Compact Geometric Representations of Hierarchies

2026-06-18 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Algorithmic Graph Theory · Depth: Expert, extended

Summary

New research provides theoretical guarantees for compact reachability embeddings, a cornerstone of modern machine learning for hierarchical data. Addressing limitations of prior dual-encoder approaches for deep hierarchies, this work proves that any directed tree admits a reachability embedding in constant dimension 3, independent of its size or depth. This generalizes to graphs with treewidth t, requiring embeddings of dimension O(t log n), where n is the number of nodes. The study also provides matching or near-matching lower bounds, showing dimension Ω(n) is necessary for general DAGs and Ω(t/log(n/t)) for treewidth t. These embeddings are empirically validated on real-world datasets like WordNet, Cora, and Gene Ontology, achieving 100% recall with significantly fewer dimensions (e.g., d=152 for WordNet) compared to prior methods.

Key takeaway

For AI Scientists designing hierarchical retrieval systems, these new compact geometric embeddings offer a significant advantage. You can achieve theoretical guarantees for exact reachability with substantially lower dimensionality, especially for tree-like structures or graphs with few cross-edges. This reduces memory and computational costs compared to prior approaches, enabling more efficient and accurate dense retrieval for complex hierarchical data.

Key insights

Compact geometric embeddings for hierarchical data achieve exact reachability with dimensions dependent on graph structure, outperforming prior methods.

Principles

• Directed trees admit 3-dimensional reachability embeddings.
• Treewidth t graphs require O(t log n) dimensions.
• Each cross-edge in a DAG adds one dimension to forest embeddings.

Method

Embeddings for directed trees use DFS discovery/finishing times. For DAGs, a 3-dimensional forest embedding is augmented with one coordinate per cross-edge. Treewidth-parameterized embeddings are constructed recursively using balanced vertex separators.

In practice

• WordNet recall: 100% at d=152, vs. prior d=512 for 95%.
• Apply to hierarchical retrieval on Cora and Gene Ontology datasets.

Topics

• Reachability Embeddings
• Directed Acyclic Graphs
• Treewidth
• Graph Algorithms
• Hierarchical Retrieval
• Depth-First Search

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.