# Manifold-Constrained Sentence Embeddings via Triplet Loss: Projecting Semantics onto Spheres, Tori, and M"obius Strips

arXiv: https://arxiv.org/abs/2505.00014 · HTML: https://mlsanity.com/paper/2505.00014/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 5 references that are in the map.
- First posted on arXiv 22 Apr 2025.

## Builds on (most cited first, 5 of 5)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Character-level Convolutional Networks for Text Classification](https://mlsanity.com/paper/1509.01626/) (2015)
- [SimCSE: Simple Contrastive Learning of Sentence Embeddings](https://mlsanity.com/paper/2104.08821/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)

## Abstract (arXiv)

Recent advances in representation learning have emphasized the role of embedding geometry in capturing semantic structure. Traditional sentence embeddings typically reside in unconstrained Euclidean spaces, which may limit their ability to reflect complex relationships in language. In this work, we introduce a novel framework that constrains sentence embeddings to lie on continuous manifolds -- specifically the unit sphere, torus, and M"obius strip -- using triplet loss as the core training objective. By enforcing differential geometric constraints on the output space, our approach encourages the learning of embeddings that are both discriminative and topologically structured. We evaluate our method on benchmark datasets (AG News and MBTI) and compare it to classical baselines including TF-IDF, Word2Vec, and unconstrained Keras-derived embeddings. Our results demonstrate that manifold-constrained embeddings, particularly those projected onto spheres and M"obius strips, significantly outperform traditional approaches in both clustering quality (Silhouette Score) and classification performance (Accuracy). These findings highlight the value of embedding in manifold space -- where topological structure complements semantic separation -- offering a new and mathematically grounded direction for geometric representation learning in NLP.