# Discriminative, Competitive, and Collaborative Representation-Based Classification withl2-Norm Regularizations

Authors: Jianping Gou; Junyu Lu; Heping Song; Hongxing Ma; Weihua Ou; Jia Ke
Journal: Complexity
 (2019)

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## Abstract

Recently, collaborative representation-based classification (CRC) and its many variations have been widely applied for various classification tasks in pattern recognition. To further enhance the pattern discrimination of CRC, in this article we propose a novel extension of CRC, entitled discriminative, competitive, and collaborative representation-based classification (DCCRC). In the proposed DCCRC, the class discrimination information is fully utilized for promoting the true class of each testing sample to dominantly represent the testing sample during collaborative representation. The class discrimination information is well considered in the newly designed discriminativel2-norm regularization that can decrease the ability of representation from the interclasses of each testing sample. Simultaneously, a competitive l2-norm regularization is introduced to the DCCRC model with the class discrimination information with the aim of enhancing the competitive ability of representation from the true class of each testing sample. The effectiveness of the proposed DCCRC is explored by extensive experiments on the several public face databases and some real numerical UCI data sets. The experimental results demonstrate that the proposed DCCRC achieves the superior performance over the state-of-the-art representation-based classification methods.


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## Body

## 1. Introduction

Nowadays, the linear representation-based classification (RBC) often including sparse representation-based classification (SRC) [1] and collaborative representation-based classification (CRC) [2] has attracted more and more attention in pattern recognition. In both SRC and CRC, each testing sample is linearly represented by all the training samples and always classified by the class-specific representation residuals. Due to the excellent representation-based classification performance, the RBC methods have been widely used in many classification tasks, such as image classification [3–10] and face recognition [11–19].It has been well known that SRC with thel1-norm regularization of representation coefficients is a very promising kind of RBC owing to its good property of sparsity and natural discrimination [1, 20, 21]. However, it has been argued that the representation-based pattern discrimination originated from the l2-norm collaborative representation of all the training samples instead of the l1-norm sparse representation of a few training samples, and then the standard CRC was first proposed as a general extension of SRC [2]. Specifically, using the l2-norm regularization of representation coefficients, the effective discrimination benefits from the collaborative representation from all the class-specific training samples. Because of the efficient closed-form solution of CRC for the effective classification performance, a great many CRC extensions have been developed in recent years [6, 15, 17–19, 22–35]. Moreover, the possible reasons of the natural discrimination from CRC were detailedly analyzed from the perspective of class separability of data [18] and the probability [22]. Among the CRC methods, the general extensions are the weighted CRC using the localities of data as the weights that constrained the collaborative representation coefficients [17, 27–29, 31]. Since collaborative representation has the efficient and effective classification performance, several two-phase collaborative representation-based classification methods have been designed in [30–33, 36]. Moreover, such two-phase collaborative representation-based classification also has the property of sparsity for enhancing the ability of pattern discrimination [30]. Using the superiorities of sparse representation and collaborative representation, the extensions of combining both were proposed for classification in [34, 35, 37, 38]. Besides, due to good latent discrimination contained in the representation, sparse representation and collaborative representation were utilized to design the effective nearest neighbor classification [39–41].In many latest extensions of CRC, the class discrimination information of data in fact was fully employed for strengthening the power of the pattern classification [42–47]. From the point of view of probability, a probabilistic CRC (ProCRC) was developed by using the discriminative regularization of the representations between all the classes and each class [22]. Using the prior information of data the extended ProCRC (EProCRC) was proposed in [43], and using the coarse to fine representation the two-phased ProCRC was proposed in [33]. Through designing the discriminative regularization of pairs of the representations of any two classes, the new discriminative sparse representation method for classification (DSRC) was proposed in [44]. On the basis of DSRC and ProCRC, a novel discriminative CRC method was proposed to extend DSRC [45]. To overcome the issue that the phases of representation and classification in the most CRC variations are not integrated into a unified model, a collaborative and competitive representation-based classifier (CCRC) was proposed in [46]. CCRC directly includes the classification decision in its model and can enhance the training sample from each class to competitively represent each testing sample. With the aim of obtaining the similar competitive representations among all the classes, the discriminative l2-norm regularization of the representations of all the classes except any one class was designed for proposing the competitive and collaborative representation classification method (Co-CRC) [47]. As argued in these discriminative CRC extensions above, the discriminative representation was achieved for favorable classification.Based on the fact that the discrimination information of data can be explored for enhancing the power of pattern discrimination in collaborative representation, in this article we proposed a novel discriminative competitive and collaborative representation-based classification method (DCCRC) by using the discriminative representation among all the classes. The proposed DCCRC assumes that each class can discriminatively and competitively represent the testing samples. The discriminative and competitive collaborative representations among all the classes can be realized by twol2-norm regularizations in the DCCRC model. One is the newly designed l2-norm regularization of the pairs of representation from all the classes and representations from all the classes excluding any one class. The other is the competitive l2-norm regularization of representations from all the classes excluding any one class [47]. To experimentally verify the classification performance of the proposed DCCRC, we compare it to the state-of-the-art RBC methods on several face databases and some real numerical UCI data sets. The conducted experiments show that the proposed method is effective with better classification results than the competing RBC methods. In summary, our main contributions in this article are given as follows:(1)
A new discriminativel2-norm regularization is designed by using the representations from all the classes excluding any one class(2)
A novel discriminative, competitive, and collaborative representation is proposed for classification by considering the discrimination information of data(3)
The experimental analyses are reported for well demonstrating the effectiveness of the proposed DCCRCThe rest of this article is organized as follows. Section2 briefly describes the related work. Section 3 detailedly presents the proposed DCCRC and then analyzes it. Section 4 reports extensive experiments to evaluate the effectiveness of the proposed DCCRC. Finally, the conclusions of this article are given in Section 5.


## 2. The Related Work

In this section, we briefly review some related RBC models. First of all, some commonly used notations are denoted here. We suppose that the set of all the training samples fromC classes is denoted as X=x1,x2,…,xN=X1,…,XC∈Rd×N, where d is the dimensionality of the feature space and N and Ni are the numbers of all the training samples from all the classes and class i, respectively. Note that the ith column vector of X represents the training sample xi and the subset of the training samples from class i is Xi∈Rd×Ni. Besides, we also assume y∈Rd is a given testing sample used for classification. In the liner representation-based classification, the testing sample y is approximately represented as y≈x1s1+x2s2+⋯+xNsN=XS, where S=s1,s2,…,sNT=S1T,…,SCTT∈RN is the vector of all the representation coefficients corresponding to all the training samples of X and Si is the subvector of the representation coefficients from class i.


### 2.1. CRC

CRC is a typical linear representation-based classifier proposed recently [2]. In the CRC, a given testing sample y is collaboratively represented by all the training samples for classification. The CRC model is defined as(1)minSy−XS22+λS22,where λ is a positive regularization parameter. Clearly, CRC can learn the closed-form solution of S as(2)S=H1y,where H1=XTX+λI−1XT with an identity matrix I. Using the learned S=XTX+λI−1XTy, the class-specific representation residuals are determined as y−XiSi2/Si2. Finally, the given testing sample y is classified into the class with the minimum representation residual among all the classes.


### 2.2. DSRC

DSRC [44] is a discriminative sparse representation method with a l2-norm regularization of the pairs of any two class-specific representations. It can achieve the good pattern discrimination among the different classes with sparsity. The DSRC model is defined as(3)minSy−XS22+γ∑i=1C∑j=1CXiSi+XjSj22,where γ is a positive regularization parameter. Through some algebra operations, the efficient solution of S can be obtained as(4)S=H2y,where H2=1+2γXTX+2γLM−1XT and M=X1TX1⋯0⋮⋱⋮0⋯XCTXC. Using the learned S=1+2γXTX+2γLM−1XTy, we compute the class-specific representation residuals with XiSi−y22 and classify y into the class with the minimum representation residual among all the classes.


### 2.3. Co-CRC

Co-CRC [47] is a new extension of CRC that can induce each training class to discriminatively and competitively represent each testing sample. The Co-CRC model is defined as(5)minSy−XS22+β∑i=1CX−iS−i22,where β is a positive regularization parameter. The second term in equation (5) is the competitive representation constraint. According to the way of solving S [47], the learned solution of S is achieved as(6)S=H3y,where H3=XTX+β∑i=1CP−iTX−iTX−iP−i−1XT and P−iT=I1⋯0⋮⋱⋮0⋯In. Using S=XTX+β∑i=1CP−iTX−iTX−iP−i−1XTy, the class-specific representation residuals are calculated as XiSi−y22 and the testing sample y is classified into the class with the minimum representation residual among all the classes.


## 2.1. CRC

CRC is a typical linear representation-based classifier proposed recently [2]. In the CRC, a given testing sample y is collaboratively represented by all the training samples for classification. The CRC model is defined as(1)minSy−XS22+λS22,where λ is a positive regularization parameter. Clearly, CRC can learn the closed-form solution of S as(2)S=H1y,where H1=XTX+λI−1XT with an identity matrix I. Using the learned S=XTX+λI−1XTy, the class-specific representation residuals are determined as y−XiSi2/Si2. Finally, the given testing sample y is classified into the class with the minimum representation residual among all the classes.


## 2.2. DSRC

DSRC [44] is a discriminative sparse representation method with a l2-norm regularization of the pairs of any two class-specific representations. It can achieve the good pattern discrimination among the different classes with sparsity. The DSRC model is defined as(3)minSy−XS22+γ∑i=1C∑j=1CXiSi+XjSj22,where γ is a positive regularization parameter. Through some algebra operations, the efficient solution of S can be obtained as(4)S=H2y,where H2=1+2γXTX+2γLM−1XT and M=X1TX1⋯0⋮⋱⋮0⋯XCTXC. Using the learned S=1+2γXTX+2γLM−1XTy, we compute the class-specific representation residuals with XiSi−y22 and classify y into the class with the minimum representation residual among all the classes.


