A P-Norm Robust Feature Extraction Method for Identifying Differentially Expressed Genes

In current molecular biology, it becomes more and more important to identify differentially expressed genes closely correlated with a key biological process from gene expression data. In this paper, based on the Schatten p-norm and Lp-norm, a novel p-norm robust feature extraction method is proposed to identify the differentially expressed genes. In our method, the Schatten p-norm is used as the regularization function to obtain a low-rank matrix and the Lp-norm is taken as the error function to improve the robustness to outliers in the gene expression data. The results on simulation data show that our method can obtain higher identification accuracies than the competitive methods. Numerous experiments on real gene expression data sets demonstrate that our method can identify more differentially expressed genes than the others. Moreover, we confirmed that the identified genes are closely correlated with the corresponding gene expression data.


Introduction
With the development of DNA microarray technology, it is possible for biologists to monitor the expression of thousands of genes simultaneously [1,2]. Besides, these genes have been detected more comprehensively than ever before. A great challenge of the current bioinformatics is to explain the microarray gene expression data to gain insight into biological processes. A large number of studies have been reported to identify the characteristic genes from gene expression data. Feature extraction is a typical application of gene expression data.
A prominent feature of gene expression data is that the number of samples is far less than the number of genes. Generally speaking, on each experiment, gene expression data always contain thousands or even more than 10,000 genes, while the number of samples is generally less than 100. Statistically, it is called the small-sample-size problem, which makes many feature extraction methods lose effectiveness. The number of genes in expression data is so huge that it is quite difficult to analyze the gene expression data. Fortunately, opposed to the whole as the regularization function to make W be a low-rank matrix which can solve the small-sample-size problem in gene expression data. Eventually, the differentially expressed genes can be identified according to the optimized W. The briefly introduction of PRFE is as follows: Firstly, the gene expression data matrix X is decomposed into two matrices W (the product of U and D) and V T by using SVD. Secondly, the L p -norm is applied to solve the optimization problem: kWÀXVk p L p L , and the Schatten p-norm is used to approximate the rank of W: kWk p S p S . Thirdly, the differentially expressed genes are identified according to the optimized matrix W. Finally, the identified genes are appraised using the Gene Ontology tool.
To evaluate the validity of our method, both simulation data and real gene expression data sets are handled by PRFE method in the experiments. By comparing PMD and SPCA methods, all empirical results show that the novel method outperforms the competitive methods for identifying differentially expressed genes.
In summary, the main contributions of this paper are as follows: -On one hand, based on the Schatten p-norm and L p -norm, for the first time it proposes a novel idea and method PRFE for identifying differentially expressed genes.
-On the other hand, extensive experiments are conducted on gene identification.
The remainder of the paper is structured as follows. Section 2 shows the methodology of PRFE. Then how to identify differentially expressed genes using PRFE is introduced. The experimental results on simulation data and real gene expression data sets are presented in Section 3. In Section 4, the conclusion is shown.

Definitions of Lp-norm and Schatten p-norm
For a matrix W contains m rows and n columns, the L p -norm (0 < p L < 1) to the power p L can be defined as where w ij is the i-th row and j-th column element of W.
The extended Schatten p-norm (0 < p S < 1) of the matrix W to the power p S can be written as where σ i is the i-th singular value of W. When p S = 1, the Schatten 1-norm is also known as the nuclear norm or trace norm, which is usually taken as the following form: kWk Ã . When p S = 0, if we define 0 0 = 0, Eq 2 is the rank of W [10].

