This paper addresses the deep face recognition (FR) problem under open-set protocol, where the goal is to learn features that have a smaller maximal intra-class distance than a minimal inter-class distance in a suitable metric space. To achieve this, the authors propose the Angular Softmax (A-Sofmax) loss, which enables Convolutional Neural Networks (CNNs) to learn angularly discriminative features. Geometrically, A-Sofmax loss imposes discriminative constraints on a hypersphere manifold, which aligns with the prior that faces also lie on a manifold. The size of the angular margin can be quantitatively adjusted by a parameter \( m \). The authors derive specific values for \( m \) to approximate the ideal feature criterion. Extensive experiments on datasets such as Labeled Face in the Wild (LFW), Youtube Faces (YTF), and MegaFace Challenge demonstrate the superiority of A-Sofmax loss in FR tasks. The code for SphereFace is publicly available.This paper addresses the deep face recognition (FR) problem under open-set protocol, where the goal is to learn features that have a smaller maximal intra-class distance than a minimal inter-class distance in a suitable metric space. To achieve this, the authors propose the Angular Softmax (A-Sofmax) loss, which enables Convolutional Neural Networks (CNNs) to learn angularly discriminative features. Geometrically, A-Sofmax loss imposes discriminative constraints on a hypersphere manifold, which aligns with the prior that faces also lie on a manifold. The size of the angular margin can be quantitatively adjusted by a parameter \( m \). The authors derive specific values for \( m \) to approximate the ideal feature criterion. Extensive experiments on datasets such as Labeled Face in the Wild (LFW), Youtube Faces (YTF), and MegaFace Challenge demonstrate the superiority of A-Sofmax loss in FR tasks. The code for SphereFace is publicly available.