On the Importance of Separation and Labelling on the Hypersphere

Good representations strive for diversity among dissimilar features, often by mapping latent representations onto the hypersphere and enforcing separation. At the same time, the latent space structure should align with data semantics. While prior work propagates the need for separation, it is difficult to systematically analyse and determine the importance that feature cluster separation and semantic alignment have on downstream performance. In this paper, we address this gap in the understanding of hyperspherical clustering methods and provide new insights into which properties promote good performance. Firstly, we introduce a principled analysis framework which disentangles the importance of cluster separation and labelling on performance. Secondly, we give a full characterisation of the optimal cluster separation, both through theoretical analysis and practical schemes that achieve near-optimal separation. The results show that even though cluster separation contributes to performance, especially in low dimensions, performance gains obtained by matching the cluster labelling to the input data structure are more significant, and aligning cluster labelling with the underlying data structure can compensate for poor separation.