The definition of PRFE
Denote by X an m×n matrix, each row of X represents the expression level of a gene in n samples, and each column of X represents the expression level of all the m genes in one sample. As mentioned above, for gene expression researches, the gene number m is much larger than the sample number n. The PRFE method decomposes the matrix X into two matrices W (the product of U and D) and V T by using SVD where W is an m×K matrix and V T is a K×n matrix, VV T = I n . The general feature extraction minimization problem [7,8] is defined as follows: where k•k F is the Frobenius norm. The differentially expressed genes are usually identified according to W [8,15], so the Eq 4 can be easily converted to the following form: which can make it more convenient to optimize W. To improve the robustness to outliers in gene expression data, we use the L p -norm (0 < p L 1) to obtain an optimized W: When p S ! 0, relative to the trace norm kWkÃ, Schatten p-norm kWk p S p S will approximate the rank of W [16], hence, we replace the kWkÃ by Schatten p-norm (0 p S 1) kWk p S p S . Finally, the PRFE method can be used to solve the feature extraction problem as follows: where λ is the regularization parameter.

Solving the PRFE problem
Eq 7 is intractable since the two items are non smooth. Therefore, the Augmented Largrangian Multiplier (AML) method [17][18][19] is taken to solve Eq 7. In this subsection, we first introduce the AML method briefly. For a matrix A, the constrained optimization problem can be written as Suppose that the matrix B satisfies the condition that B = A, then the AML algorithm to solve Eq 8 is described as follows: Algorithm 1. AML algorithm to solve Eq 8 Set 1 < η < 2. Initialize Ω and φ > 0. while not converge do To facilitate the writing, in Eq 7 we replace the W − XV with C and replace W with D. According to AML algorithm, Eq 7 can be rewritten as follows: In Eq 9, there are three variables W, C and D which make the formula quite difficult to be solved. The alternating direction method [20] can be utilized to deal with this thorny problem exactly. The core idea to resolve Eq 9 is the case that the problem is optimized only by one variable when fixing the remaining two variables. In this way, three new but solvable problems arise.
Problem 1: When fixing W and D, Eq 9 can be written as the following form: In this case, W À XV À 1 φ O can be denote as a constant e. And note that the elements in W can be decoupled, so for each element, only the following problem need to be solved: where τ denotes 1 φ . Then we denote f(w) as the objective function in Eq 11: In Eq 12, there is only one variable w, and the convexity of the equation can be easily analyzed. When w = 0, f(w) is not differentiable, so we only consider the case of w 6 ¼ 0 in the following analysis. Then we compare the minimal solution to f(w) (w 6 ¼ 0) with f ð0Þ ¼ 1 2 e 2 to obtain the optimum solution to f(w). When w 6 ¼ 0, the first, second and third derivatives of f(w) are as follows: where sgn(w) is defined as follows: sgn(w) = 1 if w > 0, and sgn(w) = −1 if w < 0. The local minimum of f(w) can be obtained by finding the root of f 0 (w) = 0, so we analysis f 0 (w) at first. According to Eq 15, f 0 (w) is convex at w > 0 and f 0 (w) is concave at w < 0. In order to find the extrema of f 0 (w), we let f@(w) = 0 and obtain the solution: In this case, we denote a constant a (a > 0) as w (w > 0), that is f@(a) = 0 and f@(−a) = 0. Therefore, f 0 (w) can obtain the maximum f@(−a) at w < 0, and f 0 (w) can obtain the minimum f@(a) at w > 0. There are three cases to solve f(w): (a) f 0 (a) ! 0 and f 0 (−a) 0 In this case, f 0 (w) 0 when w < 0 and f 0 (w) ! 0 when w > 0, so the minimal solution to f (w) is w = 0.
(b) f 0 (−a) > 0 In this case, according to Eq 13, f 0 (a) > 0. f 0 (w) ! 0 when w > 0 and w < 0, f 0 (w) = 0 has two roots which indicate that f(w) is convex at w < −a and f(w) is concave at -a w < 0. So the minimal solution to f(w) is the root of f 0 (w) = 0 at e < w < −a.
(c) f 0 (a) < 0 In this case, according to Eq 13, f 0 (−a) < 0. f 0 (w) 0 when w < 0 and w > 0, f 0 (w) = 0 has two roots which indicate that f(w) is convex at w > a and f(w) is concave at 0 w < a. So the minimal solution to f(w) is the root of f 0 (w) = 0 at a < w < e.
In summary, the Eq 11 can be optimized by where w 1 2 (e, −a) and w 2 2 (a, e) are the roots of f 0 (w) = 0. The roots can be acquired by the Newton method initialized at e [10].
Problem 2: When fixing W and C, Eq 9 can be written as: In this case, W þ 1 φ C can be denoted as E. For simplicity, Eq 18 can be rewritten as follows: where ρ denotes l φ . Suppose D and E are decomposed into UΔV T and QSR T , respectively, where Δ and S are the singular value matrices. So Eq 19 can be written as To obtain the solution of Eq 20, we first introduce the theorem: For any two matrices A; B 2 R mÂn , then tr(A T B) tr(σ(A) T σ(B)), where σ(A) and σ(B) are the singular value matrices of A and B, respectively. According to the theorem, we have the following formula When U = Q and V T = R T , the equality holds in Eq 21, so the optimal problem in Eq 20 can be converted as the following form Suppose σ i and δ i are the i-th singular values of D and E, respectively, then Eq 22 can be written as The form of Eq 23 is the same as Eq 11, so the optimal solution to Eq 23 can be obtained in the same way with the optimal solution of Eq 11. Problem 3: When fixing C and D, Eq 9 can be written as: Denote Eq 24 can be simplified to the following form: The problem in Eq 25 is equivalent to solving a quadratic function and it is easy to obtain the solution In summary, the brief algorithm of PRFE is shown as follows.
Algorithm 2. PRFE method Input: Data matrix: X 2 R mÂn Schatten p-norm value: p S L p -norm value: p L Regularization parameter: λ Output: Optimized matrix W 2 R mÂn The data matrix X is decomposed into W and V T by SVD, where W is the product of U and D. W ( XV. Solve the problem in Eq 7 by AML method. Set 1 < η < 2. Initialize C, D, Ω, Ψ and φ > 0. while not converge do Update C by the optimal solution to Eq 10 Update D by the optimal solution to Eq 18 Update W by Eq 26

