浅谈人脸识别中的loss 损失函数

我会带着你远行 2023-06-04 02:52 12阅读 0赞

# 浅谈人脸识别中的loss 损失函数 #

2019-04-17 17:57:33 [liguiyuan112][] 阅读数 641更多

分类专栏： [AI][] [人脸识别][Link 1]

本文链接： [https://blog.csdn.net/u012505617/article/details/89355690][https_blog.csdn.net_u012505617_article_details_89355690]

在人脸识别中，算法的提高主要体现在损失函数的设计上，损失函数会对整个网络的优化有着导向性的作用。我们看到许多常用的损失函数，从传统的softmax loss到cosface, arcface 都有这一定的提高，这篇文章自己就来整理下这几个算法。

无论是SphereFace、CosineFace还是ArcFace的损失函数，都是基于Softmax loss来进行修改的。

<table> 
 <tbody> 
  <tr> 
   <td>Base line</td> 
   <td>Softmax loss</td> 
  </tr> 
  <tr> 
   <td>各种延伸的算法</td> 
   <td>Triplet loss, center loss</td> 
  </tr> 
  <tr> 
   <td>最新算法</td> 
   <td>A-Softmax Loss(SphereFace),&nbsp; Cosine Margin Loss, Angular Margin Loss, Arcface</td> 
  </tr> 
 </tbody> 
</table>

## 1. Softmax loss ##

![\\large L\_1 = -\\frac\{1\}\{m\}\{\\sum\\limits\_\{i=1\}^m\}\\log\\left(\\frac\{e^\{W^T\_\{y\_i\}x\_i+b\_\{y\_i\}\}\}\{ \{\\sum\\limits\_\{j=1\}^n\}e^\{W^T\_jx\_i+b\_j\} \}\\right)][large L_1 _ -_frac_1_m_sum_limits_i_1_m_log_left_frac_e_W_T_y_i_x_i_b_y_i_ _sum_limits_j_1_n_e_W_T_jx_i_b_j_ _right]

这就是softmax loss函数，![\{W^T\_\{j\}x\_i+b\_\{j\}\}][W_T_j_x_i_b_j]表示全连接层的输出。在计算Loss下降的过程中，我们让![\{W^T\_\{j\}x\_i+b\_\{j\}\}][W_T_j_x_i_b_j] 的比重变大，从而使得log() 括号内的数更变大来更接近1，就会 log(1) = 0，整个loss就会下降。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3UwMTI1MDU2MTc_size_16_color_FFFFFF_t_70][]

这种方式只考虑了能否正确分类，却没有考虑类间距离。所以提出了center loss 损失函数。([paper][])

## 2. Center loss ##

![\\large L\_C = -\\frac\{1\}\{2\}\{\\sum\\limits\_\{i=1\}^m\}\{||x\_i-c\_\{y\_i\}||\}^2][large L_C _ -_frac_1_2_sum_limits_i_1_m_x_i-c_y_i_2]

![\\large \\Delta\{c\_j\}=\\frac\{\{\\sum\\limits\_\{i=1\}^m\}\{\\delta\{(y\_i=j)\}\\cdot\{(c\_j-x\_i)\}\}\}\{1+\{\\sum\\limits\_\{i=1\}^m\}\{\\delta\{(y\_i=j)\}\}\}][large _Delta_c_j_frac_sum_limits_i_1_m_delta_y_i_j_cdot_c_j-x_i_1_sum_limits_i_1_m_delta_y_i_j]

center loss 考虑到不仅仅是分类要对，而且要求类间有一定的距离。上面的公式中![\\large c\_\{y\_i\}][large c_y_i]表示某一类的中心，![\\large x\_i][large x_i]表示每个人脸的特征值。作者在softmax loss的基础上加入了![\\large L\_C][large L_C]，同时使用参数![\\large \\lambda][large _lambda]来控制类内距离，整体的损失函数如下：

