Title: Hypersphere Face Uncertainty Learning. Abstract: An emerging line of research has found that \emph{hyperspherical} spaces better match the underlying geometry of facial images, as evidenced by the state-of-the-art facial recognition methods which benefit empirically from hyperspherical representations. Yet, these approaches rely on deterministic embeddings and hence suffer from the \emph{feature ambiguity dilemma}, whereby ambiguous or noisy images are mapped into poorly learned regions of representation space, leading to inaccuracies~\citep{shi2019probabilistic}. PFE is the first attempt to circumvent this dilemma. However, we theoretically and empirically identify two main failure cases of PFE when it is applied to hyperspherical deterministic embeddings aforementioned. To address these issues, in this paper, we propose a novel framework for face uncertainty learning in hyperspherical space. Mathematically, we extend the \emph{von Mises Fisher} density to its $r$-radius counterpart and derive an optimization objective in closed form. For feature comparison, we also derive a closed-form mutual likelihood score for latents lying on hypersphere. Extensive experimental results on multiple challenging benchmarks confirm our hypothesis and theory, and showcase the superior performance of our framework against prior probabilistic methods and conventional hyperspherical deterministic embeddings both in risk-controlled recognition tasks and in face verification and identification tasks.