## 2.3. Co-CRC

Co-CRC [47] is a new extension of CRC that can induce each training class to discriminatively and competitively represent each testing sample. The Co-CRC model is defined as(5)minSy−XS22+β∑i=1CX−iS−i22,where β is a positive regularization parameter. The second term in equation (5) is the competitive representation constraint. According to the way of solving S [47], the learned solution of S is achieved as(6)S=H3y,where H3=XTX+β∑i=1CP−iTX−iTX−iP−i−1XT and P−iT=I1⋯0⋮⋱⋮0⋯In. Using S=XTX+β∑i=1CP−iTX−iTX−iP−i−1XTy, the class-specific representation residuals are calculated as XiSi−y22 and the testing sample y is classified into the class with the minimum representation residual among all the classes.


## 3. The Proposed DCCRC

In this section, we detailedly present the proposed DCCRC method. The basic idea of DCCRC is first given, and then the DCCRC model and its solving procedure are described, finally the essential properties that DCCRC holds are analyzed.


### 3.1. Idea of DCCRC

The proposed DCCRC contains two assumptions that are originally inspired by the competitive and collaborative representation [47]. For clear descriptions, the collaborative representation of the given testing sample y using all training samples is rewritten as y≈XS=X1S1+X2S2+⋯+XCSC=XiSi+X−iS−i, where X−i represents the training samples excluding samples from class i with the corresponding vector S−i of the representation coefficients. The one assumption originates from the expectation that the true class of given testing sample y can dominantly represent y and the other classes have little contribution to representing it (i.e., X−iS−i22=0). Unfortunately, the true class of the testing sample y is not known and any one of all the training classes could be chosen as the true class of y. In fact, we only make the training samples from one class to competitively represent the testing sample y as soon as possible and the contribution to representing y from other classes is as poor as possible in ideal case. Accordingly, with this good expectation, the testing sample y is well represented as XiSi from class i by simultaneously minimizing the representation X−iS−i from other classes. Thus, in the proposed method we introduce the competitive constraint X−iS−i22 that was first designed in [47].In collaborative representation, all the training samples approximately represent the testing sampley as soon as possible, i.e., y≈XS. During the process of representation, if y belongs to class i with dominant representation XiSi, the representation X−iS−i from the other classes tends to be very small. In specific, XiSi tends to be equivalent to the representation XS in some degree. In such an ideal case, the approximate equalities can be learned, y≈XiSi≈XS. Borrowing the idea of degrading the correlations among classes by minimizing the discriminative constraint XiSi+XjSj22 [44], we also assume that the correlation between the representation XiSi from class i and the representation X−iS−i from the other classes is as small as possible. That is to say, if class i can dominantly represent y with XiSi and all the training samples can well represent y with XS, the correlation between XS and X−iS−i should be small. Similar to the definition of XiSi+XjSj22 [44], we design the another new discriminative constraint XS+X−iS−i22. It is obvious that to minimize XS+X−iS−i22 can minimize XS22, X−iS−i22, and XSTX−iS−i. Minimizing X−iS−i22 satisfies the first assumption. If y≈XiSi≈XS, minimizing XSTX−iS−i approximately equals to minimize XiSiTX−iS−i that can well degrade the correlation between the representation from one class and the representation from the other classes.


### 3.2. Model of DCCRC

In this section, we first introduce the objective function of the proposed DCCRC model and then present the procedures of solving it in details. The given testing sampley is represented by collaborative representation of all the training samples, and the DCCRC model on the basis of its idea is defined as follows:(7)minSy−XS22+λ1∑i=1CX−iS−i22+λ2∑i=1CXS+X−iS−i22+λ3S22,where λ1,λ2,andλ3 is the positive regularization parameters. In equation (7), the second term ∑i=1CX−iS−i22, first designed in [47], is the competitive constraint that can make each class competitively and discriminatively represent the testing sample y among all the classes. The third term ∑i=1CXS+X−iS−i22 is the discriminative constraint that not only makes each class competitively represent the testing sample y but also degrades the representation correlations between one class and the other classes for more discrimination. Note that when λ1=λ2=0, DCCRC is the same as CRC, and when λ2=λ3=0, DCCRC is the same as Co-CRC.In order to achieve the solution of the representation coefficient vectorS, equation (7) should be further reformulated as(8)minSy−XS22+λ1∑i=1CX¯−iS22+λ2∑i=1CXS+X¯−iS22+λ3S22,where Xi^=0,0…0,Xi,0…0 and X¯−i=X−Xi^=X1,X2,…,Xi−1,0,Xi+1,…,XC. To simply solve S, let F1S=y−XS22+λ1∑i=1CX¯−iS22+λ3S22 and F2S=λ2∑i=1CXS+X¯−iS22. Firstly, the derivative of F1S with respect to S is calculated as(9)∂F1S∂S=y−XS22+λ1∑i=1CX¯−iS22+λ3S22∂S=−2XTy−XS+2λ1∑i=1CX¯−iTX¯−iS+2λ3S.Since∑i=1CX¯−iTX¯−iS can be rewritten as(10)∑i=1CX¯−iTX¯−iS=∑i=1CX−Xi^TX−Xi^S=∑i=1CXTX−2XTXi^+X^iTXi^S=CXTX−∑i=1CX^iTXi^S=CXTX−GS,where XTXi^=X^iTXi^, and G is defined as(11)G=X1TX1…0⋮⋱⋮0…XCTXC.Using equations (9) and (10), ∂F1S/∂S is reformulated as(12)∂F1S∂S=−2XTy−XS+2λ1CXTX−GS+2λ3S.Then, the derivative ofF2S with respect to S is calculated as(13)∂F2S∂S=∂λ2∑i=1CXS+X¯−iS22∂S=2λ2∑i=1CX+X¯−iTX+X¯−iS=2λ2∑i=1CXTX+2∑i=1CXTX¯−i+∑i=1CX¯−iTX¯−iS.In equation (13), using X¯−i=X−Xi^, ∑i=1CX¯TX¯−i can be reformulated as(14)∑i=1CXTX¯−i=XTCX−X^1+X^2+⋯+X^C=C−1XTX.Using equations (10) and (14), equation (13) can be finally rewritten as(15)∂F2S∂S=∂λ2∑i=1CXS+X¯−iS22∂S=2λ222C−2XTX−GS.Clearly, the objective function of DCCRC isF1S+F2S. Using equations (10) and (15), the derivative of the proposed function with respect to S is(16)∂F1S+F2S∂S=−2XTy−XS+2λ1CXTX−GS+2λ222C−2XTX−GS+2λ3S.Finally, we set∂F1S+F2S/∂S=0, and the solution of the representation coefficient vector S in equation (7) is obtained as(17)S=1−λ1+4λ2C+4λ2XTX+λ1+λ2G+λ3I−1XTy.After obtaining the representation coefficient vectorS, we calculate the class-specific representation residuals and determine the class label cy of the testing sample y as(18)cy=argminciy−XiSi2,i=1,2,…,C.That is to say, the given testing sampley is classified into the class with the minimal representation residuals among all the classes. According to the descriptions of the proposed DCCRC model above, the proposed DCCRC is briefly summarized in Algorithm 1.Algorithm 1:The proposed DCCRC method.
Input: The given training set X=X1,X2,…,XC and testing sample y with three regularized parameters λ1, λ2, and λ3.
Output: Determine the class label of y.
(1)
Normalize each sample ofX and y.
(2)
CalculateXTX and G with equation (11).
(3)
Solve the representation vector with equation (17).
(4)
Calculate the class-specific representation residuals withy−XiSi2.
(5)
Predict the label ofy with equation (18).


### 3.3. Analysis of DCCRC

In this section, we first further analyze the terms∑i=1CX−iS−i22 and ∑i=1CXS+X−iS−i22 in the proposed DCCRC method, in order to explain the more power of pattern discrimination. And then, the analyses of differences between the proposed DCCRC and Co-CRC, DSRC are emphasized.Using the way of analyzing the competitive representation [47], X−iS−i22 in term ∑i=1CX−iS−i22 can be rewritten as XS−XiSi22, and we can obtain the equality as(19)XS−XiSi22=XS22−2XSTXiSi+XiSi22≥XS22+XiSi22−2XS2XiSi2.Assume the angle betweenXS and XiSi is α. Using equation (19), cosα can be obtained as(20)cosα=XSTXiSiXS2XiSi2≤1.According to equation (20), when cosα=1, XS approaches XiSi with the same direction. In this ideal case, the given testing sample y is dominantly represented by XiSi from class i that y truly belongs to. Thus, to minimize ∑i=1CX−iS−i22 could have two advantages. One is that each class competitively represents the testing sample y. Another one is that the true class of y could competitively represent it and the other classes poorly represent it.Moreover,XS+X−iS−i22 in term ∑i=1CXS+X−iS−i22 can be reformulated as XS−2XiSi22. Through simple algebra of XS−2XiSi22, we can also achieve equation (20). This fact means that to minimize ∑i=1CXS+X−iS−i22 has the very similar superiorities of minimizing ∑i=1CX−iS−i22. And also, we can rewrite XS+X−iS−i22 as XiSi+2X−iS−i22=XiSi22+4X−iS−i22+4XiSiTX−iS−i. To minimize XS+X−iS−i22 is to simultaneously minimize XiSi22, X−iS−i22, and XiSiTX−iS−i. We can see that except minimizing X−iS−i22, minimizing XiSiTX−iS−i can degrade the correlation between XiSi and X−iS−i [44]. That is to say, to minimize ∑i=1CXS+X−iS−i22 could degrade the correlation between one class and the other classes, in order to enhance the power of pattern discrimination and competitive representations among all the classes. Thus, the terms ∑i=1CX−iS−i22 and ∑i=1CXS+X−iS−i22 can obtain the competitive and discriminative collaborative representation for favorable classification. Besides, the pattern discrimination among all the classes can be intuitively verified in the next section.The differences between the proposed DCCRC and Co-CRC, DSRC can be analyzed by comparing their corresponding models (i.e., equation (7) for DCCRC, equation (5) for Co-CRC, and equation (3) for DSRC). According to equations (3) and (7), DCCRC is very different from DSRC, but both have similar discriminative terms. The term XiSi+XjSj22 in DSRC can degrade the correlations between any two classes for favorable pattern discrimination, but the term XS+X−iS−i22 in DCCRC can degrade the correlations between any one class and the other classes for competitively enhancing the discriminative representation from each class for classification. Besides, compared to DSRC, the proposed DCCRC also has the competitive constraint and the regularization of the representation coefficients. Furthermore, the proposed DCCRC is the extension of Co-CRC because DCCRC and Co-CRC have the same competitive constraint ∑i=1CX−iS−i22. In contrast with Co-CRC, the proposed DCCRC also has the designed discriminative constraint XS+X−iS−i22 and the regularization of the representation coefficients, in order that DCCRC further enhances the competitive representations among all the classes. Thus, the proposed DCCRC has more pattern discrimination than DSRC and Co-CRC that can be experimentally verified in the next experimental section.


