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Mapping the Circumplex of Affect: Geometric Analysis of Emotion Representations via Hyperspherical Contrastive Learning

Paper · 2026 · Yusuke Yamauchi and Akiko Aizawa

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.

 Fields, topics and concepts

📚 Shelves 5The field of science this work belongs to (OpenAlex fields).Computer Science · 18,614 worksArtificial Intelligence · 17,965 worksComputer Vision and Pattern Recognition · 3,486 worksPsychology · 1,649 worksExperimental and Cognitive Psychology · 930 works

🧭 Topics 1Research subjects and languages the work is about.Sentiment & emotion · 1,035 works

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. DOI 10.18653/v1/2026.acl-long.772

@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}
}

References

62 references, 12 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

  1. Allen Institute for AI. 2025. Olmo-3-7b-instruct model card. https://huggingface.co/allenai/ Olmo-3-7B-Instruct. Accessed: 2025-12-27. ↗
  2. Andy Arditi, Oscar Obeso, Aaquib Syed, Daniel Paleka, Nina Rimsky, Wes Gurnee, and Neel Nanda. 2024. Refusal in language models is mediated by a single direction. ArXiv, abs/2406.11717.
  3. Yauhen Babakhin, Radek Osmulski, Ronay Ak, Gabriel Moreira, Mengyao Xu, Benedikt Schifferer, Bo Liu, and Even Oldridge. 2025. Llama-embed-nemotron-8b: A universal text embedding model for multilingual and cross-lingual tasks. Preprint, arXiv:2511.07025. — [arXiv:2511.07025](https://arxiv.org/abs/2511.07025) find
  4. Leonard Bereska and Stratis Gavves. 2024. Mechanistic interpretability for AI safety - a review. Transactions on Machine Learning Research. Survey Certification, Expert Certification.
  5. Trenton Bricken, Adly Templeton, Joshua Batson, Brian Chen, Adam Jermyn, Tom Conerly, Nick Turner, Cem Anil, Carson Denison, Amanda Askell, Robert Lasenby, Yifan Wu, Shauna Kravec, Nicholas Schiefer, Tim Maxwell, Nicholas Joseph, Zac Hatfield-Dodds, Alex Tamkin, Karina Nguyen, and 6 others. 2023. To… find
  6. Hongjun Choi, Anirudh Som, and Pavan Turaga. 2020. Amc-loss: Angular margin contrastive loss for improved explainability in image classification. Preprint, arXiv:2004.09805. — [arXiv:2004.09805](https://arxiv.org/abs/2004.09805) find
  7. Enric Junqué de Fortuny. 2025. The super emotion dataset. arXiv preprint arXiv:2505.15348. — [arXiv:2505.15348](https://arxiv.org/abs/2505.15348) find
  8. Jan de Leeuw. 2005. Modern multidimensional scaling: Theory and applications (second edition). Journal of Statistical Software, 14. find
  9. Google DeepMind. 2025. gemma3-27b-it model card. https://huggingface.co/google/ gemma-3-27b-it. Accessed: 2025-12-17. ↗
  10. Dorottya Demszky, Dana Movshovitz-Attias, Jeongwoo Ko, Alan Cowen, Gaurav Nemade, and Sujith Ravi. 2020. Goemotions: A dataset of fine-grained emotions. arXiv preprint arXiv:2005.00547. find
  11. Jiankang Deng, Jia Guo, Jing Yang, Niannan Xue, Irene Kotsia, and Stefanos Zafeiriou. 2022. Arcface: Additive angular margin loss for deep face recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):5962–5979. find
  12. Inderjit S Dhillon and Dharmendra S Modha. 2001. Concept decompositions for large sparse text data using clustering. Machine learning, 42(1):143–175. find
  13. Paul Ekman. 1992. An argument for basic emotions. Cognition & emotion, 6(3-4):169–200.
  14. Nelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec, Zac Hatfield-Dodds, Robert Lasenby, Dawn Drain, Carol Chen, Roger Grosse, Sam McCandlish, Jared Kaplan, Dario Amodei, Martin Wattenberg, and Christopher Olah. 2022. Toy models of superposition. Preprint, a… find
  15. Joshua Engels, Eric J Michaud, Isaac Liao, Wes Gurnee, and Max Tegmark. 2025. Not all language model features are one-dimensionally linear. In The Thirteenth International Conference on Learning Representations. find
  16. Kawin Ethayarajh. 2019. How contextual are contextualized word representations? Comparing the geometry of BERT, ELMo, and GPT-2 embeddings. In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Process… find
  17. Patrick Feeney and Michael C Hughes. 2023. Sincere: Supervised information noise-contrastive estimation revisited. arXiv preprint arXiv:2309.14277. — [arXiv:2309.14277](https://arxiv.org/abs/2309.14277) find
  18. Michael Freenor and Lauren Alvarez. 2025. Steering embedding models with geometric rotation: Mapping semantic relationships across languages and models. Preprint, arXiv:2510.09790. — [arXiv:2510.09790](https://arxiv.org/abs/2510.09790) find