Identifying differentially expressed genes by PRFE
The gene expression data can be denoted as a matrix X of size m×n, each row of X represents the expression level of a gene in n samples, and each column of X represents the expression level of all the m genes in one sample. Fig 1 shows the graphical depiction of processing the matrix X by PRFE. Following the convention in [15], the PRFE method decomposes the matrix X into two matrices W and V T , where s j (j = 1,2,Á Á Á, n) is the sample expression profile, g i is the gene transcriptional responses, w k is an eigensample of column of W, v k is an engienpattern of row of V T , v T j is the j-th column of V T and contains the coordinates of the j-th sample in X. To identify differentially expressed genes from X, the critical information of the differentially expressed genes in s j needs to be studied. Since v T j includes the positional information of the j-th sample in X, according to the formula the important information of differentially expressed genes in s j can be captured by w k . That is, the differentially expressed genes are identified based on W.
After W has been optimized by PRFE method, a novelW can be obtained. Therefore, according toW, the differentially expressed genes are identified.W can be denoted as follows: : ð28Þ Following the description in [21], the differentially expressed genes are usually grouped into two classes: up-regulated genes and down-regulated genes, which can be reflected by the positive items and negative items inW. Here, we only consider the absolute value of the items iñ W to identify the differentially expressed genes. Then, the matrix is summed by rows to obtain the evaluating vector Ŵ [22] Generally speaking, the larger the item in Ŵ is, the more differential the gene is. Therefore, we sort the elements in Ŵ in descending order and take the top h (h ( m is a number that can be defined according to the corresponding requirement) genes as the differentially expressed ones.