## 3.1. Idea of DCCRC

The proposed DCCRC contains two assumptions that are originally inspired by the competitive and collaborative representation [47]. For clear descriptions, the collaborative representation of the given testing sample y using all training samples is rewritten as y≈XS=X1S1+X2S2+⋯+XCSC=XiSi+X−iS−i, where X−i represents the training samples excluding samples from class i with the corresponding vector S−i of the representation coefficients. The one assumption originates from the expectation that the true class of given testing sample y can dominantly represent y and the other classes have little contribution to representing it (i.e., X−iS−i22=0). Unfortunately, the true class of the testing sample y is not known and any one of all the training classes could be chosen as the true class of y. In fact, we only make the training samples from one class to competitively represent the testing sample y as soon as possible and the contribution to representing y from other classes is as poor as possible in ideal case. Accordingly, with this good expectation, the testing sample y is well represented as XiSi from class i by simultaneously minimizing the representation X−iS−i from other classes. Thus, in the proposed method we introduce the competitive constraint X−iS−i22 that was first designed in [47].In collaborative representation, all the training samples approximately represent the testing sampley as soon as possible, i.e., y≈XS. During the process of representation, if y belongs to class i with dominant representation XiSi, the representation X−iS−i from the other classes tends to be very small. In specific, XiSi tends to be equivalent to the representation XS in some degree. In such an ideal case, the approximate equalities can be learned, y≈XiSi≈XS. Borrowing the idea of degrading the correlations among classes by minimizing the discriminative constraint XiSi+XjSj22 [44], we also assume that the correlation between the representation XiSi from class i and the representation X−iS−i from the other classes is as small as possible. That is to say, if class i can dominantly represent y with XiSi and all the training samples can well represent y with XS, the correlation between XS and X−iS−i should be small. Similar to the definition of XiSi+XjSj22 [44], we design the another new discriminative constraint XS+X−iS−i22. It is obvious that to minimize XS+X−iS−i22 can minimize XS22, X−iS−i22, and XSTX−iS−i. Minimizing X−iS−i22 satisfies the first assumption. If y≈XiSi≈XS, minimizing XSTX−iS−i approximately equals to minimize XiSiTX−iS−i that can well degrade the correlation between the representation from one class and the representation from the other classes.


## 3.2. Model of DCCRC

In this section, we first introduce the objective function of the proposed DCCRC model and then present the procedures of solving it in details. The given testing sampley is represented by collaborative representation of all the training samples, and the DCCRC model on the basis of its idea is defined as follows:(7)minSy−XS22+λ1∑i=1CX−iS−i22+λ2∑i=1CXS+X−iS−i22+λ3S22,where λ1,λ2,andλ3 is the positive regularization parameters. In equation (7), the second term ∑i=1CX−iS−i22, first designed in [47], is the competitive constraint that can make each class competitively and discriminatively represent the testing sample y among all the classes. The third term ∑i=1CXS+X−iS−i22 is the discriminative constraint that not only makes each class competitively represent the testing sample y but also degrades the representation correlations between one class and the other classes for more discrimination. Note that when λ1=λ2=0, DCCRC is the same as CRC, and when λ2=λ3=0, DCCRC is the same as Co-CRC.In order to achieve the solution of the representation coefficient vectorS, equation (7) should be further reformulated as(8)minSy−XS22+λ1∑i=1CX¯−iS22+λ2∑i=1CXS+X¯−iS22+λ3S22,where Xi^=0,0…0,Xi,0…0 and X¯−i=X−Xi^=X1,X2,…,Xi−1,0,Xi+1,…,XC. To simply solve S, let F1S=y−XS22+λ1∑i=1CX¯−iS22+λ3S22 and F2S=λ2∑i=1CXS+X¯−iS22. Firstly, the derivative of F1S with respect to S is calculated as(9)∂F1S∂S=y−XS22+λ1∑i=1CX¯−iS22+λ3S22∂S=−2XTy−XS+2λ1∑i=1CX¯−iTX¯−iS+2λ3S.Since∑i=1CX¯−iTX¯−iS can be rewritten as(10)∑i=1CX¯−iTX¯−iS=∑i=1CX−Xi^TX−Xi^S=∑i=1CXTX−2XTXi^+X^iTXi^S=CXTX−∑i=1CX^iTXi^S=CXTX−GS,where XTXi^=X^iTXi^, and G is defined as(11)G=X1TX1…0⋮⋱⋮0…XCTXC.Using equations (9) and (10), ∂F1S/∂S is reformulated as(12)∂F1S∂S=−2XTy−XS+2λ1CXTX−GS+2λ3S.Then, the derivative ofF2S with respect to S is calculated as(13)∂F2S∂S=∂λ2∑i=1CXS+X¯−iS22∂S=2λ2∑i=1CX+X¯−iTX+X¯−iS=2λ2∑i=1CXTX+2∑i=1CXTX¯−i+∑i=1CX¯−iTX¯−iS.In equation (13), using X¯−i=X−Xi^, ∑i=1CX¯TX¯−i can be reformulated as(14)∑i=1CXTX¯−i=XTCX−X^1+X^2+⋯+X^C=C−1XTX.Using equations (10) and (14), equation (13) can be finally rewritten as(15)∂F2S∂S=∂λ2∑i=1CXS+X¯−iS22∂S=2λ222C−2XTX−GS.Clearly, the objective function of DCCRC isF1S+F2S. Using equations (10) and (15), the derivative of the proposed function with respect to S is(16)∂F1S+F2S∂S=−2XTy−XS+2λ1CXTX−GS+2λ222C−2XTX−GS+2λ3S.Finally, we set∂F1S+F2S/∂S=0, and the solution of the representation coefficient vector S in equation (7) is obtained as(17)S=1−λ1+4λ2C+4λ2XTX+λ1+λ2G+λ3I−1XTy.After obtaining the representation coefficient vectorS, we calculate the class-specific representation residuals and determine the class label cy of the testing sample y as(18)cy=argminciy−XiSi2,i=1,2,…,C.That is to say, the given testing sampley is classified into the class with the minimal representation residuals among all the classes. According to the descriptions of the proposed DCCRC model above, the proposed DCCRC is briefly summarized in Algorithm 1.Algorithm 1:The proposed DCCRC method.
Input: The given training set X=X1,X2,…,XC and testing sample y with three regularized parameters λ1, λ2, and λ3.
Output: Determine the class label of y.
(1)
Normalize each sample ofX and y.
(2)
CalculateXTX and G with equation (11).
(3)
Solve the representation vector with equation (17).
(4)
Calculate the class-specific representation residuals withy−XiSi2.
(5)
Predict the label ofy with equation (18).


## 3.3. Analysis of DCCRC

In this section, we first further analyze the terms∑i=1CX−iS−i22 and ∑i=1CXS+X−iS−i22 in the proposed DCCRC method, in order to explain the more power of pattern discrimination. And then, the analyses of differences between the proposed DCCRC and Co-CRC, DSRC are emphasized.Using the way of analyzing the competitive representation [47], X−iS−i22 in term ∑i=1CX−iS−i22 can be rewritten as XS−XiSi22, and we can obtain the equality as(19)XS−XiSi22=XS22−2XSTXiSi+XiSi22≥XS22+XiSi22−2XS2XiSi2.Assume the angle betweenXS and XiSi is α. Using equation (19), cosα can be obtained as(20)cosα=XSTXiSiXS2XiSi2≤1.According to equation (20), when cosα=1, XS approaches XiSi with the same direction. In this ideal case, the given testing sample y is dominantly represented by XiSi from class i that y truly belongs to. Thus, to minimize ∑i=1CX−iS−i22 could have two advantages. One is that each class competitively represents the testing sample y. Another one is that the true class of y could competitively represent it and the other classes poorly represent it.Moreover,XS+X−iS−i22 in term ∑i=1CXS+X−iS−i22 can be reformulated as XS−2XiSi22. Through simple algebra of XS−2XiSi22, we can also achieve equation (20). This fact means that to minimize ∑i=1CXS+X−iS−i22 has the very similar superiorities of minimizing ∑i=1CX−iS−i22. And also, we can rewrite XS+X−iS−i22 as XiSi+2X−iS−i22=XiSi22+4X−iS−i22+4XiSiTX−iS−i. To minimize XS+X−iS−i22 is to simultaneously minimize XiSi22, X−iS−i22, and XiSiTX−iS−i. We can see that except minimizing X−iS−i22, minimizing XiSiTX−iS−i can degrade the correlation between XiSi and X−iS−i [44]. That is to say, to minimize ∑i=1CXS+X−iS−i22 could degrade the correlation between one class and the other classes, in order to enhance the power of pattern discrimination and competitive representations among all the classes. Thus, the terms ∑i=1CX−iS−i22 and ∑i=1CXS+X−iS−i22 can obtain the competitive and discriminative collaborative representation for favorable classification. Besides, the pattern discrimination among all the classes can be intuitively verified in the next section.The differences between the proposed DCCRC and Co-CRC, DSRC can be analyzed by comparing their corresponding models (i.e., equation (7) for DCCRC, equation (5) for Co-CRC, and equation (3) for DSRC). According to equations (3) and (7), DCCRC is very different from DSRC, but both have similar discriminative terms. The term XiSi+XjSj22 in DSRC can degrade the correlations between any two classes for favorable pattern discrimination, but the term XS+X−iS−i22 in DCCRC can degrade the correlations between any one class and the other classes for competitively enhancing the discriminative representation from each class for classification. Besides, compared to DSRC, the proposed DCCRC also has the competitive constraint and the regularization of the representation coefficients. Furthermore, the proposed DCCRC is the extension of Co-CRC because DCCRC and Co-CRC have the same competitive constraint ∑i=1CX−iS−i22. In contrast with Co-CRC, the proposed DCCRC also has the designed discriminative constraint XS+X−iS−i22 and the regularization of the representation coefficients, in order that DCCRC further enhances the competitive representations among all the classes. Thus, the proposed DCCRC has more pattern discrimination than DSRC and Co-CRC that can be experimentally verified in the next experimental section.


