Each light source can be associated with a specific type of relation, and the shadows cast by these light sources can capture the hierarchical structures corresponding to those relations. By considering the subset relations between these shadows, the framework can effectively encode and differentiate between various relation types within the same embedding space. This approach allows for a more comprehensive representation of complex relational datasets with diverse types of relationships.

What are the limitations of the current shadow cones constructions, and how can they be addressed to further improve the representation power?

One limitation of the current shadow cones constructions, particularly the umbral cones, is the non-convexity of their boundaries due to the hypercycles that define them. This non-convexity can pose challenges in optimization and may limit the representation power of the framework. To address this limitation and improve the representation power, one approach could be to explore alternative boundary definitions that maintain convexity while still capturing the essence of the shadow cones. By refining the boundary characterization to ensure convexity, the framework can provide more stable optimization properties and potentially enhance the accuracy of the hierarchical embeddings.

Can the insights from the shadow cones framework be applied to other geometric spaces beyond hyperbolic space to model hierarchical structures?

Yes, the insights from the shadow cones framework can be applied to other geometric spaces beyond hyperbolic space to model hierarchical structures. The concept of using shadows cast by objects to represent partial orders and hierarchical relations is a general principle that can be adapted to different geometric spaces. By defining subset relations between shadows formed by light sources and objects in alternative geometric spaces, similar hierarchical structures can be encoded. For example, in Euclidean space, the framework could be modified to use shadows cast by planar objects to model hierarchical relationships. By adapting the framework to different geometric spaces, it can offer a versatile and intuitive way to model hierarchical structures across various domains.