基于Schatten p-范数的低秩表示与结构约束的人脸识别方法
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本文主要探讨了"基于Schatten p-范数的结构受限判别词典学习在人脸识别中的应用"。这项研究发表在《数字信号处理》(Digital Signal Processing)杂志的95卷第10期，2019年，文章编号为102573。该期刊由Elsevier出版，可以通过其网站<https://www.elsevier.com/locate/dsp>获取最新内容。
低秩表示是当前人脸识别技术中的一个重要概念，它通过捕捉人脸图像数据中的潜在结构信息来提高识别性能。作者Heyou Chang、Fanlong Zhang等人，分别来自南京小庄大学的信任云计算与大数据分析实验室、东南大学的图像科学与技术实验室以及南京审计大学的技术学院等机构，他们合作进行这项研究。他们的工作重点在于提出一种新的方法，即结构受限的判别词典学习，其中Schatten p-范数起到了关键作用。
Schatten p-范数是一种矩阵范数，尤其适用于衡量矩阵的低秩特性，这对于处理大规模的数据集和提取其中的有用模式非常有效。在人脸识别中，低秩表示意味着通过选择具有低维度子空间的特征表示，可以减少冗余信息并增强特征的区分度。这种约束下的词典学习，即通过优化算法寻找既能保持人脸图像特征的结构一致性，又能实现良好分类性能的词典元素，有助于提高人脸识别系统的鲁棒性和准确性。
文章的关键词包括"低秩表示"、"结构受限"、"判别词典学习"和"Schatten p-范数"，这些词汇直接反映了研究的核心内容和技术手段。作者们在文中可能详细讨论了如何设计算法来估计和利用这些低秩特征表示，以及如何通过实验证明这种方法在实际人脸识别任务中的优越性，比如在不同光照条件、姿态变化或人脸遮挡等情况下的鲁棒性提升。
这篇研究论文深入探讨了如何结合低秩表示理论与Schatten p-范数，设计出一个适用于人脸识别的高效结构受限词典学习方法。这对于推动人脸识别技术的发展，尤其是在复杂环境下的实时和准确识别具有重要的理论和实践价值。读者可以期待从中了解到如何将这一方法应用于实际系统，以提升人脸识别系统的性能和实用性。
H. Chang et al. / Digital Signal Processing 95 (2019) 102573 3
2.4. D
2
L
2
R
2
By learning a sub-dictionary for each class separately, enforc-
ing
a low-rank constraint on the sub-dictionary and incorporating
the Fisher criterion into the model, S. Li et al. proposed D
2
L
2
R
2
method, which can be formulated as follows
min
D,Z
C
i
X
i
−DZ
i
2
F
+
X
i
−D
i
Z
i
i
2
F
+
j=1, j=i
D
i
Z
i
j
2
F
+λ
1
Z
1
+λ
2
F (Z) +α
C
i
D
i
∗
(4)
where F (Z) = tr(S
W
(Z)) − tr(S
B
(Z)) + ηZ
2
F
, S
W
(Z) =
C
i
=1
z
k
∈Z
i
(z
k
−
ˆ
z
i
)(z
k
−
ˆ
z
i
)
T
is the within-class scatter matrix
of Z, S
B
(Z) =
C
i
=1
n
i
(
ˆ
z
i
−
ˆ
z)(
ˆ
z
i
−
ˆ
z)
T
is the between-class scatter
matrix of Z.
ˆ
z
i
and
ˆ
z denote the mean sample of Z
i
and Z, n
i
is
the number of samples in ith class.
2.5. SCLRDL
Y. Liu et al. imposed both structure and low-rank restriction on
the coefficient matrix, and proposed SCLRDL method. The objective
function can be written as follows
min
D,Z
i
,E
i
C
i
αZ
∗
+
C
i
βE
i
l
s.t. X
i
=DZ
i
+E
i
,
π
i
(Z
i
) =0
(5)
where E
l
can be ·
2
F
for Gaussian noise and ·
1
for random
corruptions.
π
i
(Z
i
) = 0 makes all rows in Z
i
to be zeros except
the corresponding to the ith category.
