Title: Neural Collapse by Design: Learning Class Prototypes on the Hypersphere
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
Keywords: Supervised Learning, Neural Collapse, Supervised Contrastive Learning

Abstract: Neural Collapse (NC) describes the global optimum of supervised learning, yet standard cross-entropy (CE) training rarely attains its geometry in practice. This is due to unconstrained radial degrees of freedom: cross-entropy is invariant to joint rescaling of features and weights, leaving radial directions underconstrained thus preventing convergence to a unique geometry. We show that constraining optimization to the unit hypersphere removes this degeneracy and reveals a unifying view of normalized softmax classifier learning (CL) and supervised contrastive learning (SCL) as the same prototype-contrast principle: both optimize angular similarity to class prototypes, using explicit learned weights for normalized softmax and implicit class means for SCL. Despite this shared foundation, existing objectives suffer from small effective negative sets and interference between positive and negative terms, which slows convergence to NC. We address these issues with two objectives: NTCE, which contrasts class prototypes against all batch instances to expand the negative set from K classes to M samples; and NONL, which normalizes only over negatives to decouple intra-class alignment from inter-class repulsion. Theoretically, we prove that SCL already learns an optimal prototype classifier under NC, eliminating the need for post-hoc typically hours-scale linear probing. Empirically, across four benchmarks including ImageNet-1K, our methods surpass CE accuracy, reach $\ge$95\% on NC metrics, and match NC structure with substantially fewer iterations. Moreover, SCL with class-mean prototypes matches linear-probing accuracy while requiring no training. These results reframe supervised learning as prototype-based classification on the hypersphere, closing the theory–practice gap while simplifying training and accelerating convergence.