Currently, low-rank representation has achieved good clustering results as a typical representation model for learning the subspace structure of gene expression data (Xia et al., 2018; Shi et al., 2019; Wang et al., 2019; Zheng et al., 2019; Lu et al., 2020; Sun et al., 2021; Huang and Wu, 2022). In this paper, the proposed SSCAC algorithm constructs a self-training subspace clustering framework based on the low-rank representation with distance penalty (LRRADP) (Fei et al., 2017) by using the high coordination between Gaussian fields and harmonic functions (GFHF) (Zhu et al., 2003) and low-rank representation.

In a semi-supervised learning framework, the dataset is usually formulated as X = [ x ₁, x ₂, …, x _l , x _l+1, …, x _n ] = [ X _L , X _U ] ∈ R ^m×n , where x i i = 1 l and x i i = l + 1 n are labeled and unlabeled data, X _L and X _U are the labeled and unlabeled datasets, c is the number of clusters, the corresponding label set is L _a = {1, …, c}, the label of datum x _i is y _i ∈ L _a . In order to make different newly labeled data act on semi-supervised gene clustering with different strengths, this paper introduces label confidence to GFHF semi-supervised clustering, and the proposed SSCAC objective function is: 1 2 ∑ i , j = 1 n F i − F j 2 W i j + λ ∞ ∑ i = 1 l F i − μ i Y i 2 (1) where F = F 1 T , … , F n T T ∈ R n × c is the label prediction matrix, vector F _i denotes the attribution of x _i to each cluster; Y = Y 1 T , … , Y n T T ∈ B n × c is the binary label indication matrix, vector Y _i corresponds to the label of x _i , if the label y _i = k then Y _ik = 1, otherwise Y _ik = 0; λ _∞ is a very large constant; μ _i ∈ (0, 1] is the label confidence of the labeled datum x _i (i = 1, 2, …, l); W _ij is the element of the LRRADP affinity matrix W obtained by: W = Z + Z T / 2 (2)

In LRRADP, Z ∈ R ^n×n is the low-rank representation matrix and Z _i is the vector of coefficients of datum x _i represented by other data; E ∈ R ^m×n is the noise matrix. The iterative update equations are as follows (Fei et al., 2017): Z p + 1 = arg m i n Z Z p * + β p 2 X − X Z p − E p + Λ 1 , p / β p 2 2 + ‖ Z p − H p + Λ 2 , p / β p ‖ 2 2 (3) E p + 1 = arg min E E p 1 + β p 2 X − X Z p − E p + Λ 1 , p / β p 2 2 (4) where ‖ Z ‖_* = ∑ _i σ _i ( Z ) is the nuclear norm of Z , which is used as a convex approximation of matrix rank, σ _i ( Z ) denotes the i-th singular value of Z ; ‖.‖₁ and ‖.‖₂ are the l ₁-norm and l ₂-norm, respectively; auxiliary variable H , Lagrange multipliers Λ ₁, Λ ₂ and penalty parameter β are determined by the following equations: H p + 1 = arg min H λ 2 t r Ξ D ⊗ H p + β p 2 Z p − H p + Λ 2 , p / β p 2 2 (5) Λ 1 , p + 1 = Λ 1 , p + β p X p + 1 − X p + 1 Z p + 1 − E p + 1 (6) Λ 2 , p + 1 = Λ 2 , p + β p Z p + 1 − H p + 1 (7) β p + 1 = min β max , ρ β p (8)

In the update equations, λ ₁ > 0 and λ ₂ > 0 are balance parameters to trade off among the low-rank representation, noise and adaptive distance penalty.

