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? , - . — , . . — , , , .

- .

:

• (backbone) — , , , — , . InsightFace — Open Source .
• (embedding) — , . 128 512. , 512. : . , , , — . , ( ) 1, — -1 ( 0).
• Embedding- — , , ( ) .
• — , ( ) , — , — . $$$W^TX + b$ , $$$X$ — ( ), $$$W$ — , $$$b$ — . softmax.

, , . — , , . — ( — ), — . .

: (metric learning) .

, . — Triplet Loss:

$$

$$Loss = \sum_{i=0}^N[||f_{i}^a - f_{i}^p||_{2}^2 - [||f_{i}^a - f_{i}^n||_{2}^2 + \alpha],$$

:

• $$$f^a$ — anchor, , ;
• $$$f^p$ — positive, ;
• $$$f^n$ — negative, ;
• $$$\alpha$ — , .

, , $$$\alpha$ - .

, ( — ), , , embedding- .

Triplet Loss , “ ”, , , SotA , . , Triplet Loss (fine-tuning) . , .

, , , , , . :



— , ( 512), — ( , ). — () , . , “ ” . : , , . . , :

: , , , . , . , , Triplet Loss — , , (hard sampling), .

— , , : Softmax Cross Entropy — . Softmax Loss.

Softmax Loss

Softmax, . :

• $$$W$ — ()
• $$$X$ — embedding-,
• $$$b$ — bias,

, ( , embedding- ): $$$z =W^TX + b$ . Softmax( $$$\sigma$ ) :

$$

$$\sigma(z)_j = \frac{e^{z_j}} {\sum_{i=0}^C e^{z_i} }$$

— Cross Entropy, Softmax Loss :

$$

$$L_{Softmax} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{W^T_{y_i}x_i + b_{y_i}}}{\sum_{j=1}^C e^{W_j^Tx_i + b_j}},$$

$$$y_i$ — . , , 42, 42- .

— , .

: $$$W^TX + b$ — : $$$b = 0$ , $$$W^TX$ . , . :

• $$$X$ $$$F\times B$ , $$$B$ — ( ), $$$F$ — ( 512).
• $$$W$ $$$F\times C$ , $$$C$ — .

$$$W^T$ $$$X$ $$$C\times B$ , .

, :

$$

$$cos(\theta) = \frac {dot(u,v)}{||u||||v||}$$

, : — , . , , ( $$$||W_i|| = 1$ ), $$$X$ s (scale), :

$$

$$W^TX = s*cos(\theta)$$

Skala adalah hyperparameter pertama dari dua. Memperbaiki itu untuk semua vektor mengarah pada fakta bahwa mereka sekarang berada di hipersfer. Dalam versi 2D, terlihat seperti ini (satu warna - satu kelas):



Gambar dari artikel ArcFace , contoh mainan untuk demonstrasi: setiap kelas disorot dengan warnanya sendiri, setiap titik pada lingkaran adalah gambar yang diambil secara terpisah, vektor tengah setiap kelas dihubungkan ke tengah untuk kejelasan. Perhatikan bahwa kelas-kelas tersebut terhubung "tanpa celah".

Mari kita tulis ulang Softmax Loss (sekarang disebut Softmax Loss Normalized, N-Softmax) dengan pengamatan berikut ini:

$$

$$L_{N Softmax} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(\theta_{y_i})}}{e^{s*cos(\theta_{y_i})} + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$$

. N-Softmax .

Margin-Based Loss

, , — . softmax loss, . , (decision boundary), ( ). . ? () , . — (decision margin). — margin — , : scale — . 2D ( ArcFace):

— margin, — margin

, .

, margin ( $$$m$ ) :

1. . $$$cos(\theta)$ $$$cos(m*\theta)$ . : Large-Margin Softmax Loss SphereFace. , , Margin-based loss. :

   
   
   
   
   
   
   
   
   
   
   
   

   $$

   $$L_{SphereFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(m*\theta_{y_i})}}{e^{s*cos(m*\theta_{y_i})} + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$$
   
   
   
   
   
   
   

2. margin . $$$cos(\theta)$ $$$cos(\theta) - m$ . : AM-Softmax CosFace , , , . :

   
   
   
   
   
   
   
   
   
   
   
   

   $$

   $$L_{CosFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(\theta_{y_i}) - m}}{e^{s*cos(\theta_{y_i}) - m} + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$$
   
   
   
   
   
   
   

3. margin : $$$cos(\theta)$ → $$$cos(\theta + m)$ . ArcFace. ArcFace :

   
   
   
   
   
   
   
   