  19. Florian Graf, Christoph Hofer, Marc Niethammer, and Roland Kwitt. 2021. Dissecting supervised contrastive learning. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pages 3821–3830. PMLR. find
  20. Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, Amy Yang, Angela Fan, Anirudh Goyal, Anthony Hartshorn, Aobo Yang, Archi Mitra, Archie Sravankumar, Artem Korenev, Arthur Hinsvark, and 542 o… — The Llama 3 Herd of Models
  21. Yuting Guo and Jinho D Choi. 2021. Enhancing cognitive models of emotions with representation learning. In Proceedings of the Workshop on Cognitive Modeling and Computational Linguistics, pages 141–148. find
  22. Keito Inoshita and Rushia Harada. 2025. Persona-based synthetic data generation using multi-stage conditioning with large language models for emotion recognition. arXiv preprint arXiv:2507.13380. — [arXiv:2507.13380](https://arxiv.org/abs/2507.13380) find
  23. Prannay Khosla, Piotr Teterwak, Chen Wang, Aaron Sarna, Yonglong Tian, Phillip Isola, Aaron Maschinot, Ce Liu, and Dilip Krishnan. 2020. Supervised contrastive learning. Advances in neural information processing systems, 33:18661–18673. find
  24. Sean Lee, Aamir Shakir, Darius Koenig, and Julius Lipp. 2024. Open source strikes bread - new fluffy embedding model. find
  25. Kenneth Li, Oam Patel, Fernanda Viégas, Hanspeter Pfister, and Martin Wattenberg. 2023. Inference-time intervention: Eliciting truthful answers from a language model. Advances in Neural Information Processing Systems, 36:41451–41530.
  26. Ziming Liu, Ouail Kitouni, Niklas S Nolte, Eric Michaud, Max Tegmark, and Mike Williams. 2022. Towards understanding grokking: An effective theory of representation learning. Advances in Neural Information Processing Systems, 35:34651–34663. find
  27. Ilya Loshchilov, Cheng-Ping Hsieh, Simeng Sun, and Boris Ginsburg. 2025. nGPT: Normalized transformer with representation learning on the hypersphere. In The Thirteenth International Conference on Learning Representations. find
  28. Vardan Papyan, X. Y. Han, and David L. Donoho. 2020. Prevalence of neural collapse during the terminal phase of deep learning training. Proceedings of the National Academy of Sciences, 117(40):24652– 24663. find
  29. J MacQueen. 1967. Multivariate observations. In Proceedings ofthe 5th Berkeley Symposium on Mathematical Statisticsand Probability, volume 1, pages 281–297. find
  30. Core Francisco Park, Andrew Lee, Ekdeep Singh Lubana, Yongyi Yang, Maya Okawa, Kento Nishi, Martin Wattenberg, and Hidenori Tanaka. 2025. ICLR: In-context learning of representations. In The Thirteenth International Conference on Learning Representations. find
  31. Albert Mehrabian and James A. Russell. 1974. An Approach to Environmental Psychology. The MIT Press, Cambridge, MA. find
  32. Yu Meng, Jiaxin Huang, Guangyuan Wang, Chao Zhang, Honglei Zhuang, Lance Kaplan, and Jiawei Han. 2019. Spherical text embedding. Advances in neural information processing systems, 32. find
  33. Meta AI. 2024. Llama-3.3-70b-instruct model card. https://www.llama.com/docs/ model-cards-and-prompt-formats/llama3_3/. Accessed: 2025-12-27. ↗
  34. Microsoft. 2025. Phi-4-mini-instruct model card. https://huggingface.co/microsoft/ Phi-4-mini-instruct. Accessed: 2025-12-27. ↗
  35. Mistral AI. 2025. Ministral-3-14b model card. https://huggingface.co/mistralai/ Ministral-3-14B-Base-2512. Accessed: 2025-12-27. ↗
  36. Niklas Muennighoff, Nouamane Tazi, Loic Magne, and Nils Reimers. 2023. MTEB: Massive text embedding benchmark. In Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics, pages 2014–2037, Dubrovnik, Croatia. Association for Computational Linguistic…
  37. Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt. 2023. Progress measures for grokking via mechanistic interpretability. In The Eleventh International Conference on Learning Representations.
  38. OLMo Team, Allyson Ettinger, David Heineman, Amanda Bertsch, Bailey Kuehl, Dirk Groeneveld, Hamish Ivison, and 1 others. 2025. Olmo 3 technical report. Technical report, Allen Institute for AI. Accessed: 2025-12-17. find
  39. OpenAI. 2025. Gpt-5 nano model overview. https://openai.com/ja-JP/index/gpt-5-system-card/. Accessed: 2025-12-27. ↗
  40. Juri Opitz, Lucas Moeller, Andrianos Michail, Sebastian Padó, and Simon Clematide. 2025. Interpretable text embeddings and text similarity explanation: A survey. In Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing, pages 22314–22330, Suzhou, China. Association f… find