In the SSCAC objective function defined by Eq. 1, the first term is the same as that of the original GFHF, which ensures the smoothness of data labels on the LRRADP graph. The second term is the label fitness term, which incorporates the label confidence μ _i and applies it to the label indication vector Y _i of the labeled datum x _i . Actually, the objective function of GFHF is a special case of that of SSCAC with μ _i = 1 for each labeled datum x _i (i = 1, 2, …, l). That is, the SSCAC objective function is the extension of that of GHFH, which further considers the label confidences of the labeled data and can be applied to self-training clustering. Minimizing Eq. 1 can achieve both the manifold smoothness of the partition results in subspaces and the maximum matching between the predicted label and the label of labeled data under the effect of label confidence.

In the self-training process of SSCAC, newly labeled data are selected based on density-pointing relationships between data (Wu et al., 2018; Li et al., 2019) and added to the labeled dataset X _L to guide the next iteration of self-training learning. The newly labeled data selection strategy will be detailed in the next Subsection. The rules for setting the label confidence μ _i in Eq. 1 are as follows: 1) if x _i is an initially labeled datum, set the label confidence μ _i = 1 with complete confidence in its label accuracy; 2) if x _i is a newly labeled datum of the current iteration of self-training, μ _i is initialized to a random number within (0,1], and then adaptively adjusted based on the semi-supervised learning value of x _i . The specific strategy is detailed in Section 2.3; 3) only the label confidences of the newly labeled data selected in the current iteration are adjusted, the adjusted confidences remain unchanged in the subsequent iterations of self-training.

The advantage of adding the label confidence in Eq. 1 is that the value can effectively regulate the supervision strength of newly labeled data on semi-supervised gene clustering, which improves the clustering accuracy on gene expression data. The analysis is as follows: 1) if the newly labeled datum x _i is mislabeled, i.e., the position of 1 in the label indication vector Y _i does not match that of the actual label, the label prediction vector F _i will be predicted in the wrong direction under the effect of the second term of Eq. 1, and the larger the label confidence μ _i , the larger the prediction bias. In the first term of Eq. 1, the elements corresponding to data in the same subspace in the LRRADP similarity matrix W are relatively large and those corresponding to data in different subspaces are small, so that labels are mainly propagated among data in the same subspace, then the mislabeled datum x _i will lead to the label prediction bias of unlabeled data in the same gene clustering. Therefore, reducing the label confidence of mislabeled datum x _i can effectively mitigate its negative impact on semi-supervised gene clustering; 2) if the newly labeled datum x _i has correct label, the second term of Eq. 1 can guide F _i to obtain correct prediction, and then realize correct label propagation for unlabeled data in the same subspace under the effect of the first term of Eq. 1. Obviously, increasing the label confidence of correctly labeled datum is beneficial to improve the partition accuracy of unlabeled data.

The matrix form of the SSCAC objective function is: t r F T L F + t r F − μ ⊗ Y T U F − μ ⊗ Y (9) where L ∈ R ^n×n is the graph Laplacian matrix, L = D − W , D is a diagonal matrix, D _ii = ∑ _j W _i,j ; U ∈ R ^n×n is also a diagonal matrix, the first l and the remaining n − l diagonal elements are λ _∞ and 0, respectively; ⊗ denotes the Hadamard product; μ ∈ R ^n×c , if the label of x _i (i = 1, 2, …, l) is k(k = 1, 2, …, c), then the k-th element in the i-th row vector is the label confidence of x _i , and all the other elements in the row vector are 0. For each unlabeled datum, all elements in the corresponding row vector are set to 0. By setting the derivative of Eq. 9 with respect to F to zero, the following equation can be easily obtained: F = L + U − 1 U μ Y (10)

Then the predicted label of the unlabeled datum x _i can be assigned by: y ̂ i = arg max k F i k (11)

In the self-training process, how to select newly labeled data from the unlabeled dataset X _U and iteratively expand the labeled dataset X _L is an important issue. Most self-training learning methods (Nie et al., 2012; Xia et al., 2018) rely entirely on the performance of learning models and ignore the potential density information in datasets. Relatively speaking, the strategy based on the data density relationships is not restricted by the distribution of initially labeled data and entire data space (Wu et al., 2018), and is more suitable for self-training learning on non-spherically distributed data.