Discussion of the selection of p value in PRFE
In Eq 7, the values of p L and p S in PRFE method are specified within 0 < p L 1 and 0 p S 1, respectively. However, the special values of p L and p S are more interesting to be selected for solving the problem in Eq 7.
To improve the robustness to outliers in gene expression data, the L p -norm is taken as the error function. In PRFE, the value of p L should be in the range of (0, 1], and it does not mean that the smaller value of p L can acquire the better performance. Conversely, we suggest taking the L 1 -norm to improve the robustness to outliers since the error function is convex while p L = 1 [10]. The Schatten p-norm is used as the regularization function to obtain a low-rank matrix. As mentioned in Subsection 2.1 and 2.2, when p S ! 0, the Schatten p-norm approximates the rank function. That is, the Schatten p-norm can achieve more accurate to approximate rank in the range of [0, 1). However, in this case the Schatten p-norm is not convex. When p S = 1, the Schatten p-norm is convex while it cannot achieve accurate to approximate rank [10]. Thus, we have the flexibility to choose different p S values corresponding to the different situations.
In order to verify the reasonableness of the values of p L and p S , the simulation experiments are given in Subsection 3.1.

Results and Discussion
This section shows the experimental results on simulation data and real gene expression data sets. For simplicity, the regularization parameter λ in Eq 7 is taken as 1 in whole experiments [10]. To demonstrate the effectiveness of our method for recognizing the differentially expressed genes, the PMD [7], SPCA [6], CIPMD [9] and SVM-RFE [23]are used for comparison.

Results on simulation data
3.1.1 Data source. We describe here a general scheme to generate simulation data. Suppose we want to generate data from R p such that the q (q < p) leading eigenvectors of the covariance matrix S are sparse. Denote the first q eigenvectors as v 1 ,Á Á Á, v q , which are specified to be sparse and orthonormal. The remaining p − q eigenvectors are not specified to be sparse. Denote the positive eigenvalues of S in decreasing order as c 1 ,Á Á Á,c p .
We first need to generate the other p − q orthonormal eigenvectors of S. To this end, form a full-rank matrix is not of full-rank for one random draw, we can draw another set of vectors. Then we apply the Gram-Schmidt orthogonalization method to V Ã to obtain an orthogonal , which is actually the matrix Q from the QR decomposition of V Ã . Given the orthogonal matrix V, we form the covariance matrix S using the following eigen decomposition expression The first q eigenvectors of S are the pre-specified sparse vectors v 1 ,Á Á Á, v q . To generate data from the covariance matrix S, let S be a random draw from N(0, I p ) and X ¼ VC = , then cov(X) = S, as described in [24].
3.1.2 Simulation results. In order to evaluate the performance of five methods, the experiment is repeated for 30 times and the average identification accuracies are reported. For fair comparison, 500 genes are identified by the five methods with their unique parameters. Since PMD, CIPMD and SPCA are sparse methods, α 1 , α 2 and γ are the control-sparsity parameters of PMD, CIPMD and SPCA, respectively. Because p L and p S are the most important parameters of PRFE method, their impacts on our method should be studied at first. According to Subsection 2.5, we suggest taking p L = 1 to improve the robustness to outliers and taking p S = 0 to approximate the rank function. Therefore, when p L = 1, we test the performance of our model with different values of p S in the range of {0, 0.1,Á Á Á, 1} and define this special case as PRFE p L = 1. Similarly, when p S = 0, we investigate the performance of our method with different values of p L in the range of {0.1, 0.2,Á Á Á, 1} and define this special case as PRFE p S = 0. Fig 2 shows the average identification accuracies of the five methods with different parameters while the simulation data are 3000×10. In Fig 2, it can be clearly seen that either PRFE p L = 1 or PRFE p S = 0 is superior to the other four methods in spite of PMD, SPCA, CIPMD and SVM-RFE can also reach higher identification accuracies. This result clearly justifies the serviceability of the PRFE method to introduce p L -norm and p S -norm in gene identification. To be precise, while the parameters are larger than 0.4, PMD and CIPMD reach their highest point and becomes stable. The accuracies of SPCA is monotonically decreasing when the parameter are larger than 0.1. Due to SVM-RFE is not a sparse method, so it is not sensitive to the parameters. The accuracies of PRFE p L = 1 is also monotonically decreasing in all of the parameters, this verifies the Schatten p-norm can achieve more accurate to approximate rank when p S ! 0. The accuracies of PRFE p S = 0 is increasing with the increasing parameters which can demonstrate that L p -norm, as the error function, can acquire a better performance when p L = 1. The number of samples in gene expression data has an influence on the identification accuracy when we recognize differentially expressed genes using feature extraction methods. The five methods are tested with different sample numbers to find the regular pattern that how the sample numbers affect the identification accuracy. Here, for the PRFE method, we select p L = 1 and p S = 0 since PRFE can obtain the best result in this case. PMD and CIPMD can reach its highest point and becomes stable when the parameters are larger than 0.4, so we select 0.4 as the sparse parameter for PMD and CIPMD. For SPCA, we choose 0.1 as the sparse parameter since SPCA can acquire its best result when parameter is 0.1. Fig 3 shows the average identification accuracies of different methods with different sample numbers. It is obvious to be seen that with the increasing of sample numbers, the accuracies of the four methods are increased. The accuracies of SVM-RFE method is monotonically decreasing. The proposed method can dominate the other methods in all the sample numbers. Moreover, the accuracies of our method are close to 100% when n ! 60.
To further investigate the performance of the methods, the average receiver operator characteristic (ROC) curve is shown in Fig 4 with the optimal parameter of different methods. Fig 4 shows that PRFE and the competitive methods can identify differentially expressed genes effectively. However, through the True Positive Rate and False Positive Rate we can find that PRFE have the best outcome. Since we add a noise matrix into simulation data, so the false positive and false negative appear. The area under curve (AUC) statistics are listed in Table 1 with the optimal parameter of different methods. From Table 1 we can conclude that the ascending order of accuracy given by the five methods is: SPCA, PMD, SVM-RFE, CIPMD and PRFE.