![\\large L\_2=L\_S+L\_C= -\\frac\{1\}\{m\}\{\\sum\\limits\_\{i=1\}^m\}\\log\\left(\\frac\{e^\{W^T\_\{y\_i\}x\_i+b\_\{y\_i\}\}\}\{ \{\\sum\\limits\_\{j=1\}^n\}e^\{W^T\_jx\_i+b\_j\} \}\\right)+\\frac\{\\lambda\}\{2\}\{\\sum\\limits\_\{i=1\}^m\}\{||x\_i-c\_\{y\_i\}||\}^2][large L_2_L_S_L_C_ -_frac_1_m_sum_limits_i_1_m_log_left_frac_e_W_T_y_i_x_i_b_y_i_ _sum_limits_j_1_n_e_W_T_jx_i_b_j_ _right_frac_lambda_2_sum_limits_i_1_m_x_i-c_y_i_2]

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3UwMTI1MDU2MTc_size_16_color_FFFFFF_t_70 1][]

# #

## 3. Triplet Loss ##

## ![2019031115593590.png][] ##

三元组损失函数，三元组由Anchor， Negative， Positive这三个组成。从上图可以看到，一开始Anchor离Positive比较远，我们想让Anchor和Positive尽量的靠近（同类距离），Anchor和Negative尽量的远离（类间距离）。

![\\large L\_3 = \{\\sum\\limits\_\{i\}^N\}\{\\left \[ ||f(x\_i^a) - f(x\_i^p)||^2\_2 - ||f(x\_i^a)-f(x\_i^n)||\_2^2 \\right + \\alpha \]\}][large L_3 _ _sum_limits_i_N_left _ _f_x_i_a_ - f_x_i_p_2_2 - _f_x_i_a_-f_x_i_n_2_2 _right _ _alpha]

表达式左边为同类距离 ，右边为不同的类之间的距离。使用梯度下降法优化的过程就是让类内距离不断下降，类间距离不断提升，这样损失函数才能不断地缩小。

上面的几个算法都是比较传统老旧的，下面说一下比较新的算法。

--------------------

## 4. L-softmax ##

前面Softmax loss函数没有考虑类间距离，Center loss函数可以使类内变得紧凑，但没有类间可分，而Triplet loss函数比较耗时，就产生了一下新的算法。

L-softmax函数开始就做了比较精细的改动，从softmax 函数log里面的![\\large e^\{W^T\_\{y\_i\}x\_i+b\_\{y\_i\}][large e_W_T_y_i_x_i_b_y_i]转化到![\\large e^\{||W\_\{yi\}|| ||x\_i||\\psi\{(\\theta\_\{y\_i\})\}\}][large e_W_yi_ _x_i_psi_theta_y_i]。L-softmax函数不仅希望类间距离拉的更大，还能够把类内距离压缩的更紧凑。

![\\LARGE L\_4 = \\frac\{1\}\{N\}\\sum\_\{i=1\}^N L\_i = \\frac\{1\}\{N\}\\sum\_\{i=1\}^N -log(\\frac\{e^\{f\_y\_i\}\}\{\\sum\_\{j\}e^\{f\_i\}\})][LARGE L_4 _ _frac_1_N_sum_i_1_N L_i _ _frac_1_N_sum_i_1_N -log_frac_e_f_y_i_sum_j_e_f_i]

![\\LARGE L\_i = -log(\\frac\{e^\{||W\_\{yi\}|| ||x\_i||\\psi\{(\\theta\_\{y\_i\})\}\}\} \{e^\{||W\_\{yi\}|| ||x\_i||\\psi\{(\\theta\_\{y\_i\})\}\} + \\sum\_\{ j\\neq y\_i\}\{e^\{||W\_j|| ||x\_i||cos(\\theta\_j)\}\}\})][LARGE L_i _ -log_frac_e_W_yi_ _x_i_psi_theta_y_i_ _e_W_yi_ _x_i_psi_theta_y_i_ _ _sum_ j_neq y_i_e_W_j_ _x_i_cos_theta_j]

把其中的cosθ改成了cos(mθ)，

![\\large \\psi(\\theta) = \\left\\\{\\begin\{matrix\} \\cos (m\\theta ), 0\\leqslant \\theta \\leqslant \\frac\{\\pi \}\{m\}& & \\\\ D(\\theta), \\frac\{\\pi\}\{m\}\\leqslant \\theta \\leqslant \\pi & & \\end\{matrix\}\\right.][large _psi_theta_ _ _left_begin_matrix_ _cos _m_theta _ 0_leqslant _theta _leqslant _frac_pi _m_ _ _ D_theta_ _frac_pi_m_leqslant _theta _leqslant _pi _ _ _end_matrix_right.]