In the self-training process of SSCAC, newly labeled data are selected based on density-pointing relationships between data (Wu et al., 2018; Li et al., 2019). The strategy utilizes clustering by fast search and find of density peaks (DPC) (Rodriguez and Laio, 2014), and for each datum x _i , its local density ρ _i can be defined as: ρ i = ∑ j χ d i j − d c , χ x = 1 ,   x < 0 0 ,   x ≥ 0 (12) where d _ij is the Euclidean distance between x _i and x _j , and d _c is the cut-off distance. It can be seen that the value of local density ρ _i is the number of data whose distance from x _i is less than d _c . In addition, DPC defines the minimum distance between x _i and other data with higher local densities as follows: δ i = max j d i j , ∀ j ,   ρ i ≥ ρ j min j : ρ i < ρ j d i j ,  others (13)

The newly labeled data selection strategy calculates ρ _i and δ _i for each datum x _i and make x _i point to its nearest datum x _j with a higher local density, then x _j is called the “next” datum of x _i and x _i is the “previous” datum of x _j . Then, the strategy constructs the density-pointing relationships of low-density data to high-density data by selecting the “next” and “previous” unlabeled data of labeled data in batches and set their selection orders. Specifically, all the “next” data of data in the original labeled dataset X _L are firstly selected from the unlabeled dataset X _U , and their selection orders are set to 1. That is, these data are viewed as the ones that should be labeled in the first iteration of self-training and used as the newly labeled data to expand the labeled dataset. In the next iteration, all the “next” data of the newly labeled data of the previous iteration are selected from X _U , and their selection orders increase by 1. This step repeats until there exists no “next” data of the newly labeled data of the previous iteration in X _U . If there are still unselected data in X _U , the selection orders of these remaining data can be set according to the “previous” relationships using the similar process. It can be seen that the unlabeled data with the same selection orders form the newly labeled dataset of the same iteration of self-training, on which basis the proposed SSCAC algorithm can expand the labeled dataset X _L iteratively and realize self-training clustering.

According to the analysis of the SSCAC objective function in the previous subsection, it is obvious that the value of different newly labeled data should vary for semi-supervised learning. If the newly labeled datum x _i is mislabeled, its incorrect label will propagate to the unlabeled data in the same subspace, making these data together with x _i have significant differences in the label prediction vectors from those of the correctly labeled data in that subspace. In this case, Eq. 1 will inevitably result in a large function value, and x _i can be regarded as a low-value newly labeled datum. Conversely, the newly labeled datum x _i with correct label can propagate its correct label in the subspace it belongs to, so that the unlabeled data in this subspace will obtain similar label prediction vectors to those of the correctly labeled data. In this case, the objective function value of Eq. 1 will be relatively small, and x _i can be regarded as a high-value newly labeled datum. Therefore, the SSCAC algorithm proposed in this paper measures the semi-supervised learning value of newly labeled data by the objective function value of Eq. 1, and on this basis, achieves the adaptive adjustment of label confidence.

Gravitational search algorithm (GSA) (Rashedi et al., 2009) is an optimization method based on the law of gravity, which is easy to implement and requires fewer parameters. It has been proven in the literature that GSA outperforms heuristic search algorithms such as PSO and GA (Mirjalili et al., 2012; Kumar et al., 2013). The search particles in GSA are a set of individuals that attract each other and generate motion in the solution space, the position of the individual is the solution of the optimization problem. Under the influence of gravity, the individuals move toward the individuals with heavier masses, which correspond to better solutions. To distinguish from the iterations of self-training learning, the iteration index of GSA is referred to as time in this paper. In the r-th iteration of self-training, let I be the number of newly labeled data, X r = x 1 r , x 2 r , … , x i r , … , x I r be the set of newly labeled data, we use the label confidences of these newly labeled data to compose the label confidence vector. Specifically, the label confidence vector can be represented as the positions of particles when optimized by GSA, the position of GSA particle a at time t is defined by: μ a r t = μ 1 , a t , μ 2 , a t , … , μ i , a t , … , μ I , a t , a = 1,2 , … , N (14) where N is the population size, μ _i,a (t) is the label confidence of the i-th newly labeled datum x i r in particle a at time t, which is initialized to a random number within (0,1].

