Does a Shared Temperature Imply a Shared Angular Scale in Probabilistic Contrastive Learning?

Abstract

In probabilistic contrastive learning, a shared temperature is commonly interpreted as a shared similarity scale, but this interpretation does not hold for high-dimensional distributional class representations. We study the exact von Mises-Fisher (vMF) probabilistic score used by ProCo when representation dimension and class concentration grow jointly. We prove that the score retains a class-dependent leading angular gain $g_c=A_c/\tau$, where $A_c$ is the mean resultant length. This gain enters Softmax competition, pairwise decision boundaries, and feature gradients. On real CIFAR-LT, ImageNet-LT, and iNaturalist representations, the theory accurately predicts boundary movements and local gradient changes under the full vMF score. Classwise temperature adjustment also changes the cosine-zero intercept and finite-dimensional response. We construct intercept-preserving and Pure Angular controls to separate the leading gain from these accompanying changes. Complete gain equalization yields a shared-scale cosine prototype rule at leading order; a finite-dimensional margin condition guarantees agreement of the two classifiers. Across 16 frozen representation settings, prediction agreement is 98.43-99.99%, with disagreements concentrated at small cosine margins. In controlled contrastive-only training with the training-frequency prior, Pure Angular editing improves both learned representations at all tested CIFAR-10/100 imbalance factors and retains positive changes on ImageNet-LT. Thus vMF concentration not only describes class distributions, but also forms a decision and learning scale in high-dimensional probabilistic contrastive learning.