m倍θ起到了增加 margin 的效果，让类内距离更加紧凑，同时类间距离变大。m越大类间距离就越大，因为在(0, π)区间cos函数单调递减，m越大 cos(mθ)趋向于0。

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3UwMTI1MDU2MTc_size_16_color_FFFFFF_t_70 2][]

# #

## 5. SphereFace(A-Softmax) ##

A-softmax 是在 L-softmax 函数上做了一个很小的修改，A-softmax 在考虑 margin时添加两个限制条件：将权重W归一化 ![||W|| = 1][W_ _ 1]，b = 0。这使得模型的预测仅取决于 W 和 X 之间的角度。

![\\LARGE L\_5 = -\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}log( \\frac\{e^\{||x\_i||\\cos(m\\theta\_\{y\_i\})\}\} \{e^\{||x\_i||\\cos(m\\theta\_\{y\_i\})\} + \\sum\_\{j \\neq y\_i\}\{e^\{||x\_i||cos(\\theta\_j)\}\}\})][LARGE L_5 _ -_frac_1_N_sum_i_1_N_log_ _frac_e_x_i_cos_m_theta_y_i_ _e_x_i_cos_m_theta_y_i_ _ _sum_j _neq y_i_e_x_i_cos_theta_j]

## 6. CosFace ##

cosface的loss函数如下：

![\\LARGE L\_6 = -\\frac\{1\}\{N\} \\sum\_\{i=1\}^\{N\} log( \\frac\{e^\{s(cos(\\theta\_\{yi\})-m)\}\}\{e^\{s(cos(\\theta\_\{yi\})-m)\}+ \\sum\_\{j=1, j\\neq y\_i\}^k e^\{scos \\theta\_j\}\})][LARGE L_6 _ -_frac_1_N_ _sum_i_1_N_ log_ _frac_e_s_cos_theta_yi_-m_e_s_cos_theta_yi_-m_ _sum_j_1_ j_neq y_i_k e_scos _theta_j]

上式中，s为超球面的半径，m为margin。

## 7. ArcFace ##

对比arcface和cosface这两个函数，发现arcface是直接在角度空间中最大化分类界限，而cosface是在余弦空间中最大化分类界限，这样修改是因为角度距离比余弦距离在对角度的影响更加直接。

![\\LARGE L\_7= -\\frac\{1\}\{N\} \\sum\_\{i=1\}^\{N\} log(\\frac\{e^\{s(cos(\\theta\_\{yi\}+m))\}\}\{e^\{s(cos(\\theta\_\{yi\}+m))\}+\\sum\_\{j=1,j\\neq y\_i\}^k e^\{scos\\theta\_j\}\})][LARGE L_7_ -_frac_1_N_ _sum_i_1_N_ log_frac_e_s_cos_theta_yi_m_e_s_cos_theta_yi_m_sum_j_1_j_neq y_i_k e_scos_theta_j]

分类的决策边界如下：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3UwMTI1MDU2MTc_size_16_color_FFFFFF_t_70 3][]

arcface算法流程如下：

![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3UwMTI1MDU2MTc_size_16_color_FFFFFF_t_70 4][]

