Fotograma de la película "Ivan Vasilievich cambia de profesión"

? , - . — , . . — , , , .

- .

:

• (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), :

$$

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

La escala es nuestro primero de dos hiperparámetros. Arreglarlo para todos los vectores conduce al hecho de que ahora están ubicados en la hiperesfera. En una versión 2D, se ve así (un color - una clase):



Imagen del artículo de ArcFace , un ejemplo de juguete para demostración: cada clase está resaltada con su propio color, cada punto del círculo es una imagen tomada por separado, el vector medio de cada clase está conectado al centro para mayor claridad. Tenga en cuenta que las clases están conectadas "sin espacios".

Reescribamos Softmax Loss (ahora llamado Normalized Softmax Loss, N-Softmax) con estas observaciones en mente:

$$

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

Hemos dividido la suma en el denominador en dos términos para facilitar la explicación. Todas las funciones de pérdida importantes para el reconocimiento facial se basan en 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