人脸识别

model	conference	paper	first author	institute	loss fun
DeepFace	CVPR 2014	DeepFace: Closing the Gap to Human-Level Performance in Face Verification	Yaniv Taigman	Facebook AI Lab	sofrmax loss + contrastive loss
DeepID1	CVPR 2014	Deep Learning Face Representation from Predicting 10,000 Classes	Yi Sun	The Chinese University of Hongkong	sofrmax loss + contrastive loss
DeepID2	NIPS 2014	Deep Learning Face Representation by Joint Identification-Verification	Yi Sun	The Chinese University of Hongkong	sofrmax loss + contrastive loss
FaceNet	CVPR 2015	FaceNet: A Unified Embedding for Face Recognition and Clustering	Florian Schroff	google	triplet loss
CenterLoss	ECCV 2016	A Discriminative Feature Learning Approach for Deep Face Recognition	Yandong Wen	Shenzhen key lab of computer Vision and Pattern recognition	center loss
L-sofrmaxLoss	ICML 2016	Large-Margin Softmax Loss for Convolutional Neural Networks	Weiyang Liu & Yandong Wen	Peking Uiversity, South China University of Technology	L-softmax loss
SphereFace	CVPR 2017	SphereFace: Deep Hypersphere Embedding for Face Recognition	Weiyang Liu	Georgia Institute of Technology	A-softmax loss
CosFace	CVPR 2018	CosFace: Large Margin Cosine Loss for Deep Face Recognition	Hao Wang	Tencent AI Lab	large margin cosine loss
ArcFace	CVPR 2019	ArcFace: Additive Angular Margin Loss for Deep Face Recognition	Jiankang Deng & Jia Guo	Imperial College London, InsightFace	additive angular margin loss

人脸识别属于度量学习的范畴，学习到的人脸特征具有以下特点

• Intra-class Compactness

• Inter-class Discrepancy

Comparison of open-set and closed-set recognition

Contrastive Loss

$$\begin{array}{l} \operatorname{Ident}\left(f, t, \theta_{i d}\right)=-\sum_{i=1}^{n}-p_{i} \log \hat{p}_{i}=-\log \hat{p}_{t} \\ \operatorname{Verif}\left(f_{i}, f_{j}, y_{i j}, \theta_{v e}\right)=\left\{\begin{array}{ll} \frac{1}{2}\left\|f_{i}-f_{j}\right\|_{2}^{2} & \text { if } y_{i j}=1 \\ \frac{1}{2} \max \left(0, m-\left\|f_{i}-f_{j}\right\|_{2}\right)^{2} & \text { if } y_{i j}=-1 \end{array}\right. \end{array}$$

Triplet Loss

$$\sum_{i}^{N}\left[\left\|f\left(x_{i}^{a}\right)-f\left(x_{i}^{p}\right)\right\|_{2}^{2}-\left\|f\left(x_{i}^{a}\right)-f\left(x_{i}^{n}\right)\right\|_{2}^{2}+\alpha\right]_{+}$$

Center Loss

$$\begin{aligned} \mathcal{L}_{C}=& \frac{1}{2} \sum_{i=1}^{m}\left\|\boldsymbol{x}_{i}-\boldsymbol{c}_{y_{i}}\right\|_{2}^{2} \\ \Delta \boldsymbol{c}_{j}=& \frac{\sum_{i=1}^{m} \delta\left(y_{i}=j\right) \cdot\left(\boldsymbol{c}_{j}-\boldsymbol{x}_{i}\right)}{1+\sum_{i=1}^{m} \delta\left(y_{i}=j\right)} \\ \mathcal{L}=& \mathcal{L}_{S}+\lambda \mathcal{L}_{C} \\ =-& \sum_{i=1}^{m} \log \frac{e^{W_{y_{i}}^{T} \boldsymbol{x}_{i}+b_{y_{i}}}}{\sum_{j=1}^{n} e^{W_{j}^{T} \boldsymbol{x}_{i}+b_{j}}}+\frac{\lambda}{2} \sum_{i=1}^{m}\left\|\boldsymbol{x}_{i}-\boldsymbol{c}_{y_{i}}\right\|_{2}^{2} \end{aligned}$$

