Standbild aus dem Film "Ivan Vasilievich wechselt seinen Beruf"

? , - . — , . . — , , , .

- .

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

, , . — , , . — ( — ), — . .

: (metric learning) .

L o s s = \sum_{i = 0}^{N} [| | f_{i}^{a} - f_{i}^{p} | |_{2}^{2} - [| | f_{i}^{a} - f_{i}^{n} | |_{2}^{2} + α],

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

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

, , $\alpha$ - .

, ( — ), , , embedding- .

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

, , , , , . :

Einstufung

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

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

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

Softmax Loss

Softmax, . :

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

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

σ (z)_{j} = \frac{e^{z_{j}}}{\sum_{i = 0}^{C} e^{z_{i}}}

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

— Cross Entropy, Softmax Loss :

L_{S o f t m a x} = - \frac{1}{N} \sum_{i = 1}^{N} l o g \frac{e^{W_{y_{i}}^{T} x_{i} + b_{y_{i}}}}{\sum_{j = 1}^{C} e^{W_{j}^{T} x_{i} + b_{j}}},

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

$y_i$ — . , , 42, 42- .

— , .

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

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

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

, :

c o s (θ) = \frac{d o t (u, v)}{| | u | | | | v | |}

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

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

W^{T} X = s * c o s (θ)

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

Die Skala ist unser erster von zwei Hyperparametern. Das Fixieren für alle Vektoren führt dazu, dass sie sich jetzt auf der Hypersphäre befinden. In der 2D-Version sieht es ungefähr so aus (eine Farbe - eine Klasse):

Bild aus dem ArcFace- Artikel , ein Spielzeugbeispiel zur Demonstration: Jede Klasse wird mit einer eigenen Farbe hervorgehoben, jeder Punkt auf dem Kreis ist ein separat aufgenommenes Bild, der mittlere Vektor jeder Klasse ist aus Gründen der Klarheit mit der Mitte verbunden. Beachten Sie, dass die Klassen "ohne Lücken" verbunden sind.

Schreiben wir den Softmax-Verlust (jetzt als normalisierter Softmax-Verlust, N-Softmax bezeichnet) unter Berücksichtigung dieser Beobachtungen neu:

L_{N S o f t m a x} = - \frac{1}{N} \sum_{i = 1}^{N} l o g \frac{e^{s * c o s (θ_{y_{i}})}}{e^{s * c o s (θ_{y_{i}})} + \sum_{j = 1, j \neq y_{i}}^{C} e^{s * c o s (θ_{j})}}

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

Wir haben die Summe im Nenner zur Vereinfachung der weiteren Erklärung in zwei Begriffe unterteilt. Alle wichtigen Verlustfunktionen für die Gesichtserkennung basieren auf N-Softmax.

Margin-Based Loss

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

— margin, — margin

, .

, margin ( $m$ ) :

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

$L_{S p h e r e F a c e} = - \frac{1}{N} \sum_{i = 1}^{N} l o g \frac{e^{s * c o s (m * θ_{y_{i}})}}{e^{s * c o s (m * θ_{y_{i}})} + \sum_{j = 1, j \neq y_{i}}^{C} e^{s * c o s (θ_{j})}}$
$L_{SphereFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(m*\theta_{y_i})}}{e^{s*cos(m*\theta_{y_i})} + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$
margin . $cos(\theta)$ $cos(\theta) - m$ . : AM-Softmax CosFace , , , . :

$L_{C o s F a c e} = - \frac{1}{N} \sum_{i = 1}^{N} l o g \frac{e^{s * c o s (θ_{y_{i}}) - m}}{e^{s * c o s (θ_{y_{i}}) - m} + \sum_{j = 1, j \neq y_{i}}^{C} e^{s * c o s (θ_{j})}}$
$L_{CosFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(\theta_{y_i}) - m}}{e^{s*cos(\theta_{y_i}) - m} + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$
margin : $cos(\theta)$ → $cos(\theta + m)$ . ArcFace. ArcFace :

$L_{C o s F a c e} = - \frac{1}{N} \sum_{i = 1}^{N} l o g \frac{e^{s * c o s (θ_{y_{i}} + m)}}{e^{s * c o s (θ_{y_{i}} + m)} + \sum_{j = 1, j \neq y_{i}}^{C} e^{s * c o s (θ_{j})}}$
$L_{CosFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*cos(\theta_{y_i}+m) }}{e^{s*cos(\theta_{y_i}+m) } + \sum_{j=1, j\neq y_i}^C e^{s*cos(\theta_j)}}$
, ArcFace AirFace. Margin , ArcFace, , ( $arccos(cos(\theta)) = \theta$ ). , , ( — ), , $\theta$ , $(\pi - 2\theta)/\pi$ , :

$L_{A i r F a c e} = - \frac{1}{N} \sum_{i = 1}^{N} l o g \frac{e^{s * (π - 2 (θ_{y_{i}} + m)) / π}}{e^{s * (π - 2 (θ_{y_{i}} + m)) / π} + \sum_{j = 1, j \neq y_{i}}^{C} e^{s * (π - 2 θ_{j}) / π}}$
$L_{AirFace} = - \frac {1}{N} \sum_{i=1}^N log\frac {e^{s*(\pi - 2(\theta_{y_i}+m))/\pi}}{e^{s*(\pi - 2(\theta_{y_i}+m))/\pi} + \sum_{j=1, j\neq y_i}^C e^{s*(\pi - 2\theta_j)/\pi}}$

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

Dein Gesicht ist kein König! Verlustfunktionen für Gesichtserkennungsprobleme

Softmax Loss

Margin-Based Loss

Margin & Scale

:

:

More articles:

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

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