转载于:https://www.cnblogs.com/think90/p/11619347.html

[liguiyuan112]: https://me.csdn.net/u012505617
[AI]: https://blog.csdn.net/u012505617/article/category/7593791
[Link 1]: https://blog.csdn.net/u012505617/article/category/8746104
[CC 4.0 BY-SA]: http://creativecommons.org/licenses/by-sa/4.0/
[https_blog.csdn.net_u012505617_article_details_89355690]: https://blog.csdn.net/u012505617/article/details/89355690
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[paper]: http://ydwen.github.io/papers/WenECCV16.pdf
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[2019031115593590.png]: /images/20230601/4dbac6cfbfc14ad6b17d785720e3d159.png
[large L_3 _ _sum_limits_i_N_left _ _f_x_i_a_ - f_x_i_p_2_2 - _f_x_i_a_-f_x_i_n_2_2 _right _ _alpha]: https://private.codecogs.com/gif.latex?%5Clarge%20L_3%20%3D%20%7B%5Csum%5Climits_%7Bi%7D%5EN%7D%7B%5Cleft%20%5B%20%7C%7Cf%28x_i%5Ea%29%20-%20f%28x_i%5Ep%29%7C%7C%5E2_2%20-%20%7C%7Cf%28x_i%5Ea%29-f%28x_i%5En%29%7C%7C_2%5E2%20%5Cright%20&plus;%20%5Calpha%20%5D%7D
[large e_W_T_y_i_x_i_b_y_i]: https://private.codecogs.com/gif.latex?%5Clarge%20e%5E%7BW%5ET_%7By_i%7Dx_i&plus;b_%7By_i%7D
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[LARGE L_4 _ _frac_1_N_sum_i_1_N L_i _ _frac_1_N_sum_i_1_N -log_frac_e_f_y_i_sum_j_e_f_i]: https://private.codecogs.com/gif.latex?%5CLARGE%20L_4%20%3D%20%5Cfrac%7B1%7D%7BN%7D%5Csum_%7Bi%3D1%7D%5EN%20L_i%20%3D%20%5Cfrac%7B1%7D%7BN%7D%5Csum_%7Bi%3D1%7D%5EN%20-log%28%5Cfrac%7Be%5E%7Bf_y_i%7D%7D%7B%5Csum_%7Bj%7De%5E%7Bf_i%7D%7D%29
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[W_ _ 1]: https://private.codecogs.com/gif.latex?%7C%7CW%7C%7C%20%3D%201
[LARGE L_5 _ -_frac_1_N_sum_i_1_N_log_ _frac_e_x_i_cos_m_theta_y_i_ _e_x_i_cos_m_theta_y_i_ _ _sum_j _neq y_i_e_x_i_cos_theta_j]: https://private.codecogs.com/gif.latex?%5CLARGE%20L_5%20%3D%20-%5Cfrac%7B1%7D%7BN%7D%5Csum_%7Bi%3D1%7D%5E%7BN%7Dlog%28%20%5Cfrac%7Be%5E%7B%7C%7Cx_i%7C%7C%5Ccos%28m%5Ctheta_%7By_i%7D%29%7D%7D%20%7Be%5E%7B%7C%7Cx_i%7C%7C%5Ccos%28m%5Ctheta_%7By_i%7D%29%7D%20&plus;%20%5Csum_%7Bj%20%5Cneq%20y_i%7D%7Be%5E%7B%7C%7Cx_i%7C%7Ccos%28%5Ctheta_j%29%7D%7D%7D%29
[LARGE L_6 _ -_frac_1_N_ _sum_i_1_N_ log_ _frac_e_s_cos_theta_yi_-m_e_s_cos_theta_yi_-m_ _sum_j_1_ j_neq y_i_k e_scos _theta_j]: https://private.codecogs.com/gif.latex?%5CLARGE%20L_6%20%3D%20-%5Cfrac%7B1%7D%7BN%7D%20%5Csum_%7Bi%3D1%7D%5E%7BN%7D%20log%28%20%5Cfrac%7Be%5E%7Bs%28cos%28%5Ctheta_%7Byi%7D%29-m%29%7D%7D%7Be%5E%7Bs%28cos%28%5Ctheta_%7Byi%7D%29-m%29%7D&plus;%20%5Csum_%7Bj%3D1%2C%20j%5Cneq%20y_i%7D%5Ek%20e%5E%7Bscos%20%5Ctheta_j%7D%7D%29
[LARGE L_7_ -_frac_1_N_ _sum_i_1_N_ log_frac_e_s_cos_theta_yi_m_e_s_cos_theta_yi_m_sum_j_1_j_neq y_i_k e_scos_theta_j]: https://private.codecogs.com/gif.latex?%5CLARGE%20L_7%3D%20-%5Cfrac%7B1%7D%7BN%7D%20%5Csum_%7Bi%3D1%7D%5E%7BN%7D%20log%28%5Cfrac%7Be%5E%7Bs%28cos%28%5Ctheta_%7Byi%7D&plus;m%29%29%7D%7D%7Be%5E%7Bs%28cos%28%5Ctheta_%7Byi%7D&plus;m%29%29%7D&plus;%5Csum_%7Bj%3D1%2Cj%5Cneq%20y_i%7D%5Ek%20e%5E%7Bscos%5Ctheta_j%7D%7D%29
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