Results on gene expression data sets
To evaluate the proposed method, two publically available gene expression data are adopted: gene expression data of plants responding to abiotic stresses [25,26] and the leukemia data set [27]. To compare with PRFE method, PMD, CIPMD, SVM-RFE and SPCA are also used to identify differentially expressed genes.
3.2.1 Parameters selection. As mentioned in Subsection 3.1, PRFE method can reach the best performance when p L = 1 and p S = 0. Therefore, for PRFE method, we take p L = 1 and p S = 0 to identify the differentially expressed genes on real gene expression data. PMD, CIPMD and SPCA are parse methods, whose sparse parameters have an enormous influence on the identification accuracy. According to the results on simulation data in Subsection 3.1, by choosing the sparse parameters α 1 , α 2 and γ appropriately, PMD, CIPMD and SPCA can obtain their optimal performance respectively.  Gene expression data of plants responding to abiotic stresses include two classes: roots and shoots in each stress. The Affymetix CEL files were downloaded from NASCArrays [http:// affy.arabidopsis.info/link_to_iplant.shtml] [25], reference numbers are: control, NASCArrays-137; cold stress, NASCArrays-138; osmotic stress, NASCArrays-139; salt stress, NASCArrays-140; drought stress, NASCArrays-141; UV-B light stress, NASCArrays-144; and heat stress, NASCArrays-146. There are 22810 genes in each sample and the sample number of each stress type in the raw data is listed in Table 2. The raw arrays are adjusted by using GC-RMA software [26] in order to avoid the background of optional noise and normalized by using quantile normalization. The GC-RMA results are gathered in a matrix to be processed by SPCA, PMD, CIPMD, SVM-RFE and PRFE.

