sphereface论文

we set the output dimension of FC1 layer as 2 . 譬如FC1之后输出：

feat=(256, 2) 256是batch_size，256张图片，每个图片提取两个特征

横纵坐标是两个feature，得到两个类，一共256个点。 直角坐标系映射到极坐标：$W_1、W_2$向量长度不相等。 Features learned by the original softmax loss can not be classified simply via angles ——增加角度margin。 The decision boundary in softmax loss is :

$(W_1-W_2)x+b_1-b_2=0$

If we define $x$ as a feature vector (特征向量？) and constrain $\Vert W_1\Vert =\Vert W_2\Vert =1$ and $b_1=b_2=0$, the boundary:

$\Vert x\Vert \left(cos({\theta }_1)-cos({\theta }_2)\right)=0$, where ${\theta }_i$ is the angle between $W_i$ and x

到此为止，boundary只与角度θ有关，修改softmax loss直接优化角度，让 CNNs 提取到角度可分性更高的feature。现在我们来看加了$W、b、x$的约束后的图像： 通过修改的softmax loss得到的feature。Compared to original softmax loss, the features learned by modified softmax loss are angularly distributed. 作者觉得两类分的还不够开，于是引入一个整数$m(m\ge 1)$，惩罚因子，控制分开的角度距离，边界变为：

$\Vert x\Vert (cos(m{\theta }_1)-cos({\theta }_2))=0$ , $\Vert x\Vert (cos({\theta }_1)-cos({\theta }_2))=0,$

$m$越大，$m{\theta }_1$越大，得到更大的角，两类分离越远，如下图：