# Angular Constraint Embedding via SpherePair Loss for Constrained Clustering

arXiv: https://arxiv.org/abs/2510.06907 · HTML: https://mlsanity.com/paper/2510.06907/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 60 references that are in the map.
- First posted on arXiv 8 Oct 2025.

## Builds on (most cited first, 30 of 33)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [CosFace: Large Margin Cosine Loss for Deep Face Recognition](https://mlsanity.com/paper/1801.09414/) (2018)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Unsupervised Deep Embedding for Clustering Analysis](https://mlsanity.com/paper/1511.06335/) (2015)
- [SphereFace: Deep Hypersphere Embedding for Face Recognition](https://mlsanity.com/paper/1704.08063/) (2017)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [Large-Margin Softmax Loss for Convolutional Neural Networks](https://mlsanity.com/paper/1612.02295/) (2016)
- [Contrastive Clustering](https://mlsanity.com/paper/2009.09687/) (2020)
- [Towards K-means-friendly Spaces: Simultaneous Deep Learning and Clustering](https://mlsanity.com/paper/1610.04794/) (2016)
- [Variational Deep Embedding: An Unsupervised and Generative Approach to Clustering](https://mlsanity.com/paper/1611.05148/) (2016)
- [Targeted Supervised Contrastive Learning for Long-Tailed Recognition](https://mlsanity.com/paper/2111.13998/) (2021)
- [Multi-class Classification without Multi-class Labels](https://mlsanity.com/paper/1901.00544/) (2019)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [Dissecting Supervised Contrastive Learning](https://mlsanity.com/paper/2102.08817/) (2021)
- [Deep Hyperspherical Learning](https://mlsanity.com/paper/1711.03189/) (2017)
- [A Framework for Deep Constrained Clustering -- Algorithms and Advances](https://mlsanity.com/paper/1901.10061/) (2019)
- [Neural network-based clustering using pairwise constraints](https://mlsanity.com/paper/1511.06321/) (2015)
- [Principled approach to the selection of the embedding dimension of networks](https://mlsanity.com/paper/2004.09928/) (2020)
- [A Framework for Deep Constrained Clustering](https://mlsanity.com/paper/2101.02792/) (2021)
- [Deep Conditional Gaussian Mixture Model for Constrained Clustering](https://mlsanity.com/paper/2106.06385/) (2021)
- [Clustering-driven Deep Embedding with Pairwise Constraints](https://mlsanity.com/paper/1803.08457/) (2018)
- [A Classification-Based Approach to Semi-Supervised Clustering with Pairwise Constraints](https://mlsanity.com/paper/2001.06720/) (2020)
- [AutoEmbedder: A semi-supervised DNN embedding system for clustering](https://mlsanity.com/paper/2007.05830/) (2020)
- [Semi-Supervised Constrained Clustering: An In-Depth Overview, Ranked Taxonomy and Future Research Directions](https://mlsanity.com/paper/2303.00522/) (2023)
- [Semi-Supervised Clustering with Neural Networks](https://mlsanity.com/paper/1806.01547/) (2018)
- [Semi-crowdsourced Clustering with Deep Generative Models](https://mlsanity.com/paper/1810.11971/) (2018)
- [Deep Clustering with Incomplete Noisy Pairwise Annotations: A Geometric Regularization Approach](https://mlsanity.com/paper/2305.19391/) (2023)

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

- [Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning](https://mlsanity.com/paper/2605.00265/) (2026)
- [Uncertainty-Aware Probabilistic Constrained Clustering from Entangled Pairwise Supervision](https://mlsanity.com/paper/2608.12027/) (2026)

## Abstract (arXiv)

Constrained clustering integrates domain knowledge through pairwise constraints. However, existing deep constrained clustering (DCC) methods are either limited by anchors inherent in end-to-end modeling or struggle with learning discriminative Euclidean embedding, restricting their scalability and real-world applicability. To avoid their respective pitfalls, we propose a novel angular constraint embedding approach for DCC, termed SpherePair. Using the SpherePair loss with a geometric formulation, our method faithfully encodes pairwise constraints and leads to embeddings that are clustering-friendly in angular space, effectively separating representation learning from clustering. SpherePair preserves pairwise relations without conflict, removes the need to specify the exact number of clusters, generalizes to unseen data, enables rapid inference of the number of clusters, and is supported by rigorous theoretical guarantees. Comparative evaluations with state-of-the-art DCC methods on diverse benchmarks, along with empirical validation of theoretical insights, confirm its superior performance, scalability, and overall real-world effectiveness. Code is available at hrefhttps://github.com/spherepaircc/SpherePairCC/tree/mainour repository.