Results on gene expression data of plants responding to abiotic stresses. (a) Data source and GO analysis
For fair comparison, 500 genes are identified by PMD, CIPMD and SPCA by choosing α 1 , α 2 and γ appropriately. SVM-RFE has no sparse parameters, the top 500 genes of SVM-RFE method are selected as the differentially expressed genes. And according to Subsection 2.4, the top 500 genes of PRFE method are selected as the differentially expressed genes. The identified genes are checked by Gene Ontology (GO) Term Enrichment tools which can be used to describe genes in the input or query set and to help discover what functions the genes may have in common [28]. GOTermFinder, as a web-based tool, can find the significant GO terms among plenty of genes and it is publicly available at http://go.princeton.edu/cgi-bin/ GOTermFinder [29]. Therefore, GOTermFinder offers some significant information for the biological explanation of high-throughput experiments. The threshold parameters are given as follows: maximum P-value is set to 0.01 and minimum number of gene products is set to 2. In the following, only the primary outcomes of GO Term Enrichment are shown. (b) Term responding to stress The numbers of genes and P-value of response to stress (GO:0006950) in root and shoot samples are given in Table 3.
From Table 3 we can see that all the five methods can identify the differentially expressed genes with higher sample frequency which can reflect the accuracy of the feature extraction method and lower P-value. PRFE, SPCA and PMD are unsupervised methods, so we first compare the three algorithms. In the 12 terms, there is only two of them (osmotic stress in shoot Table 4. Response to abiotic stimulus (GO:0009628). In this table, the response to abiotic stimulus on differentially expressed genes is shown, whose background frequency in TAIR is 2842/30322 (9.4%), where 2842/30322 represents having 2842 genes response to stimulus in whole 30322 genes. SF and PV represent the sample frequency and P-value, respectively. The sample frequency can reflect the identify accuracy of the diffenrent methods, e.g. 155, represents the method identifies 500 genes, in which there are 155 genes responding to abiotic stimulus. Root and shoot denote the root samples and shoot samples, respectively. samples and salt stress in root samples) that the proposed method is surpassed by PMD slightly. In the remaining 10 terms, PRFE method outperforms PMD and SPCA. Generally speaking, since supervised methods take the class labels into consideration, they usually have better performance than unsupervised methods. However, unsupervised methods have unique advantages than supervised methods. For example, when a data set has no class information, in this case the supervised methods are always helpless in analyzing the data set, but unsupervised methods like PMD, SPCA and PRFE can analyze the data without class labels effectively. Table 3 shows that PRFE outperforms CIPMD on drought stress in shoot samples, salt stress in root samples and UV-B stress in shoot samples. Furthermore, only on drought stress in shoot and root samples, osmotic stress in root samples, salt stress in shoot samples and UV-B stress in root samples SVM-RFE is superior to our method. (c) Term responding to abiotic stimulus Table 4 shows the gene numbers and P-value of response to abiotic stimulus (GO:0009628) in root and shoot samples.

Stress
As Table 4 lists, each of the five methods can acquire good performance when it is used to identify the differentially expressed genes responding to abiotic stimulus. We still analysis the unsupervised methods at first. The proposed method is superior to SPCA and PMD in 11 terms, only for the salt stress data set in root samples, PRFE method is dominated by PMD and SPCA. For the supervised methods, PRFE is superior to CIPMD on cold stress in the root and shoot samples, drought stress in the shoot samples and osmotic stress in the shoot samples. On cold stress in the shoot samples and drought stress in the shoot samples our method outperforms SVM-RFE. The leukemia data set consists of 27 cases of acute lymphoblastic leukemia (ALL) and 11 cases of acute myelogenous leukemia (AML) [27]. It is summarized by a 5000×38 matrix for further processed.  The protein encoded by this gene associates with class II major histocompatibility complex (MHC) and is an important chaperone that regulates antigen presentation for immune response. It also serves as cell surface receptor for the cytokine macrophage migration inhibitory factor (MIF) which, when bound to the encoded protein, initiates survival pathways and cell proliferation.

11
L19779_at HIST2H2AA4 Histone cluster 2, H2aa4. Histones are basic nuclear proteins that are responsible for the nucleosome structure of the chromosomal fiber in eukaryotes.

AFFX-HUMRGE/ M10098_5_at
SRP68 This gene encodes a subunit of the signal recognition particle (SRP). The SRP is a ribonucleoprotein complex that transports secreted and membrane proteins to the endoplasmic reticulum for processing.

