Convolution as inner product has been the founding basis of convolutional
neural networks (CNNs) and the key to end-to-end visual representation
learning. Benefiting from deeper architectures, recent CNNs have demonstrated
increasingly strong representation abilities. Despite such improvement, the
increased depth and larger parameter space have also led to challenges in
properly training a network. In light of such challenges, we propose
hyperspherical convolution (SphereConv), a novel learning framework that gives
angular representations on hyperspheres. We introduce SphereNet, deep
hyperspherical convolution networks that are distinct from conventional inner
product based convolutional networks. In particular, SphereNet adopts
SphereConv as its basic convolution operator and is supervised by generalized
angular softmax loss - a natural loss formulation under SphereConv. We show
that SphereNet can effectively encode discriminative representation and
alleviate training difficulty, leading to easier optimization, faster
convergence and comparable (even better) classification accuracy over
convolutional counterparts. We also provide some theoretical insights for the
advantages of learning on hyperspheres. In addition, we introduce the learnable
SphereConv, i.e., a natural improvement over prefixed SphereConv, and
SphereNorm, i.e., hyperspherical learning as a normalization method.
Experiments have verified our conclusions.

本文提出了基于超球面的学习框架 ——SphereConv，以及其在超球面上的深度学习网络 SphereNet，使用广义角 softmax 损失函数实现监督。实验证明，SphereNet 能够有效编码具有辨别力的特征表示并缓解训练困难，使优化更容易，收敛更快，并且分类准确度（甚至更好）与基于传统卷积的网络相当。

深度超球学习

Deep Hyperspherical Learning

This paper addresses deep face recognition (FR) problem under open-set
protocol, where ideal face features are expected to have smaller maximal
intra-class distance than minimal inter-class distance under a suitably chosen
metric space. However, few existing algorithms can effectively achieve this
criterion. To this end, we propose the angular softmax (A-Softmax) loss that
enables convolutional neural networks (CNNs) to learn angularly discriminative
features. Geometrically, A-Softmax loss can be viewed as imposing
discriminative constraints on a hypersphere manifold, which intrinsically
matches the prior that faces also lie on a manifold. Moreover, the size of
angular margin can be quantitatively adjusted by a parameter $m$. We further
derive specific $m$ to approximate the ideal feature criterion. Extensive
analysis and experiments on Labeled Face in the Wild (LFW), Youtube Faces (YTF)
and MegaFace Challenge show the superiority of A-Softmax loss in FR tasks. The
code has also been made publicly available.

本文提出了一种叫做 A-Softmax 的损失函数来训练卷积神经网络，以学习面部图像特征，使得最大内类距离小于最小类间距离，从而解决开放集（open-set）协议下的人脸识别问题。与之前的算法相比，该方法在 LFW、YTF 和 MegaFace Challenge 数据集上取得了更好的结果。