L-Softmax Loss

$$\begin{array}{l} \left\|\boldsymbol{W}_{1}\right\|\|\boldsymbol{x}\| \cos \left(\theta_{1}\right)>\left\|\boldsymbol{W}_{2}\right\|\|\boldsymbol{x}\| \cos \left(\theta_{2}\right) \\ \\ \begin{aligned} \left\|\boldsymbol{W}_{1}\right\|\|\boldsymbol{x}\| \cos \left(\theta_{1}\right) & \geq\left\|\boldsymbol{W}_{1}\right\|\|\boldsymbol{x}\| \cos \left(m \theta_{1}\right) \\ &>\left\|\boldsymbol{W}_{2}\right\|\|\boldsymbol{x}\| \cos \left(\theta_{2}\right) \end{aligned} \\ L_{i}=-\log \left(\frac{e^{\left\|\boldsymbol{W}_{y_{i}}\right\|\left\|\boldsymbol{x}_{i}\right\| \psi\left(\theta_{y_{i}}\right)}}{e^{\left\|\boldsymbol{W}_{y_{i}}\right\|\left\|\boldsymbol{x}_{i}\right\| \psi\left(\theta_{y_{i}}\right)}+\sum_{j \neq y_{i}} e^{\left\|\boldsymbol{W}_{j}\right\|\left\|\boldsymbol{x}_{i}\right\| \cos \left(\theta_{j}\right)}}\right) \end{array} \\ \psi(\theta)=\left\{\begin{array}{l} \cos (m \theta), \quad 0 \leq \theta \leq \frac{\pi}{m} \\ \mathcal{D}(\theta), \quad \frac{\pi}{m}<\theta \leq \pi \end{array}\right. \\ \psi(\theta)=(-1)^{k} \cos (m \theta)-2 k, \quad \theta \in\left[\frac{k \pi}{m}, \frac{(k+1) \pi}{m}\right] \\$$

SphereFace

$$\begin{aligned} &\text { Weight Norm and zero bias } \quad\left\|\boldsymbol{W}_{i}\right\|=1, b_{i}=0\\ &\text { Classification boundary } \cos \left(m \theta_{1}\right)=\cos \left(\theta_{2}\right)\\ &L_{\mathrm{ang}}=\frac{1}{N} \sum_{i}-\log \left(\frac{e^{\left\|\boldsymbol{x}_{i}\right\| \psi\left(\theta_{y_{i}, i}\right)}}{e^{\left\|\boldsymbol{x}_{i}\right\| \psi\left(\theta_{y_{i}, i}\right)}+\sum_{j \neq y_{i}} e^{\left\|\boldsymbol{x}_{i}\right\| \cos \left(\theta_{j, i}\right)}}\right) \end{aligned}$$

CosFace

Weight Norm and Feature Norm

$$\begin{aligned} W &=\frac{W^{*}}{\left\|W^{*}\right\|} \\ x &=\frac{x^{*}}{\left\|x^{*}\right\|} \\ \cos \left(\theta_{j}, i\right) &=W_{j}^{T} x_{i} \end{aligned}$$

classification boundary

$$\cos \left(\theta_{1}\right)-m>\cos \left(\theta_{2}\right) \text { and } \cos \left(\theta_{2}\right)-m>\cos \left(\theta_{1}\right)$$

loss function

$$L_{l m c}=\frac{1}{N} \sum_{i}-\log \frac{e^{s\left(\cos \left(\theta_{y_{i}, i}\right)-m\right)}}{e^{s\left(\cos \left(\theta_{y_{i}, i}\right)-m\right)}+\sum_{j \neq y_{i}} e^{s \cos \left(\theta_{j, i}\right)}}$$

其中，NSL是Normalized version of Softmax Loss。

Arcface

$$\begin{array}{l} \cos \left(\theta_{1}-m\right)>\cos \left(\theta_{2}\right) \\ L_{3}=-\frac{1}{N} \sum_{i=1}^{N} \log \frac{e^{s\left(\cos \left(\theta_{y_{i}}+m\right)\right)}}{e^{s\left(\cos \left(\theta_{y_{i}}+m\right)\right)}+\sum_{j=1, j \neq y_{i}}^{n} e^{s \cos \theta_{j}}} . \\ L_{4}=-\frac{1}{N} \sum_{i=1}^{N} \log \frac{e^{s\left(\cos \left(m_{1} \theta_{y_{i}}+m_{2}\right)-m_{3}\right)}}{\left.e^{s\left(\cos \left(m_{1} \theta_{j_{i}}+m_{2}\right)-m_{3}\right.}\right)+\sum_{j=1, j \neq y_{i}}^{n} e^{s \cos \theta_{j}}} . \end{array}$$

人脸损失可视化

采用通用的人脸损失公式，采用不同的参数如下，在minist上的可视化效果见Visualization of Face Loss

loss name	w_norm	x_norm	s	m1	m2	m3
softmax	False	False	1	1	0	0
L-softmax_v1	False	False	1	2	0	0
A-softmax_v1	True	False	1	2	0	0
A-softmax_v2	True	False	1	3	0	0
norm-softmax	True	True	1	1	0	0
CosFace_v1	True	True	4	1	0	0.1
CosFace_v2	True	True	4	1	0	0.2
ArcFace_v1	True	True	4	1	0.1	0
ArcFace_v2	True	True	4	1	0.2	0
ArcFace_v3	True	True	4	1	0.3	0