2.6. RDLRR
By exploiting the low-rankness of both the data representation
and each occlusion-induced error image simultaneously, G. Gao et
al. proposed RDLRR to decompose the data matrix X into two parts
DZ and E, where Z is a low-rank matrix in vector representation
space, while E contains a series of low-rank noise images in origi-
nal
image space. The objective function is formulated as follows
min
D,Z,W,E
i
Z
∗
+λ
n
i=1
E
i
∗
+αZ −Q
2
F
s.t. X =DZ +E, H = WZ
(6)
where H =[h
1
, h
2
, ..., h] ∈R
C×n
and W represent class label ma-
trix
and classifiers parameters, respectively. h
i
=[0, 0, ..., 1, ...,
0, 0]
T
is the label vector of sample x
i
, where the position of ele-
ment
1 indicates the class of x
i
. Q is the same with that in DSLR.
n is the number of sample data.
3. Structure-constrained discriminative dictionary learning
based on Schatten p-norm for face recognition
In this section, a novel dictionary-learning approach based on
the Schatten p-norm model with a structure constraint and dis-
crimination
constraint is proposed.
3.1. SDDLS
p
In face recognition with occluded data, variations between im-
ages
of the same person due to noise are nearly larger than those
due to the change of identity. To reduce the role of large variations
and enhance the role of small variations, we propose applying the
Schatten p-norm (0 < p < 1) to approximate the nonconvex rank
minimization problem. Compared with the widely used nuclear
norm, the Schatten p-norm (0 < p < 1) treats each singular value
differently, which is beneficial for achieving our goal. According to
the above discussions, we formulate the model of SDDLS
p
as fol-
lows:
min
D,Z,E,W
Z
p
S
p
+βZ
1
+λE
1
+αr(Z) +γ g(Z)
s.t. X =DZ +E
(7)
where X, D and E are data matrix, dictionary and coefficient ma-
trix,
respectively. r(Z) and g(Z) are regularization term on Z.
α, β, λ and γ are tradeoff parameters and p ∈(0, 1).
3.2. Structure-constraint term for coefficients
Given a class-specific dictionary D =[D
1
, ..., D
C
], the ideal co-
efficient
matrix should have a block-diagonal structure. Then, we
construct a regularization term r(Z) =P Z
2
F
on the representa-
tion
Z to hit the point, where is the elementwise multiplication
operator and the element in the ith row and jth column of P is
defined as
P
i, j
=
0, if d
i
and x
j
belong to the same class
1
, otherwise
P =[p
1
, ..., p
n
] is a weighted matrix, where p
i
has the form of
[1, ..., 1, 0, ..., 0, 1, ..., 1]
T
. Suppose that x
i
belongs to class c;
then, all elements in p
i
for D
c
are 0s, whereas all others are 1s.
The
term r(Z) encourages Z
j
i
(i = j) to be small values and Z
i
i
to be large values, which makes
j=i
D
j
Z
j
i
2
F
≈ 0 and X
i
≈ D
i
Z
i
i
.
Note that r(Z) is different from the last term in [14]. In [14], the
minimization of
C
j
=1, j=i
D
i
Z
j
i
2
F
cannot ensure that the values in
Z
i
j
(i = j) are small. In addition, [14]attempts to learn a structured
dictionary by minimizing the rank of each subdictionary, which
reduces the diversity in the subdictionary and weakens the repre-
sentation
ability of the dictionary. The last term in [15] forces Z to
be close to Q, which implies that the representation of the samples
from the same class should be identical. The term may adversely
affect the ability of representations. This drawback is overcome in
our proposed approach by using the term r(Z) since the regulariza-
tion
r(Z) only reacts on Z
j
i
(i = j) and encourages the generation
of a coefficient matrix with a block-diagonal structure.
3.3. Discriminative term for coefficients
To make the dictionary optimal for face recognition, we propose
to incorporate the classification error as a term into the objective
function for dictionary learning. Here, we adopt a simple linear
classifier W. Given a label matrix Y, we construct a classification
error term g(Z) =Y − WZ
2
F
to make the coding coefficient dis-
criminative
via projecting the cth class coding coefficients only to
the cth dimension of the label space.
3.4. Optimization
To make problem (7) separable, we first introduce two auxiliary
variables J and L. Then, problem (7)can be rewritten as
min
D,Z,E,W,J,L
J
p
S
p
+βL
1
+λE
1
+αP Z
2
F
+γ Y − WZ
2
F
s.t. X =DZ +E, Z =J, Z = L
(8)
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