## 4. Experiments

In this section, the extensive experiments on several face databases and some real numerical UCI data sets are conducted. In the experiments, we compare the proposed DCCRC with the state-of-the-art RBC methods including SRC [1], CRC [2], CCRC [46], Co-CRC [47], DSRC [44], ProCRC [22], and EProCRC [43]. It should be noted that all regularized parameters in the competing methods are preset as the range 10−3,10−2,…,1,10,100 for fair comparisons in the experiments. The optimal classification results of each competing method are obtained among the range of its parameters.


### 4.1. Data Sets

In this section, we briefly describe the used data sets including the AR, YaleB, IMM, Yale, and PIE29 face databases and the real UCI data sets. The YaleB database (http://vision.ucsd.edu/leekc/ExtYaleDatabase/ExtYaleB.html) was taken under different poses and uncontrolled illumination conditions. The Yale database (http://cvc.yale.edu/projects/yalefaces/yalefaces.html) was taken by different facial expressions. The AR database (http://www2.ece.ohio-state.edu/aleix/ARdatabase.html) was taken by various facial expressions and illumination conditions, and we use a subset of AR with 1400 image from 100 subjects. The IMM database (http://www.imm.dtu.dk/∼aam/datasets/datasets.html) contains 240 annotated monocular images from 40 subjects. The PIE29 database (http://www.intbox.com/public/project/4742/) was taken in different conditions including 13 postures, 43 lights, and 4 expressions. In the experiments, each image is cropped and resized into 32×32 with 256 gray levels per pixel and also the gray level values are normalized to [0, 1]. The numbers of total samples, classes, samples per class, and chosen training samples per class are shown in Table 1. As an example, the image samples of one subject from each face data base are shown in Figure 1.Table 1
The main information about the used face databases.
Data
Total samples
Classes
Samples per class
Training samples per class
YaleB
2432
38
64
12, 18, 24, 30
Yale
165
15
11
2, 3, 4, 5
AR
1400
100
14
2, 4, 6, 8
IMM
240
40
6
2, 3, 4, 5
PIE29
1632
68
24
2, 4, 6, 8Figure 1
The example of the facial images of one individual from five face database. (a) YaleB. (b) PIE29. (c) IMM. (d) Yale. (e) AR.
(a)
(b)
(c)
(d)
(e)The real used eight UCI data sets were downloaded from UC Irvine Machine Learning Repository (UCI) (http://archive.ics.uci.edu/ml). They are “Wine,” “Vehicle,” “Auto MPG,” “Statlog (Heart),” “Statlog (Australian Credit Approval),” “Credit Approval,” “Isolet,” and “Ionosphere.” Note that “Auto MPG,” “Statlog (Heart),” “Statlog (Australian Credit Approval),” “Credit Approval,” and “Ionosphere” are abbreviated as “Auto,” “Heart,” “SCredit,” “Credit,” and “Iono,” respectively. The numbers of total samples, classes, attributes, and training samples per class are displayed in Table 2. In the experiments, each sample on these UCI data sets is also normalized to 0,1. Furthermore, on these face and UCI data sets, they are randomly divided into the sets of the training and testing samples ten times, and the training samples chosen from each class are shown in Tables 1 and 2.Table 2
The main information about the used UCI data set.
Data
Total samples
Classes
Attributes
Training samples per class
Wine
178
3
13
7, 8
Vehicle
846
4
18
10, 11
Auto
392
3
8
10, 14
Credit
690
2
15
8, 10
Heart
270
2
13
6, 8
SCredit
690
2
14
35, 50
Isolet
1560
2
617
21, 26
Iono
351
2
34
15, 20


### 4.2. Experiment 1

In this section, we first conduct the experiments to analyze the competitive termλ1X−iS−i22 and the discriminative term λ2∑i=1CXS+X−iS−i22 by varying the values of the parameters λ1 and λ2 in the proposed DCCRC on the five face databases. The values of the parameters λ1, λ2, and λ3 are preset as 10−3,10−2,…,1,10,100, and the numbers of training samples per class are chosen as l=4 on AR, l=4 on IMM, l=18 on YaleB, l=2 on Yale, and l=6 on PIE29. For visual comparisons, the model minSy−XS22+λ1∑i=1CX−iS−i22+λ3S22 without λ2∑i=1CXS+X−iS−i22 is denoted as DCCRC1, and the model minSy−XS22+λ2∑i=1CXS+X−iS−i22+λ3S22 without λ1X−iS−i22 is denoted as DCCRC2. Accordingly, we compare DCCRC1 with DCCRC to demonstrate the discrimination of the term λ2∑i=1CXS+X−iS−i22 by varying the values of the parameter λ1. And we compare DCCRC2 with DCCRC to demonstrate the discrimination of the term λ1X−iS−i22 by varying the values of the parameter λ2. It should be noted that the values of the parameters λ2 and λ3 are optimal with best classification accuracies when DCCRC1 is compared with DCCRC, and the values of the parameters λ1 and λ3 are optimal with best classification accuracies when DCCRC2 is compared with DCCRC. For conveniently presenting the values of λ1 and λ2 in the figures, we use p1=lgλ1 and p2=lgλ2 (i.e., the values of p1 and p2 correspond to that of λ1 and λ2, respectively).The classification accuracies of DCCRC1 and DCCRC with varying λ1 are shown in Figure 2, and the ones of DCCRC2 and DCCRC with varying λ2 are shown in Figure 3. From the experimental results in Figure 2, we can see that DCCRC with λ2∑i=1CXS+X−iS−i22 significantly performs better than DCCRC1 without λ2∑i=1CXS+X−iS−i22, and DCCRC is more robust to the variations of λ1 than DCCRC1. As shown in Figure 3, we can also observe that DCCRC with λ1X−iS−i22 significantly performs better than DCCRC2 without λ1X−iS−i22 and DCCRC is more robust to the variations of λ2 than DCCRC2. In addition, the classification performance of DCCRC1 with variations of λ1 and DCCRC2 with variations of λ2 shows that the terms λ1X−iS−i22 and λ2∑i=1CXS+X−iS−i22 can improve the power of the pattern discrimination. The experimental results in two figures imply that the proposed DCCRC has effective and robust classification performance. As a consequence, the more pattern discrimination of the proposed DCCRC originated from the competitive and discriminative terms is well verified.Figure 2
The comparisons of DCCRC and DCCRC1 with varying values of parameter λ1 on each face database. (a) AR (l = 4). (b) IMM (l = 4). (c) YaleB (l = 18). (d) Yale (l = 2). (e) PIE29 (l = 6).
(a)
(b)
(c)
(d)
(e)Figure 3
The comparisons of DCCRC and DCCRC2 with varying values of parameter λ2 on each face database. (a) AR (l = 4). (b) IMM (l = 4). (c) YaleB (l = 18). (d) Yale (l = 2). (e) PIE29 (l = 6).
(a)
(b)
(c)
(d)
(e)And then, we visually verify the discriminative ability of the proposed DCCRC method in comparison with the competitive CRC method (i.e., Co-CRC). As discussed in Section3.3, we define the class-specific representation contribution for the given testing sample y as(21)conyi=XiSi22∑i=1CXiSi22.Clearly, both DCCRC and Co-CRC classify each testing sample into the class with the largestconyi among all the classes. Then, the pattern discrimination ability of both is intuitively represented by the representation reconstructive images for the given testing samples from class 26 in IMM and class 9 in AR. The first five representation reconstructive images of the testing samples corresponding to the top five largest representation contributions are illustrated in Figure 4. Note that the numbers in the bracket under each reconstructive image are the class and its representation contribution conyi. For example, 26,10.74 under the reconstructive image means the class 26 has the representation contribution cony26=10.74. As can be seen in Figure 4, the proposed DCCRC correctly represents and classifies the testing samples, but Co-CRC wrongly represents and classifies them. Moreover, we can observe that the first reconstructive image that is reconstructed by the class with the largest representation contribution in Co-CRC is very similar to the testing image on each face database. Through the experimental illustrations in Figure 4, the proposed DCCRC is more discriminative than Co-CRC for classification. This means the designed term λ2∑i=1CXS+X−iS−i22 is discriminative. Therefore, it can be concluded that the proposed DCCRC has the effective and robust classification due to the competitive and discriminative constraints.Figure 4
The reconstructive images of the given testing samples by the classes with the top five largest representation contributions via DCCRC and Co-CRC on the face databases (a) IMM and (b) AR.
(a)
(b)


