---
title: "Mapping the Circumplex of Affect: Geometric Analysis of Emotion"
description: "Psychologists often arrange emotions on a circle defined by valence and arousal, and this picture is commonly used to… 62 references."
authors: ["Yusuke Yamauchi", "Akiko Aizawa"]
year: 2026
venue: "Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)"
source: "https://aclanthology.org/2026.acl-long.772/"
doi: "10.18653/v1/2026.acl-long.772"
licence: "CC BY 4.0 (ACL Anthology)"
licence_url: "https://creativecommons.org/licenses/by/4.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/yamauchi-2026-mapping-circumplex-affect-geometric-analysis/"
---
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# Mapping the Circumplex of Affect: Geometric Analysis of Emotion Representations via Hyperspherical Contrastive Learning

**Authors:** [Yusuke Yamauchi](https://md.research.elamur.ai/free/en/authors/yusuke-yamauchi/) and [Akiko Aizawa](https://md.research.elamur.ai/free/en/authors/akiko-aizawa/) · **Year:** 2026 · **Venue:** [Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)](https://md.research.elamur.ai/free/en/venues/acl/) ([ACL 2026](https://md.research.elamur.ai/free/en/venues/acl/2026/)) · **Publisher:** [Association for Computational Linguistics](https://md.research.elamur.ai/free/en/publishers/association-for-computational-linguistics/) · **Licence:** [CC BY 4.0 (ACL Anthology)](https://creativecommons.org/licenses/by/4.0/)

**Links:** [Publisher's page](https://aclanthology.org/2026.acl-long.772/) · [PDF](https://aclanthology.org/2026.acl-long.772.pdf) · [DOI 10.18653/v1/2026.acl-long.772](https://doi.org/10.18653/v1/2026.acl-long.772) · [OpenAlex](https://openalex.org/W7166809595)

**Citations:** 0 citations (OpenAlex, 2026-10-14)

## Summary
Psychologists often arrange emotions on a circle defined by valence and arousal, and this picture is commonly used to interpret neural representations after training. It has rarely been built into how language models learn, so whether the circle actually suits model embeddings remained unclear. Yamauchi and Aizawa assembled an emotion-labelled text dataset from existing corpora plus synthetic text, covering 12 categories arranged on an Empirical Circumplex Model (ECM). They attached a hyperspherical nGPT projection head, with a standard GPT head as a baseline, to three families of backbones: BERT-like encoders, LLM-based encoders and decoder-only LLMs. Each head was trained with one of three contrastive objectives: SINCERE, SoftCSE and the new CircularCSE. The objectives differ in how strongly they impose a ring-shaped geometry. Ring-constrained embeddings are easier to interpret and hold up under dimensionality reduction or with few labels. They separate classes worse in high dimensions and in fine-grained label sets. This shows a real cost to forcing human-style, low-dimensional structure onto model representations.

## Key points
- **Training losses:** SINCERE, SoftCSE and the proposed CircularCSE, applied directly to normalized sentence embeddings, with no classifier layer or cross-entropy term.
- **Evaluation metrics:** V-Measure measures how well classes are separated. CD-r, the Pearson correlation with Circumplex Distance, measures agreement with the psychological model.
- **Theoretical explanation:** SINCERE places labels as an orthogonal simplex with 90° margins, whereas a 2D ring forces much narrower margins as the number of labels grows. This explains why CircularCSE separates classes less well.
- **Robustness beyond training data:** the V-Measure gap between methods is about 0.1 on the training set but shrinks to roughly 0.01 beyond the training distribution. In the learned space, "Neutral" sits near low-arousal emotions such as calmness and relief, not at the circle's centre.
- **Code:** released as the GitHub repository `yama11235/EmpiricalCircumplexModel`.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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## Cite

Cite as: Yusuke Yamauchi and Akiko Aizawa. 2026. Mapping the Circumplex of Affect: Geometric Analysis of Emotion Representations via Hyperspherical Contrastive Learning. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers).

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@inproceedings{yamauchi2026mapping,
  title = {{Mapping the Circumplex of Affect: Geometric Analysis of Emotion Representations via Hyperspherical Contrastive Learning}},
  author = {Yusuke Yamauchi and Akiko Aizawa},
  booktitle = {Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)},
  year = {2026},
  doi = {10.18653/v1/2026.acl-long.772},
  url = {https://doi.org/10.18653/v1/2026.acl-long.772}
}
```

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*Updated 2 November 2026*