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Hierarchical Geometry of Cognitive States in Transformer Embedding Spaces

Sophie Zhao

TL;DR

This paper investigates whether transformer sentence embeddings encode a graded, hierarchical organization that aligns with human-defined cognitive attributes. It analyzes fixed embeddings from three models on a 480-sentence dataset annotated with seven cognitive tiers and continuous energy scores, using linear and shallow nonlinear probes, UMAP visualizations, and nonparametric permutation tests. The results show robust decodability of both continuous and discrete annotations, with permutation tests confirming significance and confusion patterns revealing adjacent-tier locality, while TF-IDF baselines underperform. The findings suggest that embedding geometry reflects a structured, hierarchical organization beyond surface lexical cues, with implications for interpretability, safety analysis, and alignment in language models, though limitations such as annotation subjectivity and dataset scope are acknowledged.

Abstract

Recent work has shown that transformer-based language models learn rich geometric structure in their embedding spaces, yet the presence of higher-level cognitive organization within these representations remains underexplored. In this work, we investigate whether sentence embeddings encode a graded, hierarchical structure aligned with human-interpretable cognitive or psychological attributes. We construct a dataset of 480 natural-language sentences annotated with continuous ordinal energy scores and discrete tier labels spanning seven ordered cognitive categories. Using fixed sentence embeddings from multiple transformer models, we evaluate the recoverability of these annotations via linear and shallow nonlinear probes. Across models, both continuous scores and tier labels are reliably decodable, with shallow nonlinear probes providing consistent performance gains over linear probes. Lexical TF-IDF baselines perform substantially worse, indicating that the observed structure is not attributable to surface word statistics alone. Nonparametric permutation tests further confirm that probe performance exceeds chance under label-randomization nulls. Qualitative analyses using UMAP visualizations and confusion matrices reveal smooth low-to-high gradients and predominantly adjacent-tier confusions in embedding space. Taken together, these results provide evidence that transformer embedding spaces exhibit a hierarchical geometric organization aligned with human-defined cognitive attributes, while remaining agnostic to claims of internal awareness or phenomenology.

Hierarchical Geometry of Cognitive States in Transformer Embedding Spaces

TL;DR

This paper investigates whether transformer sentence embeddings encode a graded, hierarchical organization that aligns with human-defined cognitive attributes. It analyzes fixed embeddings from three models on a 480-sentence dataset annotated with seven cognitive tiers and continuous energy scores, using linear and shallow nonlinear probes, UMAP visualizations, and nonparametric permutation tests. The results show robust decodability of both continuous and discrete annotations, with permutation tests confirming significance and confusion patterns revealing adjacent-tier locality, while TF-IDF baselines underperform. The findings suggest that embedding geometry reflects a structured, hierarchical organization beyond surface lexical cues, with implications for interpretability, safety analysis, and alignment in language models, though limitations such as annotation subjectivity and dataset scope are acknowledged.

Abstract

Recent work has shown that transformer-based language models learn rich geometric structure in their embedding spaces, yet the presence of higher-level cognitive organization within these representations remains underexplored. In this work, we investigate whether sentence embeddings encode a graded, hierarchical structure aligned with human-interpretable cognitive or psychological attributes. We construct a dataset of 480 natural-language sentences annotated with continuous ordinal energy scores and discrete tier labels spanning seven ordered cognitive categories. Using fixed sentence embeddings from multiple transformer models, we evaluate the recoverability of these annotations via linear and shallow nonlinear probes. Across models, both continuous scores and tier labels are reliably decodable, with shallow nonlinear probes providing consistent performance gains over linear probes. Lexical TF-IDF baselines perform substantially worse, indicating that the observed structure is not attributable to surface word statistics alone. Nonparametric permutation tests further confirm that probe performance exceeds chance under label-randomization nulls. Qualitative analyses using UMAP visualizations and confusion matrices reveal smooth low-to-high gradients and predominantly adjacent-tier confusions in embedding space. Taken together, these results provide evidence that transformer embedding spaces exhibit a hierarchical geometric organization aligned with human-defined cognitive attributes, while remaining agnostic to claims of internal awareness or phenomenology.
Paper Structure (34 sections, 1 equation, 5 figures, 4 tables)

This paper contains 34 sections, 1 equation, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Dataset
  3. Cognitive Tier Taxonomy
  4. Sentence Construction and Annotation
  5. Continuous Energy Scores
  6. Dataset Intent
  7. Methods
  8. Sentence Embeddings
  9. Probing Analysis
  10. Regression Probes for Continuous Energy Scores
  11. Linear probe (Ridge regression).
  12. Nonlinear probe (MLP regressor).
  13. Classification Probes for Cognitive Tiers
  14. Confusion Matrix Analysis
  15. Stability Across Splits
  16. ...and 19 more sections

Figures (5)

  • Figure 1: Confusion matrix for tier classification using BAAI-bge-large-en-v1.5 embeddings.
  • Figure 2: 3D UMAP visualization of sentence embeddings colored by energy scores.
  • Figure 3: Permutation test results. Vertical lines indicate observed probe performance.
  • Figure 4: Confusion matrices for tier classification across three embedding models.
  • Figure 5: 2D UMAP visualization of sentence embeddings colored by continuous energy scores.