### 4.3. Experiment 2

In this section, we compare the proposed DCCRC to the competing methods on the face databases and the UCI data sets. The experimental results of each competing method are the averages of the classification accuracies on ten division of each data set. The best classification accuracies of each method are achieved among the range of its parameter, and the preset class-specific training samples on each data set are shown in Tables1 and 2.The classification accuracies of all the competing methods are shown in Table3 on face databases and Table 4 on the UCI data sets. Note that the best classification performance among all the methods on each data set is indicated in bold face. As shown in two tables, the classification accuracies of each competing method almost ascend with the increase of the class-specific training samples on all the data sets. On the face databases, we can see that the proposed DCCRC nearly achieves the best classification accuracies among all the competing methods, but it could not significantly improve very much in comparison with some methods. As displayed in Table 4, the proposed DCCRC significantly performs better than the other competing methods. In addition, from the classification results in two tables, we can observe that CCRC, Co-CRC, DSRC, ProCRC, and EProCRC obtain the similar competitive classification performance. The possible reason is that these methods can fully employ the class-specific representations in the collaborative representation to improve the pattern discrimination among all the classes. As a consequence, we can conclude that our DCCRC method is a promising representation-based classifier in pattern classification with effectiveness and robustness.Table 3
The classification accuracies (%) of the competing methods with the corresponding standard deviations with different class-specific training samples on each face database.
Data
l
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
YaleB
12
91.770.95
93.58 ± 0.58
93.78 ± 0.63
94.34 ± 0.51
93.72 ± 0.67
94.53 ± 0.40
94.05 ± 0.53
94.61 ± 0.38
18
95.48 ± 0.45
96.21 ± 0.41
96.55 ± 0.46
96.63 ± 0.51
96.51 ± 0.49
96.52 ± 0.46
96.09 ± 0.54
96.91 ± 0.38
24
96.81 ± 0.43
97.31 ± 0.42
97.62 ± 0.23
97.69 ± 0.23
97.59 ± 0.22
97.54 ± 0.29
97.13 ± 0.30
97.93 ± 0.24
30
97.57 ± 0.47
97.64 ± 0.38
98.12 ± 0.46
98.13 ± 0.48
98.12 ± 0.46
97.63 ± 0.43
97.46 ± 0.47
98.25 ± 0.45
AR
2
84.99 ± 0.90
86.71 ± 0.77
89.92 ± 0.70
90.15 ± 0.77
89.89 ± 0.68
90.10 ± 0.79
85.46 ± 0.80
90.18 ± 0.77
4
94.18 ± 0.49
94.48 ± 1.16
96.55 ± 0.78
96.60 ± 0.73
96.50 ± 0.80
96.48 ± 0.50
94.93 ± 0.63
96.72 ± 0.78
6
96.67 ± 0.94
96.25 ± 0.75
98.04 ± 0.87
98.02 ± 0.89
97.85 ± 0.90
97.78 ± 0.58
97.35 ± 0.46
98.08 ± 0.80
8
98.00 ± 0.87
98.00 ± 0.88
99.08 ± 0.55
99.17 ± 0.51
99.06 ± 0.64
98.38 ± 0.34
98.22 ± 0.62
99.17 ± 0.43
Yale
2
73.21 ± 4.23
71.60 ± 4.68
77.53 ± 4.36
77.53 ± 4.36
77.04 ± 4.39
77.65 ± 4.53
76.54 ± 3.59
78.52 ± 4.16
3
82.92 ± 1.56
80.69 ± 1.11
85.97 ± 1.44
86.39 ± 1.95
85.97 ± 1.44
85.56 ± 2.51
85.14 ± 2.91
86.81 ± 1.93
4
85.87 ± 2.12
81.75 ± 2.98
88.41 ± 2.37
88.89 ± 2.23
88.57 ± 2.17
87.78 ± 2.65
87.94 ± 2.93
89.37 ± 2.37
5
90.74 ± 3.95
88.33 ± 4.65
92.04 ± 2.57
92.41 ± 2.37
92.22 ± 2.43
88.89 ± 1.57
88.89 ± 6.29
92.96 ± 2.78
IMM
2
63.54 ± 2.76
60.00 ± 1.58
66.67 ± 2.67
66.77 ± 2.54
66.67 ± 2.52
66.15 ± 2.66
65.83 ± 2.19
66.88 ± 2.40
3
69.72 ± 2.34
63.89 ± 2.15
73.06 ± 2.34
73.61 ± 3.48
72.50 ± 2.47
73.33 ± 3.21
72.08 ± 2.09
73.61 ± 3.15
4
74.38 ± 4.24
68.96 ± 5.09
77.08 ± 3.42
78.33 ± 3.59
77.08 ± 3.68
77.08 ± 2.92
75.83 ± 3.23
78.54 ± 3.57
5
82.92 ± 4.01
81.67 ± 4.08
85.00 ± 2.24
85.42 ± 3.68
84.58 ± 2.92
83.33 ± 3.42
83.33 ± 3.42
86.25 ± 3.45
PIE29
4
89.90 ± 0.70
90.94 ± 0.57
91.01 ± 0.71
91.37 ± 0.55
90.97 ± 0.74
91.56 ± 0.48
90.50 ± 0.60
91.68 ± 0.58
6
92.12 ± 0.77
92.29 ± 0.40
92.68 ± 0.64
92.81 ± 0.65
92.60 ± 0.70
92.76 ± 0.67
92.45 ± 0.41
92.91 ± 0.70
8
92.94 ± 0.48
92.74 ± 0.75
93.49 ± 0.54
93.49 ± 0.60
93.51 ± 0.54
93.22 ± 0.66
93.14 ± 0.56
93.77 ± 0.60
10
93.68 ± 0.61
94.03 ± 0.78
94.47 ± 0.53
94.54 ± 0.54
94.39 ± 0.47
93.99 ± 0.85
93.89 ± 0.92
94.60 ± 0.54Table 4
The classification accuracies (%) of the competing methods with the corresponding standard deviations with different class-specific training samples on each UCI data set.
Data
l
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
Vehicle
10
56.382.00
62.11 ± 3.27
60.02 ± 3.60
60.07 ± 3.62
60.02 ± 3.62
53.55 ± 4.14
57.94 ± 2.21
63.03 ± 3.44
11
57.03 ± 1.13
64.56 ± 1.77
63.22 ± 1.69
63.24 ± 1.69
63.22 ± 1.69
54.34 ± 7.00
61.22 ± 4.23
65.01 ± 2.70
Auto
10
70.54 ± 2.24
73.02 ± 2.09
73.47 ± 1.66
73.67 ± 1.93
73.68 ± 1.90
69.18 ± 5.38
70.02 ± 2.86
73.89 ± 1.59
14
73.80 ± 4.06
75.28 ± 3.03
75.66 ± 3.14
75.63 ± 3.18
75.62 ± 3.17
71.16 ± 7.51
74.16 ± 4.13
75.93 ± 3.60
Credit
8
61.25 ± 5.80
67.98 ± 2.44
68.72 ± 1.04
68.55 ± 0.84
68.55 ± 0.82
59.91 ± 5.39
62.85 ± 6.82
69.50 ± 3.05
10
64.90 ± 4.28
68.78 ± 1.12
69.25 ± 2.19
69.31 ± 2.02
69.22 ± 2.01
61.88 ± 3.92
63.82 ± 6.59
72.42 ± 2.79
Wine
7
66.88 ± 4.79
77.58 ± 6.10
87.13 ± 3.17
88.41 ± 2.64
72.10 ± 3.67
62.80 ± 3.62
76.56 ± 6.17
89.81 ± 2.06
8
70.52 ± 5.39
82.8 ± 1.93
89.22 ± 4.17
90.00 ± 3.00
77.79 ± 3.96
66.88 ± 7.01
80.91 ± 6.17
91.04 ± 2.57
Heart
6
58.76 ± 5.69
61.94 ± 5.16
64.65 ± 2.89
64.42 ± 2.77
64.42 ± 2.77
61.24 ± 2.2
63.64 ± 8.63
67.13 ± 4.04
8
63.54 ± 4.11
70.87 ± 3.32
71.42 ± 2.65
71.42 ± 2.65
71.42 ± 2.65
69.53 ± 5.49
66.14 ± 4.40
73.23 ± 2.83
SCredit
35
63.10 ± 2.77
71.97 ± 2.87
72.00 ± 2.77
72.03 ± 2.71
72.23 ± 2.77
57.58 ± 3.35
63.74 ± 4.81
75.84 ± 3.81
50
66.07 ± 3.51
74.64 ± 2.83
74.41 ± 1.38
74.44 ± 1.34
74.47 ± 1.52
59.32 ± 2.60
65.53 ± 4.88
76.95 ± 4.32
Isolet
21
71.41 ± 3.02
71.38 ± 1.48
69.83 ± 0.94
69.47 ± 1.10
69.47 ± 1.10
66.36 ± 2.69
72.89 ± 2.39
73.19 ± 2.50
26
78.10 ± 1.70
79.22 ± 2.02
77.80 ± 2.20
77.53 ± 2.29
77.55 ± 2.32
75.81 ± 3.01
79.50 ± 2.27
79.88 ± 2.28
Iono
15
83.49 ± 4.25
88.66 ± 3.88
87.54 ± 3.50
87.35 ± 3.39
87.35 ± 3.43
82.06 ± 3.84
81.99 ± 3.18
89.72 ± 4.29
20
86.37 ± 1.94
91.38 ± 1.27
88.49 ± 2.05
88.04 ± 1.55
87.97 ± 1.45
82.06 ± 1.45
83.09 ± 1.91
92.99 ± 1.12