Based on the SSCAC objective function given in Eq. 9, the GSA fitness function of particle a at time t is defined as: fi t n e s s a t = t r F T L F + t r F − μ a r t ⊗ Y T U F − μ a r t ⊗ Y (15)

For the i-th newly labeled datum x i r , the force acting on particle a from particle b at time t is expressed as: f i , a b t = G t M a t × M b t R a b t + ε μ i , b t − μ i , a t (16) where G(t) is gravitational constant at time t, M _a (t) and M _b (t) are the inertial masses of particle a and particle b, R _ab (t) is the Euclidean distance between particle a and particle b, and ɛ is a tiny constant to avoid zero denominator. The following equation can be used to determine the gravitational constant G(t): G t = G 0 e − α t T (17) where G ₀ is the initial value of the gravitational coefficient, α is the decay coefficient, G ₀ and α are usually taken as 100 and 20 (Rashedi et al., 2009), and T is the maximum time.

During the motion of a particle, the inertial mass M _a (t) of particle a can be updated according to the adapted value: m a t = fi t n e s s a t − w o r s t t b e s t t − w o r s t t (18) M a t = m a t ∑ b = 1 N m b t (19) where m _a (t) is the intermediate variable, best(t) and worst(t) are the best and worst fitness values among all particles at time t, respectively. In this paper, the particle position that makes the fitness value Eq. 15 obtain the minimum value is selected as the label confidence of the newly labeled data. Here, best(t) and worst(t) are respectively given by: b e s t t = min b ∈ 1 , … , N fi t n e s s b t (20) w o r s t t = max b ∈ 1 , … , N fi t n e s s b t (21)

According to Newtonian gravity and the laws of motion, the gravitational force on particle a in the i-th dimension at time t is the sum of the gravitational forces from all other particles. f i , a t = ∑ b = 1 , b ≠ a N r a n d b f i , a b t (22) where rand _b is a random number within [0,1]. According to Newton’s second law, the acceleration of particle a in the i-th dimension is: s i , a t = f i , a t M a t (23)

Therefore, the velocity and position of particle a in the i-th dimension at the next time are updated by: v i , a t + 1 = r a n d a × v i , a t + s i , a t μ i , a t + 1 = μ i , a t + v i , a t + 1 (24) where rand _a is a random number within [0,1], the initial velocity v _i,a (0) is 0.

When time t reaches T, the position of the particle that obtains the minimum fitness value is used as the label confidence vector μ ^r for the newly labeled data X ^r at the r-th iteration of self-training. Then, we can update μ , U and Y in Eq. 9 based on the obtained label confidence vector μ ^r , the newly labeled data X ^r , their predicted labels respectively, and guide the subsequent iterations of self-training. It can be seen that the proposed strategy can adaptively adjust the label confidence based on the semi-supervised learning value of the newly labeled data. By reducing the label confidences of low-value newly labeled data, we can effectively reduce their effect on semi-supervised learning and thus alleviate the problem of mistaken reinforcement in the self-training gene clustering.

For a set of gene expression data X = [ x ₁, x ₂, …, x _l , x _l+1, …, x _n ] = [ X _L , X _U ] ∈ R ^m×n , the detailed procedure of SSCAC is given in Algorithm 1, and the framework of SSCAC is shown in Figure 1. In SSCAC, the stopping condition is set to X _U = ∅ or the clustering accuracy no longer increases as suggested in the literature (Qu et al., 2019).

