# Combined Hyperbolic and Euclidean Soft Triple Loss Beyond the Single Space Deep Metric Learning

arXiv: https://arxiv.org/abs/2510.05643 · HTML: https://mlsanity.com/paper/2510.05643/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 34 references that are in the map.
- First posted on arXiv 7 Oct 2025.

## Builds on (most cited first, 23 of 23)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Finetuned Language Models Are Zero-Shot Learners](https://mlsanity.com/paper/2109.01652/) (2021)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [ImageNet-21K Pretraining for the Masses](https://mlsanity.com/paper/2104.10972/) (2021)
- [Deep Metric Learning via Lifted Structured Feature Embedding](https://mlsanity.com/paper/1511.06452/) (2015)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Multi-Similarity Loss with General Pair Weighting for Deep Metric Learning](https://mlsanity.com/paper/1904.06627/) (2019)
- [A Metric Learning Reality Check](https://mlsanity.com/paper/2003.08505/) (2020)
- [No Fuss Distance Metric Learning using Proxies](https://mlsanity.com/paper/1703.07464/) (2017)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Proxy Anchor Loss for Deep Metric Learning](https://mlsanity.com/paper/2003.13911/) (2020)
- [SoftTriple Loss: Deep Metric Learning Without Triplet Sampling](https://mlsanity.com/paper/1909.05235/) (2019)
- [Registration based Few-Shot Anomaly Detection](https://mlsanity.com/paper/2207.07361/) (2022)
- [Rademacher Complexity for Adversarially Robust Generalization](https://mlsanity.com/paper/1810.11914/) (2018)
- [Hyperbolic Vision Transformers: Combining Improvements in Metric Learning](https://mlsanity.com/paper/2203.10833/) (2022)
- [From Trees to Continuous Embeddings and Back: Hyperbolic Hierarchical Clustering](https://mlsanity.com/paper/2010.00402/) (2020)
- [Recall@k Surrogate Loss with Large Batches and Similarity Mixup](https://mlsanity.com/paper/2108.11179/) (2021)
- [HIER: Metric Learning Beyond Class Labels via Hierarchical Regularization](https://mlsanity.com/paper/2212.14258/) (2022)
- [Learning Semantic Proxies from Visual Prompts for Parameter-Efficient Fine-Tuning in Deep Metric Learning](https://mlsanity.com/paper/2402.02340/) (2024)
- [Multi Proxy Anchor Family Loss for Several Types of Gradients](https://mlsanity.com/paper/2110.03997/) (2021)

## Abstract (arXiv)

Deep metric learning (DML) aims to learn a neural network mapping data to an embedding space, which can represent semantic similarity between data points. Hyperbolic space is attractive for DML since it can represent richer structures, such as tree structures. DML in hyperbolic space is based on pair-based loss or unsupervised regularization loss. On the other hand, supervised proxy-based losses in hyperbolic space have not been reported yet due to some issues in applying proxy-based losses in a hyperbolic space. However, proxy-based losses are attractive for large-scale datasets since they have less training complexity. To address these, this paper proposes the Combined Hyperbolic and Euclidean Soft Triple (CHEST) loss. CHEST loss is composed of the proxy-based losses in hyperbolic and Euclidean spaces and the regularization loss based on hyperbolic hierarchical clustering. We find that the combination of hyperbolic and Euclidean spaces improves DML accuracy and learning stability for both spaces. Finally, we evaluate the CHEST loss on four benchmark datasets, achieving a new state-of-the-art performance.