   
   
   
   

   $$

   $$L_{CosFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(\theta_{y_i}+m) }}{e^{s*cos(\theta_{y_i}+m) } + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$$
   
   
   
   
   
   
   

4. , ArcFace AirFace. Margin , ArcFace, , ( $$$arccos(cos(\theta)) = \theta$ ). , , ( — ), , $$$\theta$ , $$$(\pi - 2\theta)/\pi$ , :

   
   
   
   
   
   
   
   
   
   
   
   

   $$

   $$L_{AirFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*(\pi - 2(\theta_{y_i}+m))/\pi}}{e^{s*(\pi - 2(\theta_{y_i}+m))/\pi} + \sum_{j=1, j\neq y_i}^C e^{s*(\pi - 2\theta_j)/\pi}}$$
   
   
   
   
   
   
   

margin — , — ( ArcFace).

Margin & Scale

- , , — scale (s) margin (m), . , AM Softmax $$$s = 30$ , $$$m = 0.35$ , ArcFace — $$$s = 64$ , a $$$m = 0.5$ , CosFace (, , AM Softmax) $$$s = 64$ , a $$$m = 0.35$ . , , “ ”, .

— AdaCos, — scale margin . :

• Margin scale — , .
• scale margin, margin scale.
• scale .
• — scale
  
  2 20 , , Y — , , X — . , , , :
  
  
  
  

  scale ( , , margin 0), — margin scale=30. , scale, “” , margin X. , — scale margin, - , ? AdaCos scale ( ). , : $$$s = \sqrt{2}*\ln{(C-1)}$ , $$$C$ — . s [10, 25], .
  
  
  
  AdaCos — scale. , scale , . , , .
  
  
  
  

  
  
  
  
  
  
  
  

  margin , , , ? . X , Y — ( N-softmax $$$cos(\theta)$ ):
  
  
  
  

  
  
  
  
  

  : CosFace N-softmax , ArcFace — . SphereFace, $$$\frac {\pi}{margin}$ , $$$\frac{\pi}{3}$ . ArcFace — target logit, , , . , , ( $$$\frac{5\pi}{6}$ ). , (, ) , :
  
  

  
  
  
  
  

  # cosine - cos(theta)
  # phi - cos(theta + m)
  # th - cos(math.pi - m)
  # mm - sin(math.pi - m) * m
  if easy_margin:
    phi = torch.where(cosine > 0, phi, cosine)  
  else:
    phi = torch.where(cosine > th, phi, cosine - mm)
        
        
  
          
          
          
        
  
      
          
          
          
        
        
  
          
          
          
        
  
      
      

  
  
  
  
  

  easy_margin , :

  
  
  
  
  
  
  
  
  
  
  

  Easy margin , $$$\frac{\pi}{2}$ N-Softmax ( $$$cos(\theta)$ ), not easy margin . , , , $$$\frac{\pi}{2}$ , “”, , , .

  
  
  
  
  
  
  

  , . — " " ArcFace. , ( 4 8 ):

  
  
  
  
  
  
  

  Loss	LFW	MegaFace, Rank1 @ $$$10^6$	MegaFace, Tar @ Far $$$10^{-6}$
  AM-Softmax/CosFace	99.33	0.9833	0.9841
  ArcFace	99.83	0.9836	0.9848
  SphereFace	99.42	0.9743	0.9766

  
  
  

  ( 3 ), LFW, — - :

  
  
  
  
  
  
  

  Loss	Resnet50-MSC	MobileNet-MSC	Resnet50-Casia	MobileNet-Casia
  AM-Softmax/CosFace	99.3	97.65	99.34	98.46
  ArcFace	99.15	98.43	99.35	99.01
  SphereFace	99.02	96.86	99.1	97.83

  
  
  

  , ArcFace SotA .

  
  
  
  
  
  
  
  
  
  

  . (margin) , . AM Softmax ( ) ArcFace ( ). , , , AirFace.

  
  
  
  
  
  
  
  
  
  

  :

  
  
  

  SphereFace https://arxiv.org/abs/1704.08063
  
  AM Softmax https://arxiv.org/abs/1801.05599
  
  CosFace https://arxiv.org/abs/1801.09414
  
  ArcFace https://arxiv.org/abs/1801.07698
  
  AirFace https://arxiv.org/abs/1907.12256

  
  
  
  
  
  
  

  :

  
  
  

  Deep Face Recognition: A Survey https://arxiv.org/abs/1804.06655
  
  A Performance Evaluation of Loss Functions for Deep Face Recognition https://arxiv.org/abs/1901.05903