AFFX-HUMRGE/ M10098_3_at
SRP68 This gene encodes a subunit of the signal recognition particle (SRP). The SRP is a ribonucleoprotein complex that transports secreted and membrane proteins to the endoplasmic reticulum for processing.
14 M91036_rna1_at HBG2 The gamma globin genes (HBG1 and HBG2) are normally expressed in the fetal liver, spleen and bone marrow. For fair comparison, 100 genes are identified by the five methods. The Gene Ontology (GO) enrichment of functional annotation of the identified genes by five methods is detected by ToppFun [30] which is publicly available at http://toppgene.cchmc.org/enrichment.jsp. Here, GO: Biological Process is the main objective to analysis. The P-value is set to 0.01 and number of gene limits is set to 2 by ToppFun.
(b) Terms relate to leukemia data Table 5 lists the top 10 closely related terms corresponding to different methods. From Table 5 it can be clearly found that PRFE method outperforms PMD and CIPMD in all 10 terms. Our method can identify the same number of genes as SPCA in the following three terms: defense response, regulation of immune system process and leukocyte activation. However, we have lower P-values than SPCA in these three terms. Though in the term: cell activation our method is surpassed by SPCA, PRFE outperforms SPCA in the remaining terms. SVM-RFE method performs best in all the five methods. But in the term: response to reactive oxygen species, only PRFE and PMD can identify differentially expressed genes, in addition, PRFE can identify more genes than PMD.
To further study the performance of the methods, a Venn diagram is shown in Fig 5. From Fig 5 we can see that both PRFE and SVM-RFE identify less 'unique' differentially expressed genes than PMD, SPCA and CIPMD. There are 17 genes shared by all the five methods. The detailed information of the 5 'unique' differentially expressed genes extracted by PRFE are shown in Table 6. From Table 6 we can see that the 5 'unique' differentially expressed genes extracted by PRFE and neglected by other methods are associated with leukemia. Therefore, this suggests that PRFE is an excellent method for identifying differentially expressed genes on leukemia data set.
(c) Genes correlate with leukemia data To further study the correlation between the identified genes and leukemia data, they are verified based on the literatures. For simplicity, the top 30 genes identified by PRFE are taken into consideration. Depending on [31], there are 50 genes most closely correlated with the leukemia data set distinction in the known samples. Among these 50 genes, 3 genes are contained in the top 30 genes identified by PRFE. The Affymetrix ID and Gene Symbol of 3 genes are given as follows: M13792_at (ADA), M69043_at (NFKBIA), Y00787_s_at (IL8). The article [31] was written by Golub et al. in 1999, at that time, only 50 genes were found to be associated LGALS1 The galectins are a family of beta-galactoside-binding proteins implicated in modulating cell-cell and cell-matrix interactions. This gene product may act as an autocrine negative growth factor that regulates cell proliferation.

26
X78992_at ZFP36L2 This gene is a member of the TIS11 family of early response genes. Family members are induced by various agonists such as the phorbol ester TPA and the polypeptide mitogen EGF.

27
X12671_rna1_at HNRNPA1 The protein encoded by this gene has two repeats of quasi-RRM domains that bind to RNAs. It is one of the most abundant core proteins of hnRNP complexes and it is localized to the nucleoplasm.

28
M33680_at CD81 The protein encoded by this gene is a member of the transmembrane 4 superfamily, also known as the tetraspanin family. Most of these members are cell-surface proteins that are characterized by the presence of four hydrophobic domains.

29
Y00787_s_at IL8 Gene expression profiling study of contribution of GM-CSF and IL-8 to the CD44-induced differentiation of acute monoblastic leukemia. with the leukemia data set. As time goes on, many other genes were found to be closely correlated with leukemia. According to [32], there are 210 genes is related to leukemia. All the 30 genes identified by our method can be found in [32]. The detailed information of the 30 genes are shown in Table 7.

Conclusion
In this paper, based on the Schatten p-norm and L p -norm, we propose a novel feature extraction method named as PRFE to identify differentially expressed genes in gene expression data sets. The method combined the Schatten p-norm and L p -norm to provide an effective way for gene identification. Numerous experiments on simulation data and real gene expression data sets demonstrate that the proposed method has a better performance than the other state-ofthe-art gene identification methods. Moreover, the identified genes are confirmed that they are closely correlated with the corresponding gene expression data.