## 一、从前辈说起

Softmax函数，或称归一化指数函数[1]，是逻辑函数的一种推广。 它能将一个含任意实数的K维向量      “压缩”到另一个K维实向量     中，使得每一个元素的范围都在

## 二、SphereFace

pytorch代码实现：

# SphereFace
class SphereProduct(nn.Module):
r"""Implement of large margin cosine distance: :
Args:
in_features: size of each input sample
out_features: size of each output sample
m: margin
cos(m*theta)
"""
def __init__(self, in_features, out_features, m=4):
super(SphereProduct, self).__init__()
self.in_features = in_features
self.out_features = out_features
self.m = m
self.base = 1000.0
self.gamma = 0.12
self.power = 1
self.LambdaMin = 5.0
self.iter = 0
self.weight = Parameter(torch.FloatTensor(out_features, in_features))
nn.init.xavier_uniform(self.weight)
# duplication formula
# 将x\in[-1,1]范围的重复index次映射到y\[-1,1]上
self.mlambda = [
lambda x: x ** 0,
lambda x: x ** 1,
lambda x: 2 * x ** 2 - 1,
lambda x: 4 * x ** 3 - 3 * x,
lambda x: 8 * x ** 4 - 8 * x ** 2 + 1,
lambda x: 16 * x ** 5 - 20 * x ** 3 + 5 * x
]
"""
执行以下代码直观了解mlambda
import matplotlib.pyplot as  plt
mlambda = [
lambda x: x ** 0,
lambda x: x ** 1,
lambda x: 2 * x ** 2 - 1,
lambda x: 4 * x ** 3 - 3 * x,
lambda x: 8 * x ** 4 - 8 * x ** 2 + 1,
lambda x: 16 * x ** 5 - 20 * x ** 3 + 5 * x
]
x = [0.01 * i for i in range(-100, 101)]
print(x)
for f in mlambda:
plt.plot(x,[f(i) for i in x])
plt.show()
"""
def forward(self, input, label):
# lambda = max(lambda_min,base*(1+gamma*iteration)^(-power))
self.iter += 1
self.lamb = max(self.LambdaMin, self.base * (1 + self.gamma * self.iter) ** (-1 * self.power))
# --------------------------- cos(theta) & phi(theta) ---------------------------
cos_theta = F.linear(F.normalize(input), F.normalize(self.weight))
cos_theta = cos_theta.clamp(-1, 1)
cos_m_theta = self.mlambda[self.m](cos_theta)
theta = cos_theta.data.acos()
k = (self.m * theta / 3.14159265).floor()
phi_theta = ((-1.0) ** k) * cos_m_theta - 2 * k
NormOfFeature = torch.norm(input, 2, 1)
# --------------------------- convert label to one-hot ---------------------------
one_hot = torch.zeros(cos_theta.size())
one_hot = one_hot.cuda() if cos_theta.is_cuda else one_hot
one_hot.scatter_(1, label.view(-1, 1), 1)
# --------------------------- Calculate output ---------------------------
output = (one_hot * (phi_theta - cos_theta) / (1 + self.lamb)) + cos_theta
output *= NormOfFeature.view(-1, 1)
return output
def __repr__(self):
return self.__class__.__name__ + '(' \
+ 'in_features=' + str(self.in_features) \
+ ', out_features=' + str(self.out_features) \
+ ', m=' + str(self.m) + ')'

## 三、CosFace

# CosFace
r"""Implement of large margin cosine distance: :
Args:
in_features: size of each input sample
out_features: size of each output sample
s: norm of input feature
m: margin
cos(theta) - m
"""
def __init__(self, in_features, out_features, s=30.0, m=0.40):
self.in_features = in_features
self.out_features = out_features
self.s = s
self.m = m
self.weight = Parameter(torch.FloatTensor(out_features, in_features))
nn.init.xavier_uniform_(self.weight)
def forward(self, input, label):
# --------------------------- cos(theta) & phi(theta) ---------------------------
cosine = F.linear(F.normalize(input), F.normalize(self.weight))
phi = cosine - self.m
# --------------------------- convert label to one-hot ---------------------------
one_hot = torch.zeros(cosine.size(), device='cuda')
# one_hot = one_hot.cuda() if cosine.is_cuda else one_hot
one_hot.scatter_(1, label.view(-1, 1).long(), 1)
# -------------torch.where(out_i = {x_i if condition_i else y_i) -------------
output = (one_hot * phi) + ((1.0 - one_hot) * cosine)
# you can use torch.where if your torch.__version__ is 0.4
output *= self.s
# print(output)
return output
def __repr__(self):
return self.__class__.__name__ + '(' \
+ 'in_features=' + str(self.in_features) \
+ ', out_features=' + str(self.out_features) \
+ ', s=' + str(self.s) \
+ ', m=' + str(self.m) + ')'

## 四、ArcFace

# ArcFace
class ArcMarginProduct(nn.Module):
r"""Implement of large margin arc distance: :
Args:
in_features: size of each input sample
out_features: size of each output sample
s: norm of input feature
m: margin
cos(theta + m)
"""
def __init__(self, in_features, out_features, s=30.0, m=0.50, easy_margin=False):
super(ArcMarginProduct, self).__init__()
self.in_features = in_features
self.out_features = out_features
self.s = s
self.m = m
# Parameter 的用途：
# 将一个不可训练的类型Tensor转换成可以训练的类型parameter
# 并将这个parameter绑定到这个module里面
# net.parameter()中就有这个绑定的parameter，所以在参数优化的时候可以进行优化的
# https://www.jianshu.com/p/d8b77cc02410
# 初始化权重
self.weight = Parameter(torch.FloatTensor(out_features, in_features))
nn.init.xavier_uniform_(self.weight)
self.easy_margin = easy_margin
self.cos_m = math.cos(m)
self.sin_m = math.sin(m)
self.th = math.cos(math.pi - m)
self.mm = math.sin(math.pi - m) * m
def forward(self, input, label):
# --------------------------- cos(theta) & phi(theta) ---------------------------
# torch.nn.functional.linear(input, weight, bias=None)
# y=x*W^T+b
cosine = F.linear(F.normalize(input), F.normalize(self.weight))
sine = torch.sqrt(1.0 - torch.pow(cosine, 2))
# cos(a+b)=cos(a)*cos(b)-size(a)*sin(b)
phi = cosine * self.cos_m - sine * self.sin_m
if self.easy_margin:
# torch.where(condition, x, y) → Tensor
# condition (ByteTensor) – When True (nonzero), yield x, otherwise yield y
# x (Tensor) – values selected at indices where condition is True
# y (Tensor) – values selected at indices where condition is False
# return:
# A tensor of shape equal to the broadcasted shape of condition, x, y
# cosine>0 means two class is similar, thus use the phi which make it
phi = torch.where(cosine > 0, phi, cosine)
else:
phi = torch.where(cosine > self.th, phi, cosine - self.mm)
# --------------------------- convert label to one-hot ---------------------------
# one_hot = torch.zeros(cosine.size(), requires_grad=True, device='cuda')
# 将cos(\theta + m)更新到tensor相应的位置中
one_hot = torch.zeros(cosine.size(), device='cuda')
# scatter_(dim, index, src)
one_hot.scatter_(1, label.view(-1, 1).long(), 1)
# -------------torch.where(out_i = {x_i if condition_i else y_i) -------------
output = (one_hot * phi) + ((1.0 - one_hot) * cosine)
# you can use torch.where if your torch.__version__ is 0.4
output *= self.s
# print(output)
return output