### 4.4. Experiment 3

In this section, we conduct the experiments on IMM and Yale to compare the proposed DCCRC with the competing methods under the situations of the testing samples with the corruptions. In the experiments, the numbersl of the class-specific training samples are preset as l=3 on IMM and l=6 on Yale, and the remaining samples per class are regarded as the testing samples. And the classification results of each competing method are the averages of recognition accuracies on ten training and testing divisions of data. Moreover, the testing samples per class are randomly corrupted by randomly adding the pixels and the block occlusion with a panda. That is to say, the pixel corruptions are that some pixels of each testing image are randomly replaced by the uncertain gray scale values between 0 and 255, and some part of each testing image is randomly occluded by the panda. The ratios of the corrupted size to the original size of each testing image are from 0.1 to 0.4 with a step 0.1. As an example, the testing samples with the random pixels from one class are shown in Figure 5 and with the random occlusions shown in Figure 6.Figure 5
The testing images with the random pixel corruptions from one class. (a) IMM. (b) Yale.
(a)
(b)Figure 6
The testing images with the random block occlusions from one class. (a) IMM. (b) Yale.
(a)
(b)The classification results of the competing methods on IMM and Yale with varying ratios of the random corruptions are shown in Table5 for random occlusions and in Table 6 for random corrected pixels. Note that the best classification performance among all the methods in two tables is indicated in bold face. As listed in two tables, the classification accuracies of each competing method descend with the increase of the ratios of the corrupted size of each testing image. From these experimental results, we can see that the proposed DCCRC is nearly the most robust among all the competing methods, since it outperforms the other competing methods. Thus, the proposed method has more effectiveness and robustness under the situations of data with noises.Table 5
The classification accuracies (%) of the competing methods with the corresponding standard deviations under the situations of the random block occlusions.
Data
Ratios
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
IMM
0.1
62.17 ± 1.80
63.00 ± 1.00
65.17 ± 1.33
65.50 ± 0.85
62.33 ± 0.97
63.67 ± 1.35
62.67 ± 0.97
66.00 ± 0.82
0.2
54.33 ± 1.43
56.17 ± 1.80
59.50 ± 1.87
58.67 ± 1.63
55.50 ± 3.27
55.33 ± 1.55
57.50 ± 1.18
59.67 ± 1.45
0.3
45.67 ± 2.44
47.67 ± 1.62
50.00 ± 2.36
50.83 ± 2.47
48.17 ± 2.44
49.50 ± 2.51
49.83 ± 1.22
51.50 ± 2.26
0.4
37.33 ± 3.14
35.83 ± 1.90
40.50 ± 2.15
41.00 ± 2.44
38.33 ± 2.36
39.67 ± 1.72
39.33 ± 2.20
41.00 ± 2.44
Yale
0.1
82.67 ± 2.83
86.67 ± 2.11
86.50 ± 2.25
86.50 ± 1.94
85.50 ± 2.53
85.00 ± 3.18
84.17 ± 3.23
87.33 ± 2.31
0.2
76.00 ± 2.83
79.17 ± 3.46
79.00 ± 1.73
79.17 ± 1.76
78.67 ± 1.76
75.33 ± 1.15
75.00 ± 1.60
79.33 ± 1.89
0.3
64.83 ± 3.52
68.83 ± 2.49
70.33 ± 2.56
69.50 ± 3.01
69.17 ± 2.94
62.50 ± 3.23
65.50 ± 3.49
70.83 ± 1.94
0.4
53.07 ± 2.13
56.80 ± 1.36
57.87 ± 2.32
56.27 ± 4.49
56.53 ± 4.81
45.33 ± 2.53
50.67 ± 2.39
58.40 ± 2.29Table 6
The classification accuracies (%) of the competing methods with the corresponding standard deviations under the situations of the random pixel corruptions.
Data
Ratios
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
IMM
0.1
65.17 ± 1.11
64.67 ± 0.67
65.33 ± 1.55
65.67 ± 1.86
63.17 ± 2.32
62.17 ± 2.39
63.50 ± 1.11
66.00 ± 2.07
0.2
58.67 ± 1.35
59.33 ± 1.33
58.17 ± 1.22
58.17 ± 1.22
57.33 ± 1.33
53.00 ± 1.63
57.17 ± 0.85
59.67 ± 1.80
0.3
51.33 ± 1.35
52.17 ± 1.55
49.33 ± 1.62
49.67 ± 1.80
48.67 ± 2.27
43.33 ± 2.74
48.172.07
53.17 ± 3.00
0.4
40.50 ± 3.64
40.50 ± 3.10
39.83 ± 2.26
40.00 ± 2.11
39.83 ± 2.66
32.50 ± 2.04
32.17 ± 3.60
40.83 ± 2.30
Yale
0.1
87.73 ± 2.29
89.87 ± 1.81
90.67 ± 1.69
90.67 ± 1.69
89.33 ± 1.89
90.93 ± 1.96
89.87 ± 2.17
90.93 ± 1.96
0.2
83.47 ± 1.81
85.87 ± 1.36
86.93 ± 1.31
87.47 ± 1.07
85.07 ± 1.00
86.13 ± 1.81
86.40 ± 1.77
88.00 ± 1.19
0.3
79.73 ± 4.25
80.27 ± 2.59
83.73 ± 2.72
83.20 ± 2.61
79.73 ± 1.96
83.20 ± 2.87
83.20 ± 3.44
84.53 ± 2.32
0.4
69.07 ± 2.44
74.67 ± 2.23
75.73 ± 1.55
74.67 ± 2.53
70.13 ± 3.22
72.00 ± 1.19
76.00 ± 1.89
76.27 ± 1.55


## 4.1. Data Sets

In this section, we briefly describe the used data sets including the AR, YaleB, IMM, Yale, and PIE29 face databases and the real UCI data sets. The YaleB database (http://vision.ucsd.edu/leekc/ExtYaleDatabase/ExtYaleB.html) was taken under different poses and uncontrolled illumination conditions. The Yale database (http://cvc.yale.edu/projects/yalefaces/yalefaces.html) was taken by different facial expressions. The AR database (http://www2.ece.ohio-state.edu/aleix/ARdatabase.html) was taken by various facial expressions and illumination conditions, and we use a subset of AR with 1400 image from 100 subjects. The IMM database (http://www.imm.dtu.dk/∼aam/datasets/datasets.html) contains 240 annotated monocular images from 40 subjects. The PIE29 database (http://www.intbox.com/public/project/4742/) was taken in different conditions including 13 postures, 43 lights, and 4 expressions. In the experiments, each image is cropped and resized into 32×32 with 256 gray levels per pixel and also the gray level values are normalized to [0, 1]. The numbers of total samples, classes, samples per class, and chosen training samples per class are shown in Table 1. As an example, the image samples of one subject from each face data base are shown in Figure 1.Table 1
The main information about the used face databases.
Data
Total samples
Classes
Samples per class
Training samples per class
YaleB
2432
38
64
12, 18, 24, 30
Yale
165
15
11
2, 3, 4, 5
AR
1400
100
14
2, 4, 6, 8
IMM
240
40
6
2, 3, 4, 5
PIE29
1632
68
24
2, 4, 6, 8Figure 1
The example of the facial images of one individual from five face database. (a) YaleB. (b) PIE29. (c) IMM. (d) Yale. (e) AR.
(a)
(b)
(c)
(d)
(e)The real used eight UCI data sets were downloaded from UC Irvine Machine Learning Repository (UCI) (http://archive.ics.uci.edu/ml). They are “Wine,” “Vehicle,” “Auto MPG,” “Statlog (Heart),” “Statlog (Australian Credit Approval),” “Credit Approval,” “Isolet,” and “Ionosphere.” Note that “Auto MPG,” “Statlog (Heart),” “Statlog (Australian Credit Approval),” “Credit Approval,” and “Ionosphere” are abbreviated as “Auto,” “Heart,” “SCredit,” “Credit,” and “Iono,” respectively. The numbers of total samples, classes, attributes, and training samples per class are displayed in Table 2. In the experiments, each sample on these UCI data sets is also normalized to 0,1. Furthermore, on these face and UCI data sets, they are randomly divided into the sets of the training and testing samples ten times, and the training samples chosen from each class are shown in Tables 1 and 2.Table 2
The main information about the used UCI data set.
Data
Total samples
Classes
Attributes
Training samples per class
Wine
178
3
13
7, 8
Vehicle
846
4
18
10, 11
Auto
392
3
8
10, 14
Credit
690
2
15
8, 10
Heart
270
2
13
6, 8
SCredit
690
2
14
35, 50
Isolet
1560
2
617
21, 26
Iono
351
2
34
15, 20


