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The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures

Andres Anabalon, Hugo Garces, Julio Oliva, Jose Cifuentes

TL;DR

The work addresses representing hierarchical information with a geometry-based embedding in three-dimensional Minkowski spacetime, using only local causal relations between token pairs. The authors introduce an iterative causality-enforcement algorithm that adjusts time coordinates to realize a perfect, low-dimensional embedding of WordNet hierarchies, including ambigious cases, by exploiting causal structure rather than distances alone. Their results demonstrate near-perfect embeddings for large WordNet subsets (82,115 nouns) and the mammal subtree, with 3D embeddings and near-null geodesics suggesting conformal-like properties and deep links to relativistic geometry. The findings imply that hierarchical meaning can be faithfully represented geometrically, offering robust, interpretable concept spaces for NLP systems and potential pathways to improved semantic retrieval and generation constrained by causal structure.

Abstract

We show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. The correlation of data ends up purely encoded in the causal structure. Our model relies solely on oriented token pairs -- local hierarchical signals -- with no access to global symbolic structure. We apply our method to the corpus of \textit{WordNet}. We provide a perfect embedding of the mammal sub-tree including ambiguities (more than one hierarchy per node) in such a way that the hierarchical structures get completely codified in the geometry and exactly reproduce the ground-truth. We extend this to a perfect embedding of the maximal unambiguous subset of the \textit{WordNet} with 82{,}115 noun tokens and a single hierarchy per token. We introduce a novel retrieval mechanism in which causality, not distance, governs hierarchical access. Our results seem to indicate that all discrete data has a perfect geometrical representation that is three-dimensional. The resulting embeddings are nearly conformally invariant, indicating deep connections with general relativity and field theory. These results suggest that concepts, categories, and their interrelations, namely hierarchical meaning itself, is geometric.

The Geometry of Meaning: Perfect Spacetime Representations of Hierarchical Structures

TL;DR

The work addresses representing hierarchical information with a geometry-based embedding in three-dimensional Minkowski spacetime, using only local causal relations between token pairs. The authors introduce an iterative causality-enforcement algorithm that adjusts time coordinates to realize a perfect, low-dimensional embedding of WordNet hierarchies, including ambigious cases, by exploiting causal structure rather than distances alone. Their results demonstrate near-perfect embeddings for large WordNet subsets (82,115 nouns) and the mammal subtree, with 3D embeddings and near-null geodesics suggesting conformal-like properties and deep links to relativistic geometry. The findings imply that hierarchical meaning can be faithfully represented geometrically, offering robust, interpretable concept spaces for NLP systems and potential pathways to improved semantic retrieval and generation constrained by causal structure.

Abstract

We show that there is a fast algorithm that embeds hierarchical structures in three-dimensional Minkowski spacetime. The correlation of data ends up purely encoded in the causal structure. Our model relies solely on oriented token pairs -- local hierarchical signals -- with no access to global symbolic structure. We apply our method to the corpus of \textit{WordNet}. We provide a perfect embedding of the mammal sub-tree including ambiguities (more than one hierarchy per node) in such a way that the hierarchical structures get completely codified in the geometry and exactly reproduce the ground-truth. We extend this to a perfect embedding of the maximal unambiguous subset of the \textit{WordNet} with 82{,}115 noun tokens and a single hierarchy per token. We introduce a novel retrieval mechanism in which causality, not distance, governs hierarchical access. Our results seem to indicate that all discrete data has a perfect geometrical representation that is three-dimensional. The resulting embeddings are nearly conformally invariant, indicating deep connections with general relativity and field theory. These results suggest that concepts, categories, and their interrelations, namely hierarchical meaning itself, is geometric.
Paper Structure (5 sections, 2 equations, 3 figures)

This paper contains 5 sections, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Natural Hierarchy
  3. The algorithm
  4. Experiments
  5. Conclusions

Figures (3)

  • Figure 1: Events with their light cones. There are only two natural hierarchies in this figure $D \rightarrow C \rightarrow A$ and $B \rightarrow A$. Their transitive closure also follows from the causal diagram.
  • Figure 2: Left panel: Perfect embedding of two hierarchies — depth 8 (blue dots) and depth 9 (red dots) — both merging at the last 5 nodes. Past light cones are drawn for reference of the null surface at each token. It can be seen that geodesics connecting tokens are almost null. Right panel: Same embedding after a spatial rotation.
  • Figure 3: The figure shows a perfect embedding of the mammal sub‑tree. Whenever a token has a second hierarchy, that branch is highlighted in blue. The three‑dimensional layout offers a human‑friendly, visualization of the entire hierarchy.