  41. Kiho Park, Yo Joong Choe, and Victor Veitch. 2024. The linear representation hypothesis and the geometry of large language models. In Proceedings of the 41st International Conference on Machine Learning, volume 235 of Proceedings of Machine Learning Research, pages 39643–39666. PMLR.
  42. Robert Plutchik. 1980. Emotion: A Psychoevolutionary Synthesis. Harper & Row, New York, USA. find
  43. Qwen. 2025. Qwen3-30b-a3b-instruct-2507 model card. https://huggingface.co/Qwen/ Qwen3-30B-A3B-Instruct-2507. Accessed: 2025-12-27. ↗
  44. Hannah Rashkin, Eric Michael Smith, Margaret Li, and Y-Lan Boureau. 2019. Towards empathetic open-domain conversation models: A new benchmark and dataset. In Proceedings of the 57th annual meeting of the association for computational linguistics, pages 5370–5381. find
  45. Luis Rei and Dunja Mladenić. 2023. Detecting fine-grained emotions in literature. Applied Sciences, 13(13). find
  46. Benjamin Reichman, Adar Avsian, and Larry Heck. 2025. Emotions where art thou: Understanding and characterizing the emotional latent space of large language models. arXiv preprint arXiv:2510.22042. — [arXiv:2510.22042](https://arxiv.org/abs/2510.22042) find
  47. Andrew Rosenberg and Julia Hirschberg. 2007. V-measure: A conditional entropy-based external cluster evaluation measure. In Proceedings of the 2007 joint conference on empirical methods in natural language processing and computational natural language learning (EMNLP-CoNLL), pages 410–420.
  48. James A. Russell. 1980. A circumplex model of affect. Journal of Personality and Social Psychology, 39(6):1161–1178. find
  49. Shalom H Schwartz. 2012. An overview of the Schwartz theory of basic values. Online Readings in Psychology and Culture, 2(1):11. find
  50. Phillip Shaver, Judith Schwartz, Donald Kirson, and Cary O’connor. 1987. Emotion knowledge: further exploration of a prototype approach. Journal of personality and social psychology, 52(6):1061. find
  51. Chris Van Pelt and Alex Sorokin. 2012. Designing a scalable crowdsourcing platform. In Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, pages 765–766. find
  52. Liang Wang, Nan Yang, Xiaolong Huang, Linjun Yang, Rangan Majumder, and Furu Wei. 2024. Multilingual e5 text embeddings: A technical report. arXiv preprint arXiv:2402.05672.
  53. Xiangyu Wang and Chengqing Zong. 2025. Learning emotion category representation to detect emotion relations across languages. IEEE Transactions on Pattern Analysis and Machine Intelligence, 47(6):4752– 4767. find
  54. Jerry S Wiggins. 1979. A psychological taxonomy of trait-descriptive terms: The interpersonal domain. Journal of Personality and Social Psychology, 37(3):395–412. find
  55. Jingyuan Yang, Jie Li, Leida Li, Xiumei Wang, and Xinbo Gao. 2021. A circular-structured representation for visual emotion distribution learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4237–4246. find
  56. Michelle Yik, James A Russell, and James H Steiger. 2011. A 12-point circumplex structure of core affect. Emotion, 11(4):705. find
  57. Yanzhao Zhang, Mingxin Li, Dingkun Long, Xin Zhang, Huan Lin, Baosong Yang, Pengjun Xie, An Yang, Dayiheng Liu, Junyang Lin, Fei Huang, and Jingren Zhou. 2025. Qwen3 embedding: Advancing text embedding and reranking through foundation models. arXiv preprint arXiv:2506.05176.
  58. Bo Zhao, Maya Okawa, Eric J. Bigelow, Rose Yu, Tomer Ullman, Ekdeep Singh Lubana, and Hidenori Tanaka. 2025. Emergence of hierarchical emotion organization in large language models. Preprint, arXiv:2507.10599. — [arXiv:2507.10599](https://arxiv.org/abs/2507.10599) find
  59. Chenxi Zhao, Jinglei Shi, Liqiang Nie, and Jufeng Yang. 2024a. To err like human: Affective bias-inspired measures for visual emotion recognition evaluation. Advances in Neural Information Processing Systems, 37:134747–134769. find
  60. Haiyan Zhao, Hanjie Chen, Fan Yang, Ninghao Liu, Huiqi Deng, Hengyi Cai, Shuaiqiang Wang, Dawei Yin, and Mengnan Du. 2024b. Explainability for large language models: A survey. ACM Transactions on Intelligent Systems and Technology, 15(2):1–38.
  61. Sicheng Zhao, Hongxun Yao, Yue Gao, Rongrong Ji, Wenlong Xie, Xiaolei Jiang, and Tat-Seng Chua. 2016. Predicting personalized emotion perceptions of social images. In Proceedings of the 24th ACM international conference on Multimedia, pages 1385– 1394. find
  62. Haojie Zhuang, Wei Emma Zhang, Jian Yang, Weitong Chen, and Quan Z Sheng. 2024. Not all negatives are equally negative: Soft contrastive learning for unsupervised sentence representations. In Proceedings of the 33rd ACM International Conference on Information and Knowledge Management, pages 3591–360… find

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