## 4.2. Experiment 1

In this section, we first conduct the experiments to analyze the competitive termλ1X−iS−i22 and the discriminative term λ2∑i=1CXS+X−iS−i22 by varying the values of the parameters λ1 and λ2 in the proposed DCCRC on the five face databases. The values of the parameters λ1, λ2, and λ3 are preset as 10−3,10−2,…,1,10,100, and the numbers of training samples per class are chosen as l=4 on AR, l=4 on IMM, l=18 on YaleB, l=2 on Yale, and l=6 on PIE29. For visual comparisons, the model minSy−XS22+λ1∑i=1CX−iS−i22+λ3S22 without λ2∑i=1CXS+X−iS−i22 is denoted as DCCRC1, and the model minSy−XS22+λ2∑i=1CXS+X−iS−i22+λ3S22 without λ1X−iS−i22 is denoted as DCCRC2. Accordingly, we compare DCCRC1 with DCCRC to demonstrate the discrimination of the term λ2∑i=1CXS+X−iS−i22 by varying the values of the parameter λ1. And we compare DCCRC2 with DCCRC to demonstrate the discrimination of the term λ1X−iS−i22 by varying the values of the parameter λ2. It should be noted that the values of the parameters λ2 and λ3 are optimal with best classification accuracies when DCCRC1 is compared with DCCRC, and the values of the parameters λ1 and λ3 are optimal with best classification accuracies when DCCRC2 is compared with DCCRC. For conveniently presenting the values of λ1 and λ2 in the figures, we use p1=lgλ1 and p2=lgλ2 (i.e., the values of p1 and p2 correspond to that of λ1 and λ2, respectively).The classification accuracies of DCCRC1 and DCCRC with varying λ1 are shown in Figure 2, and the ones of DCCRC2 and DCCRC with varying λ2 are shown in Figure 3. From the experimental results in Figure 2, we can see that DCCRC with λ2∑i=1CXS+X−iS−i22 significantly performs better than DCCRC1 without λ2∑i=1CXS+X−iS−i22, and DCCRC is more robust to the variations of λ1 than DCCRC1. As shown in Figure 3, we can also observe that DCCRC with λ1X−iS−i22 significantly performs better than DCCRC2 without λ1X−iS−i22 and DCCRC is more robust to the variations of λ2 than DCCRC2. In addition, the classification performance of DCCRC1 with variations of λ1 and DCCRC2 with variations of λ2 shows that the terms λ1X−iS−i22 and λ2∑i=1CXS+X−iS−i22 can improve the power of the pattern discrimination. The experimental results in two figures imply that the proposed DCCRC has effective and robust classification performance. As a consequence, the more pattern discrimination of the proposed DCCRC originated from the competitive and discriminative terms is well verified.Figure 2
The comparisons of DCCRC and DCCRC1 with varying values of parameter λ1 on each face database. (a) AR (l = 4). (b) IMM (l = 4). (c) YaleB (l = 18). (d) Yale (l = 2). (e) PIE29 (l = 6).
(a)
(b)
(c)
(d)
(e)Figure 3
The comparisons of DCCRC and DCCRC2 with varying values of parameter λ2 on each face database. (a) AR (l = 4). (b) IMM (l = 4). (c) YaleB (l = 18). (d) Yale (l = 2). (e) PIE29 (l = 6).
(a)
(b)
(c)
(d)
(e)And then, we visually verify the discriminative ability of the proposed DCCRC method in comparison with the competitive CRC method (i.e., Co-CRC). As discussed in Section3.3, we define the class-specific representation contribution for the given testing sample y as(21)conyi=XiSi22∑i=1CXiSi22.Clearly, both DCCRC and Co-CRC classify each testing sample into the class with the largestconyi among all the classes. Then, the pattern discrimination ability of both is intuitively represented by the representation reconstructive images for the given testing samples from class 26 in IMM and class 9 in AR. The first five representation reconstructive images of the testing samples corresponding to the top five largest representation contributions are illustrated in Figure 4. Note that the numbers in the bracket under each reconstructive image are the class and its representation contribution conyi. For example, 26,10.74 under the reconstructive image means the class 26 has the representation contribution cony26=10.74. As can be seen in Figure 4, the proposed DCCRC correctly represents and classifies the testing samples, but Co-CRC wrongly represents and classifies them. Moreover, we can observe that the first reconstructive image that is reconstructed by the class with the largest representation contribution in Co-CRC is very similar to the testing image on each face database. Through the experimental illustrations in Figure 4, the proposed DCCRC is more discriminative than Co-CRC for classification. This means the designed term λ2∑i=1CXS+X−iS−i22 is discriminative. Therefore, it can be concluded that the proposed DCCRC has the effective and robust classification due to the competitive and discriminative constraints.Figure 4
The reconstructive images of the given testing samples by the classes with the top five largest representation contributions via DCCRC and Co-CRC on the face databases (a) IMM and (b) AR.
(a)
(b)


## 4.3. Experiment 2

In this section, we compare the proposed DCCRC to the competing methods on the face databases and the UCI data sets. The experimental results of each competing method are the averages of the classification accuracies on ten division of each data set. The best classification accuracies of each method are achieved among the range of its parameter, and the preset class-specific training samples on each data set are shown in Tables1 and 2.The classification accuracies of all the competing methods are shown in Table3 on face databases and Table 4 on the UCI data sets. Note that the best classification performance among all the methods on each data set is indicated in bold face. As shown in two tables, the classification accuracies of each competing method almost ascend with the increase of the class-specific training samples on all the data sets. On the face databases, we can see that the proposed DCCRC nearly achieves the best classification accuracies among all the competing methods, but it could not significantly improve very much in comparison with some methods. As displayed in Table 4, the proposed DCCRC significantly performs better than the other competing methods. In addition, from the classification results in two tables, we can observe that CCRC, Co-CRC, DSRC, ProCRC, and EProCRC obtain the similar competitive classification performance. The possible reason is that these methods can fully employ the class-specific representations in the collaborative representation to improve the pattern discrimination among all the classes. As a consequence, we can conclude that our DCCRC method is a promising representation-based classifier in pattern classification with effectiveness and robustness.Table 3
The classification accuracies (%) of the competing methods with the corresponding standard deviations with different class-specific training samples on each face database.
Data
l
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
YaleB
12
91.770.95
93.58 ± 0.58
93.78 ± 0.63
94.34 ± 0.51
93.72 ± 0.67
94.53 ± 0.40
94.05 ± 0.53
94.61 ± 0.38
18
95.48 ± 0.45
96.21 ± 0.41
96.55 ± 0.46
96.63 ± 0.51
96.51 ± 0.49
96.52 ± 0.46
96.09 ± 0.54
96.91 ± 0.38
24
96.81 ± 0.43
97.31 ± 0.42
97.62 ± 0.23
97.69 ± 0.23
97.59 ± 0.22
97.54 ± 0.29
97.13 ± 0.30
97.93 ± 0.24
30
97.57 ± 0.47
97.64 ± 0.38
98.12 ± 0.46
98.13 ± 0.48
98.12 ± 0.46
97.63 ± 0.43
97.46 ± 0.47
98.25 ± 0.45
AR
2
84.99 ± 0.90
86.71 ± 0.77
89.92 ± 0.70
90.15 ± 0.77
89.89 ± 0.68
90.10 ± 0.79
85.46 ± 0.80
90.18 ± 0.77
4
94.18 ± 0.49
94.48 ± 1.16
96.55 ± 0.78
96.60 ± 0.73
96.50 ± 0.80
96.48 ± 0.50
94.93 ± 0.63
96.72 ± 0.78
6
96.67 ± 0.94
96.25 ± 0.75
98.04 ± 0.87
98.02 ± 0.89
97.85 ± 0.90
97.78 ± 0.58
97.35 ± 0.46
98.08 ± 0.80
8
98.00 ± 0.87
98.00 ± 0.88
99.08 ± 0.55
99.17 ± 0.51
99.06 ± 0.64
98.38 ± 0.34
98.22 ± 0.62
99.17 ± 0.43
Yale
2
73.21 ± 4.23
71.60 ± 4.68
77.53 ± 4.36
77.53 ± 4.36
77.04 ± 4.39
77.65 ± 4.53
76.54 ± 3.59
78.52 ± 4.16
3
82.92 ± 1.56
80.69 ± 1.11
85.97 ± 1.44
86.39 ± 1.95
85.97 ± 1.44
85.56 ± 2.51
85.14 ± 2.91
86.81 ± 1.93
4
85.87 ± 2.12
81.75 ± 2.98
88.41 ± 2.37
88.89 ± 2.23
88.57 ± 2.17
87.78 ± 2.65
87.94 ± 2.93
89.37 ± 2.37
5
90.74 ± 3.95
88.33 ± 4.65
92.04 ± 2.57
92.41 ± 2.37
92.22 ± 2.43
88.89 ± 1.57
88.89 ± 6.29
92.96 ± 2.78
IMM
2
63.54 ± 2.76
60.00 ± 1.58
66.67 ± 2.67
66.77 ± 2.54
66.67 ± 2.52
66.15 ± 2.66
65.83 ± 2.19
66.88 ± 2.40
3
69.72 ± 2.34
63.89 ± 2.15
73.06 ± 2.34
73.61 ± 3.48
72.50 ± 2.47
73.33 ± 3.21
72.08 ± 2.09
73.61 ± 3.15
4
74.38 ± 4.24
68.96 ± 5.09
77.08 ± 3.42
78.33 ± 3.59
77.08 ± 3.68
77.08 ± 2.92
75.83 ± 3.23
78.54 ± 3.57
5
82.92 ± 4.01
81.67 ± 4.08
85.00 ± 2.24
85.42 ± 3.68
84.58 ± 2.92
83.33 ± 3.42
83.33 ± 3.42
86.25 ± 3.45
PIE29
4
89.90 ± 0.70
90.94 ± 0.57
91.01 ± 0.71
91.37 ± 0.55
90.97 ± 0.74
91.56 ± 0.48
90.50 ± 0.60
91.68 ± 0.58
6
92.12 ± 0.77
92.29 ± 0.40
92.68 ± 0.64
92.81 ± 0.65
92.60 ± 0.70
92.76 ± 0.67
92.45 ± 0.41
92.91 ± 0.70
8
92.94 ± 0.48
92.74 ± 0.75
93.49 ± 0.54
93.49 ± 0.60
93.51 ± 0.54
93.22 ± 0.66
93.14 ± 0.56
93.77 ± 0.60
10
93.68 ± 0.61
94.03 ± 0.78
94.47 ± 0.53
94.54 ± 0.54
94.39 ± 0.47
93.99 ± 0.85
93.89 ± 0.92
94.60 ± 0.54Table 4
The classification accuracies (%) of the competing methods with the corresponding standard deviations with different class-specific training samples on each UCI data set.
Data
l
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
Vehicle
10
56.382.00
62.11 ± 3.27
60.02 ± 3.60
60.07 ± 3.62
60.02 ± 3.62
53.55 ± 4.14
57.94 ± 2.21
63.03 ± 3.44
11
57.03 ± 1.13
64.56 ± 1.77
63.22 ± 1.69
63.24 ± 1.69
63.22 ± 1.69
54.34 ± 7.00
61.22 ± 4.23
65.01 ± 2.70
Auto
10
70.54 ± 2.24
73.02 ± 2.09
73.47 ± 1.66
73.67 ± 1.93
73.68 ± 1.90
69.18 ± 5.38
70.02 ± 2.86
73.89 ± 1.59
14
73.80 ± 4.06
75.28 ± 3.03
75.66 ± 3.14
75.63 ± 3.18
75.62 ± 3.17
71.16 ± 7.51
74.16 ± 4.13
75.93 ± 3.60
Credit
8
61.25 ± 5.80
67.98 ± 2.44
68.72 ± 1.04
68.55 ± 0.84
68.55 ± 0.82
59.91 ± 5.39
62.85 ± 6.82
69.50 ± 3.05
10
64.90 ± 4.28
68.78 ± 1.12
69.25 ± 2.19
69.31 ± 2.02
69.22 ± 2.01
61.88 ± 3.92
63.82 ± 6.59
72.42 ± 2.79
Wine
7
66.88 ± 4.79
77.58 ± 6.10
87.13 ± 3.17
88.41 ± 2.64
72.10 ± 3.67
62.80 ± 3.62
76.56 ± 6.17
89.81 ± 2.06
8
70.52 ± 5.39
82.8 ± 1.93
89.22 ± 4.17
90.00 ± 3.00
77.79 ± 3.96
66.88 ± 7.01
80.91 ± 6.17
91.04 ± 2.57
Heart
6
58.76 ± 5.69
61.94 ± 5.16
64.65 ± 2.89
64.42 ± 2.77
64.42 ± 2.77
61.24 ± 2.2
63.64 ± 8.63
67.13 ± 4.04
8
63.54 ± 4.11
70.87 ± 3.32
71.42 ± 2.65
71.42 ± 2.65
71.42 ± 2.65
69.53 ± 5.49
66.14 ± 4.40
73.23 ± 2.83
SCredit
35
63.10 ± 2.77
71.97 ± 2.87
72.00 ± 2.77
72.03 ± 2.71
72.23 ± 2.77
57.58 ± 3.35
63.74 ± 4.81
75.84 ± 3.81
50
66.07 ± 3.51
74.64 ± 2.83
74.41 ± 1.38
74.44 ± 1.34
74.47 ± 1.52
59.32 ± 2.60
65.53 ± 4.88
76.95 ± 4.32
Isolet
21
71.41 ± 3.02
71.38 ± 1.48
69.83 ± 0.94
69.47 ± 1.10
69.47 ± 1.10
66.36 ± 2.69
72.89 ± 2.39
73.19 ± 2.50
26
78.10 ± 1.70
79.22 ± 2.02
77.80 ± 2.20
77.53 ± 2.29
77.55 ± 2.32
75.81 ± 3.01
79.50 ± 2.27
79.88 ± 2.28
Iono
15
83.49 ± 4.25
88.66 ± 3.88
87.54 ± 3.50
87.35 ± 3.39
87.35 ± 3.43
82.06 ± 3.84
81.99 ± 3.18
89.72 ± 4.29
20
86.37 ± 1.94
91.38 ± 1.27
88.49 ± 2.05
88.04 ± 1.55
87.97 ± 1.45
82.06 ± 1.45
83.09 ± 1.91
92.99 ± 1.12