Algorithm 1

Note that when the stopping condition is that the clustering accuracy no longer increases, the labels of the remaining data in X _U are obtained based on F .

Step 1: Set the parameters, including maximum value of penalty parameter β _max , iteration stop parameter ξ, constant ρ, balance parameters λ ₁ and λ ₂ of the LRRADP algorithm, and population size N, maximum time T, constant ɛ of the GSA algorithm.

In this paper, comparative experiments are conducted in two benchmark gene expression datasets, as shown in Table 1. The Gal dataset (Ideker et al., 2001) is composed of gene expression measurements for 205 genes involved in galactose use in Saccharomyces cerevisiae. The gene expression profiles were measured with four replicate assays across 20 time points and the expression patterns reflect four functional categories. Yeast is a UCI dataset, which aims to predict the localization sites of proteins in cells and contains 1,484 yeast genes with eight methods of predicting protein localization sites in dimensions. Besides, we also demonstrate the applications of the proposed algorithm in other datasets, details of the datasets are tabulated in Supplementary Table S1, and the clustering results can be seen in Supplementary Tables S2, S3.

To verify the effectiveness of the SSCAC algorithm proposed in this paper for gene expression data, SSCAC is compared with three unsupervised clustering algorithms and four semi-supervised learning algorithms, including the K-means clustering based on the original gene expression data X , the K-means clustering based on the low-rank representation matrix Z (LRR + Kmeans) (Xia et al., 2018), the NCut clustering based on the LRR similarity matrix W (LRR + NCut) (Liu et al., 2013), SSC-LRR (Xia et al., 2018), STDP (Wu et al., 2018), STDPNF (Li et al., 2019), and LRRADP + GFHF (Fei et al., 2017) algorithms, where SSC-LRR, STDP, and STDPNF are self-training methods. To illustrate the effectiveness of the filter in the STDPNF algorithm, both STDP and STDPNF use KNN as the base classifier. Based on the suggestion of the literature (Xia et al., 2018), the balance parameter λ in LRR and SSC-LRR algorithms is tuned within [2⁻³, 2⁴], and the parameter value corresponding to the optimal clustering result is selected, so we set λ = 0.1 for all the datasets. In LRRADP + GFHF and SSCAC, we set the balance parameters λ ₁ = 100, λ ₂ = 1 and λ _∞ = 1 × 10⁵, and the maximum value of penalty parameter β _max = 10⁴, iteration stop parameter ξ = 10⁻⁵, constant ρ = 1.01. And we set the maximum time of the adaptive adjustment of label confidence T = 100, population size N = 50, and constant ɛ = 2.2204e − 16 in SSCAC, the cut-off distance d _c is the corresponding value of data distance sorted in ascending order of 2%, and the other parameters in comparison methods are set as suggested in the original studies. Similar to literature (Nie et al., 2012; Fei et al., 2017), the experiments in this paper form the initial labeled dataset X _L by randomly selecting 10% of the data in each dataset, and the rest of the data form the unlabeled dataset X _U . All algorithms are run 10 times with randomly selected initial labeled data, and the algorithm performance is evaluated using the mean value of the results.

To assess the partition performance, we use two popular metrics, accuracy (ACC) and Normalized mutual information (NMI).

(1) ACC is calculated by A C C = ∑ i = 1 n δ y i , m a p y ̂ i n (25) where y _i and y ̂ i denote the true label and predicted label of x _i , respectively, m a p y ̂ i denotes the mapping match between the true label and the predicted label, and δ y i , m a p y ̂ i = 1 when y i = m a p y ̂ i , otherwise, it is 0. The closer the value of ACC is to 1, the higher the partition accuracy is.

