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 \href{https://github.com/spherepaircc/SpherePairCC/tree/main}{our repository}.

Clustering is widely used in unsupervised learning method that deals with unlabeled data. Deep clustering has become a popular study area that relates clustering with Deep Neural Network (DNN) architecture. Deep clustering method downsamples high dimensional data, which may also relate clustering loss. Deep clustering is also introduced in semi-supervised learning (SSL). Most SSL methods depend on pairwise constraint information, which is a matrix containing knowledge if data pairs can be in the same cluster or not. This paper introduces a novel embedding system named AutoEmbedder, that downsamples higher dimensional data to clusterable embedding points. To the best of our knowledge, this is the first research endeavor that relates to traditional classifier DNN architecture with a pairwise loss reduction technique. The training process is semi-supervised and uses Siamese network architecture to compute pairwise constraint loss in the feature learning phase. The AutoEmbedder outperforms most of the existing DNN based semi-supervised methods tested on famous datasets.