## 4.4. Experiment 3

In this section, we conduct the experiments on IMM and Yale to compare the proposed DCCRC with the competing methods under the situations of the testing samples with the corruptions. In the experiments, the numbersl of the class-specific training samples are preset as l=3 on IMM and l=6 on Yale, and the remaining samples per class are regarded as the testing samples. And the classification results of each competing method are the averages of recognition accuracies on ten training and testing divisions of data. Moreover, the testing samples per class are randomly corrupted by randomly adding the pixels and the block occlusion with a panda. That is to say, the pixel corruptions are that some pixels of each testing image are randomly replaced by the uncertain gray scale values between 0 and 255, and some part of each testing image is randomly occluded by the panda. The ratios of the corrupted size to the original size of each testing image are from 0.1 to 0.4 with a step 0.1. As an example, the testing samples with the random pixels from one class are shown in Figure 5 and with the random occlusions shown in Figure 6.Figure 5
The testing images with the random pixel corruptions from one class. (a) IMM. (b) Yale.
(a)
(b)Figure 6
The testing images with the random block occlusions from one class. (a) IMM. (b) Yale.
(a)
(b)The classification results of the competing methods on IMM and Yale with varying ratios of the random corruptions are shown in Table5 for random occlusions and in Table 6 for random corrected pixels. Note that the best classification performance among all the methods in two tables is indicated in bold face. As listed in two tables, the classification accuracies of each competing method descend with the increase of the ratios of the corrupted size of each testing image. From these experimental results, we can see that the proposed DCCRC is nearly the most robust among all the competing methods, since it outperforms the other competing methods. Thus, the proposed method has more effectiveness and robustness under the situations of data with noises.Table 5
The classification accuracies (%) of the competing methods with the corresponding standard deviations under the situations of the random block occlusions.
Data
Ratios
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
IMM
0.1
62.17 ± 1.80
63.00 ± 1.00
65.17 ± 1.33
65.50 ± 0.85
62.33 ± 0.97
63.67 ± 1.35
62.67 ± 0.97
66.00 ± 0.82
0.2
54.33 ± 1.43
56.17 ± 1.80
59.50 ± 1.87
58.67 ± 1.63
55.50 ± 3.27
55.33 ± 1.55
57.50 ± 1.18
59.67 ± 1.45
0.3
45.67 ± 2.44
47.67 ± 1.62
50.00 ± 2.36
50.83 ± 2.47
48.17 ± 2.44
49.50 ± 2.51
49.83 ± 1.22
51.50 ± 2.26
0.4
37.33 ± 3.14
35.83 ± 1.90
40.50 ± 2.15
41.00 ± 2.44
38.33 ± 2.36
39.67 ± 1.72
39.33 ± 2.20
41.00 ± 2.44
Yale
0.1
82.67 ± 2.83
86.67 ± 2.11
86.50 ± 2.25
86.50 ± 1.94
85.50 ± 2.53
85.00 ± 3.18
84.17 ± 3.23
87.33 ± 2.31
0.2
76.00 ± 2.83
79.17 ± 3.46
79.00 ± 1.73
79.17 ± 1.76
78.67 ± 1.76
75.33 ± 1.15
75.00 ± 1.60
79.33 ± 1.89
0.3
64.83 ± 3.52
68.83 ± 2.49
70.33 ± 2.56
69.50 ± 3.01
69.17 ± 2.94
62.50 ± 3.23
65.50 ± 3.49
70.83 ± 1.94
0.4
53.07 ± 2.13
56.80 ± 1.36
57.87 ± 2.32
56.27 ± 4.49
56.53 ± 4.81
45.33 ± 2.53
50.67 ± 2.39
58.40 ± 2.29Table 6
The classification accuracies (%) of the competing methods with the corresponding standard deviations under the situations of the random pixel corruptions.
Data
Ratios
SRC
CRC
CCRC
ProCRC
EProCRC
Co-CRC
DSRC
DCCRC
IMM
0.1
65.17 ± 1.11
64.67 ± 0.67
65.33 ± 1.55
65.67 ± 1.86
63.17 ± 2.32
62.17 ± 2.39
63.50 ± 1.11
66.00 ± 2.07
0.2
58.67 ± 1.35
59.33 ± 1.33
58.17 ± 1.22
58.17 ± 1.22
57.33 ± 1.33
53.00 ± 1.63
57.17 ± 0.85
59.67 ± 1.80
0.3
51.33 ± 1.35
52.17 ± 1.55
49.33 ± 1.62
49.67 ± 1.80
48.67 ± 2.27
43.33 ± 2.74
48.172.07
53.17 ± 3.00
0.4
40.50 ± 3.64
40.50 ± 3.10
39.83 ± 2.26
40.00 ± 2.11
39.83 ± 2.66
32.50 ± 2.04
32.17 ± 3.60
40.83 ± 2.30
Yale
0.1
87.73 ± 2.29
89.87 ± 1.81
90.67 ± 1.69
90.67 ± 1.69
89.33 ± 1.89
90.93 ± 1.96
89.87 ± 2.17
90.93 ± 1.96
0.2
83.47 ± 1.81
85.87 ± 1.36
86.93 ± 1.31
87.47 ± 1.07
85.07 ± 1.00
86.13 ± 1.81
86.40 ± 1.77
88.00 ± 1.19
0.3
79.73 ± 4.25
80.27 ± 2.59
83.73 ± 2.72
83.20 ± 2.61
79.73 ± 1.96
83.20 ± 2.87
83.20 ± 3.44
84.53 ± 2.32
0.4
69.07 ± 2.44
74.67 ± 2.23
75.73 ± 1.55
74.67 ± 2.53
70.13 ± 3.22
72.00 ± 1.19
76.00 ± 1.89
76.27 ± 1.55


## 5. Conclusions

Collaborative representation-based classification is a typical technique for pattern recognition. To further improve pattern discrimination in collaborative representation, we design a new discriminative, competitive, and collaborative representation-based classification method (DCCRC) in this article. The proposed DCCRC extends the Co-CRC method and mainly designs a discriminative regularization of the collaborative representation from all the classes and the ones from all the classes excluding any one class. The proposed method fully utilizes the class-specific representations in collaborative representation and can competitively and discriminatively enhance the class-specific representations for good classification. The extensive experiments on some face databases and UCI data sets are conducted for verifying the effectiveness and robustness of the proposed DCCRC method. Through comparing DCCRC with the state-of-the-art representation-based classification methods, the proposed DCCRC outperforms the competing methods. Thus, the proposed DCCRC is an effective and robust classifier in pattern recognition. In the future work, we will employ the idea of the competitive and collaborative representation among all the classes into the other kinds of classifiers.



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*Source: 3251026-2019-12-07.xml*