SphereFace：超球面深度嵌入提升人脸识别的精度

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本文主要探讨了深度人脸识别（Deep Face Recognition，FR）在开放集协议下的问题，该协议要求理想的面部特征在适当选择的度量空间中具有较小的最大类内距离和较大的最小类间距离。现有的许多算法难以有效实现这一理想标准。为此，作者提出了Angular Softmax（A-Softmax）损失函数，这是一个关键创新，它使卷积神经网络（Convolutional Neural Networks, CNNs）能够学习到角度上具有高度区分性的特征。几何上讲，A-Softmax损失可以视为在高维超球面上施加区分性约束，这个约束与人类面部数据天然地居住在一个低维流形上的先验相契合。这种设计考虑了人脸特征在空间分布上的自然特性，使得模型能够更好地捕捉人脸之间的细微差异。A-Softmax损失通过参数m调整角度边际，使得模型的学习过程更加符合开放集场景下对特征的理想要求。为了优化这一目标，作者进一步推导出了一个特定的m值，用于逼近理想的特征准则。他们进行了广泛的实验分析，验证了A-Softmax在各类评估指标上，如识别准确率、鲁棒性和抗干扰能力等方面，相较于传统的softmax损失有显著的优势。通过在公开的面部识别数据集上进行实验，包括LFW、CASIA-WebFace和MS-Celeb-1M等，A-Softmax显著提高了模型在开放集环境中的性能，尤其是在存在大量非目标类样本的情况下。此外，论文还探讨了A-Softmax在不同网络架构（如ResNet、VGG等）以及不同深度设置下的表现，展示了其通用性和可扩展性。实验结果表明，A-Softmax不仅提高了单个图像的识别精度，而且对于多模态数据融合也有积极影响，进一步提升了人脸识别系统的整体性能。 SphereFace论文提出了一种创新的深度学习框架，即A-Softmax，它通过构造一个具有角度约束的超球面来优化人脸识别的开放集性能。该方法为解决实际应用中复杂且挑战性的深度人脸识别问题提供了一个强大的工具，对于未来人脸识别技术的发展具有重要意义。

2. Related Work

Metric learning.

Metric learning aims to learn a sim-

ilarity (distance) function. Traditional metric learning

[

] usually learns a matrix

for a distance met-

ric

−x

)

A(x

−x

)

upon the given

features

, x

. Recently, prevailing deep metric learning

[

] usually uses neural networks

to automatically learn discriminative features

, x

fol-

lowed by a simple distance metric such as Euclidean dis-

tance

−x

. Most widely used loss functions for deep

metric learning are contrastive loss [

] and triplet loss

[32, 22, 6], and both impose Euclidean margin to features.

Deep face recognition.

Deep face recognition is ar-

guably one of the most active research area in the past few

years. [

] address the open-set FR using CNNs super-

vised by softmax loss, which essentially treats open-set FR

as a multi-class classiﬁcation problem. [

] combines con-

trastive loss and softmax loss to jointly supervise the CNN

training, greatly boosting the performance. [

] uses triplet

loss to learn a uniﬁed face embedding. Training on nearly

200 million face images, they achieve current state-of-the-art

FR accuracy. Inspired by linear discriminant analysis, [

]

proposes center loss for CNNs and also obtains promising

performance. In general, current well-performing CNNs

[

] for FR are mostly built on either contrastive loss or

triplet loss. One could notice that state-of-the-art FR meth-

ods usually adopt ideas (e.g. contrastive loss, triplet loss)

from metric learning, showing open-set FR could be well

addressed by discriminative metric learning.

L-Softmax loss [

] also implicitly involves the concept

of angles. As a regularization method, it shows great im-

provement on closed-set classiﬁcation problems. Differently,

A-Softmax loss is developed to learn discriminative face em-

bedding. The explicit connections to hypersphere manifold

makes our learned features particularly suitable for open-set

FR problem, as veriﬁed by our experiments. In addition,

the angular margin in A-Softmax loss is explicitly imposed

and can be quantitatively controlled (e.g. lower bounds to

approximate desired feature criterion), while [

] can only

be analyzed qualitatively.

3. Deep Hypersphere Embedding

3.1. Revisiting the Softmax Loss

We revisit the softmax loss by looking into the decision

criteria of softmax loss. In binary-class case, the posterior

probabilities obtained by softmax loss are

exp(W

x + b

)

exp(W

x + b

) + exp(W

x + b

)

(1)

exp(W

x + b

)

exp(W

x + b

) + exp(W

x + b

)

(2)

where

is the learned feature vector.

and

are weights

and bias of last fully connected layer corresponding to class

, respectively. The predicted label will be assigned to

class 1 if

and class 2 if

. By comparing

and

, it is clear that

x+b

and

x+b

de-

termine the classiﬁcation result. The decision boundary is

−W

)x+b

−b

. We then rewrite

x+b

kkxkcos(θ

)+b

where

is the angle between

and

. Notice that if we normalize the weights and zero

the biases (

k=1

), the posterior probabilities be-

come

=kxkcos(θ

)

and

=kxkcos(θ

)

. Note that

and

share the same

, the ﬁnal result only depends on

the angles

and

. The decision boundary also becomes

cos(θ

)−cos(θ

)=0

(i.e. angular bisector of vector

and

). Although the above analysis is built on binary-calss

case, it is trivial to generalize the analysis to multi-class case.

During training, the modiﬁed softmax loss (

k=1, b

)

encourages features from the

-th class to have smaller angle

(larger cosine distance) than others, which makes angles

between

and features a reliable metric for classiﬁcation.

To give a formal expression for the modiﬁed softmax loss,

we ﬁrst deﬁne the input feature

and its label

. The

original softmax loss can be written as

L =

− log





(3)

where

denotes the

-th element (

j ∈ [1, K]

is the

class number) of the class score vector

, and

is the

number of training samples. In CNNs,

is usually the

output of a fully connected layer

, so

= W

+ b

and

= W

+ b

where

, W

are the

-th

training sample, the

-th and

-th column of

respectively.

We further reformulate L

in Eq. (3) as

= − log





= − log



kkx

k cos(θ

)+b

kkx

k cos(θ

j,i

)+b



(4)

in which

j,i

(0≤θ

j,i

≤π)

is the angle between vector

and

. As analyzed above, we ﬁrst normalize

k=1, ∀j

in each iteration and zero the biases. Then we have the

modiﬁed softmax loss:

modiﬁed

− log



k cos(θ

)

k cos(θ

j,i

)



(5)

Although we can learn features with angular boundary with

the modiﬁed softmax loss, these features are still not neces-

sarily discriminative. Since we use angles as the distance

metric, it is natural to incorporate angular margin to learned

features in order to enhance the discrimination power. To

this end, we propose a novel way to combine angular margin.

3.2. Introducing Angular Margin to Softmax Loss

Instead of designing a new type of loss function and con-

structing a weighted combination with softmax loss (similar

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