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：$W_{1}、W_{2}$W1​、W2​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 :

$(W_{1} −W_{2})x+b_{1} −b_{2}=0$(W1​−W2​)x+b1​−b2​=0

If we define $x$x as a feature vector (特征向量？) and constrain $∥W_{1}∥=∥W_{2}∥=1$∥W1​∥=∥W2​∥=1 and $b_{1} =b_{2} =0$b1​=b2​=0, the boundary:

$\left \| x \right \|\left ( cos(\theta _{1})-cos(\theta_{2}) \right )=0$∥x∥(cos(θ1​)−cos(θ2​))=0,
where $θ_{i}$θi​ is the angle between $W_{i}$Wi​ 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$W、b、x$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$m(m ≥ 1)$m(m≥1)，惩罚因子，控制分开的角度距离，边界变为：

$∥x∥(cos(mθ_{1} )−cos(θ_{2}))=0$∥x∥(cos(mθ1​)−cos(θ2​))=0 , $∥x∥(cos(θ_{1} )− cos(θ_{2} ))=0,$∥x∥(cos(θ1​)−cos(θ2​))=0,

$m$m越大，$mθ_{1}$mθ1​越大，得到更大的角，两类分离越远，如下图：