Compact Geometric Representations of Hierarchies

2026-06-16

AI Summary

This work addresses the problem of constructing low-dimensional geometric embeddings for directed acyclic graphs (DAGs) with ancestor–descendant relationships, aiming to avoid embedding dimensions that scale explosively with the number of nodes or graph depth. By leveraging structural properties—such as treewidth and the number of cross edges—and insights from geometric embedding theory, the authors propose a compact representation whose dimension depends only on these structural parameters rather than the total node count. Key contributions include a proof that any directed tree admits an exact reachability-preserving embedding in three dimensions, and an upper bound of \( O(t \log n) \) dimensions for DAGs of treewidth \( t \), accompanied by a nearly matching lower bound that reveals fundamental limits on dimensionality. Experiments on real-world datasets demonstrate that the method substantially reduces embedding dimension while maintaining high recall, outperforming existing approaches with theoretical guarantees.

Abstract

Computing geometric representations of data is a cornerstone of modern machine learning, typically achieved by training dual encoders which map queries and documents into a shared embedding space. Recent work of You et al. [NeurIPS '25] has extended this approach to hierarchical retrieval, where relevance is determined by the ancestor-descendant relationships in a Directed Acyclic Graph (DAG). While previous work has shown that valid embeddings exist when the number of descendants is small, these bounds degrade significantly for deep hierarchies, requiring dimensions as large as the total number of nodes. In this paper, we investigate compact reachability embeddings for more general graph classes and provide theoretical guarantees for representing hierarchies using embeddings whose dimension depends on structural graph parameters. We prove that for any directed tree, there exists a reachability embedding in constant dimension 3, independent of the tree's size or depth. We generalize this result to graphs characterized by treewidth $t$, constructing embeddings of dimension $O(t \log n)$, where $n$ is the number of nodes. Complementing these upper bounds, we provide matching or near-matching lower bounds, showing that dimension $Ω(n)$ is necessary for general DAGs and $Ω(t/\log(n/t))$ is required for graphs of treewidth $t$. We also obtain upper and lower bounds parameterized by the number of cross-edges in the DAG. We additionally show that our embeddings can be constructed on real world datasets, and that they give much smaller dimensions in high recall regimes compared to prior embeddings with theoretical guarantees.

Problem

• hierarchical retrieval
• reachability embeddings
• directed acyclic graph
• treewidth
• compact representation

Innovation

• reachability embeddings
• hierarchical retrieval
• treewidth
• directed acyclic graphs
• compact representations