# Educational Cone Model in Embedding Vector Spaces

arXiv: https://arxiv.org/abs/2512.04227 · HTML: https://mlsanity.com/paper/2512.04227/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 13 references that are in the map.
- First posted on arXiv 3 Dec 2025.

## Builds on (most cited first, 6 of 6)

- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [MTEB: Massive Text Embedding Benchmark](https://mlsanity.com/paper/2210.07316/) (2022)
- [BeLLM: Backward Dependency Enhanced Large Language Model for Sentence Embeddings](https://mlsanity.com/paper/2311.05296/) (2023)
- [CEFR-Based Sentence Difficulty Annotation and Assessment](https://mlsanity.com/paper/2210.11766/) (2022)
- [Sub-Sentence Encoder: Contrastive Learning of Propositional Semantic Representations](https://mlsanity.com/paper/2311.04335/) (2023)
- [Linear Cross-Lingual Mapping of Sentence Embeddings](https://mlsanity.com/paper/2305.14256/) (2023)

## Built on by (most cited first, 2 of 2)

- [Accurate and Efficient Statistical Testing for Word Semantic Breadth](https://mlsanity.com/paper/2605.08048/) (2026)
- [Predicting Disagreement with Human Raters in LLM-as-a-Judge Difficulty Assessment without Using Generation-Time Probability Signals](https://mlsanity.com/paper/2605.12422/) (2026)

## Abstract (arXiv)

Human-annotated datasets with explicit difficulty ratings are essential in intelligent educational systems. Although embedding vector spaces are widely used to represent semantic closeness and are promising for analyzing text difficulty, the abundance of embedding methods creates a challenge in selecting the most suitable method. This study proposes the Educational Cone Model, which is a geometric framework based on the assumption that easier texts are less diverse (focusing on fundamental concepts), whereas harder texts are more diverse. This assumption leads to a cone-shaped distribution in the embedding space regardless of the embedding method used. The model frames the evaluation of embeddings as an optimization problem with the aim of detecting structured difficulty-based patterns. By designing specific loss functions, efficient closed-form solutions are derived that avoid costly computation. Empirical tests on real-world datasets validated the model's effectiveness and speed in identifying the embedding spaces that are best aligned with difficulty-annotated educational texts.