(2) NMI is calculated by N M I = 2 I A , B H A + H B (26) where A and B denote the vectors consisting of the true and predicted labels corresponding to the partition results, respectively. I(A, B) denotes the mutual information measure, H(A) and H(B) denote the entropy of A and B, respectively. The value of NMI is between 0 and 1, and a larger value of NMI indicates a better partition performance.

Table 2 shows the ACC and NMI results of eight algorithms on two benchmark gene expression datasets. The optimal and suboptimal results are marked with bold and italics, respectively.

From the results in Table 2, it can be seen that.

(1) For the two benchmark gene expression datasets, the clustering results of the SSCAC algorithm proposed in this paper are significantly better than those of the comparison algorithms, indicating the effectiveness of the proposed self-training subspace clustering framework and the adaptive adjustment strategy of label confidence. In addition, the performance of the semi-supervised learning methods outperforms that of the unsupervised clustering algorithms in general, reflecting the advantages of the semi-supervised learning methods.

In order to verify the convergence of the proposed updating strategy of confidence vector in SSCAC, convergence analysis experiments regarding the number of iterations versus fitness value are done for two benchmark gene expression datasets, Gal and Yeast. As shown in Figure 2, the fitness values flatten out with increasing iteration number and finally converge in approximately 100 iterations. Then, the position of the particle that obtains the minimum fitness value is used as the label confidence vector for the newly labeled data, on which basis SSCAC yields superior clustering results.

In order to validate the effectiveness of label confidence, we also conduct an ablation study. The ablation algorithm is referred to as SSCNAC, i.e., SSCAC without label confidence. In SSCNAC, the same label confidence μ _i = 1 is implicitly set for each newly labeled datum x _i in the self-training process, thus SSCNAC is a self-training subspace clustering algorithm based on original GFHF. The parameter setting of SSCNAC is the same as that of SSCAC, and the performance of SSCNAC and SSCAC in terms of ACC and NMI is reported in Table 3. The optimal values of Table 3 are shown in bold. From Table 3, it can be seen that the proposed SSCAC algorithm achieves better clustering performance over SSCNAC. As with other self-training algorithms, SSCNAC performs self-training with complete confidence in the label accuracy of newly labeled data, and therefore suffers from the problem of mislabeling. Comparatively speaking, the proposed SSCAC algorithm introduces label confidences into the semi-supervised clustering objective function and adaptively adjusts them based on semi-supervised learning values, thus can effectively mitigate the negative impact of mislabeled data on self-training learning. This advantage of SSCAC will be further demonstrated in Table 4.

Moreover, from Tables 2, 3, we can also observe that the clustering results of SSCNAC outperform those of LRRADP + GFHF, with an average improvement of 1.25% and 4.75% in ACC and NMI, respectively. In essence, the SSCNAC algorithm with fixed-label confidence is a direct extension of LRRADP + GFHF on self-training, which gives SSCNAC the ability to learn from unlabeled data in self-training framework and therefore has better generalization performance. The above results demonstrate the positive role of unlabeled data in self-training learning and the effectiveness of the proposed self-training subspace clustering framework based on GFHF for gene expression data.

In the proposed SSCAC algorithm, λ ₁ and λ ₂ are balance parameters to trade off among the low-rank representation, noise and adaptive distance penalty. Figure 3 shows the impact of the two hyper-parameters on the performace of SSCAC. As can be observed, the proposed SSCAC algorithm is comparatively unaffected by hyper-parameters that are close to the ideal. To be more precise, we advise setting λ ₁ = 100 and λ ₂ = 1.

In order to analyze the impact of initially labeled data size on algorithm performance, we increase the initially labeled data ratio from 10% to 90% and conducted experiments, all algorithms are run 10 times. The average ACC curves of semi-supervised SSC-LRR, STDP, STDPNF, LRRADP + GFHF, SSCNAC and SSCAC algorithms are given in Figure 4.

It can be seen from Figure 4 that, in general, the partition accuracy of each algorithm increases along with the size of initially labeled data, the reason is that the increase in available label information helps to obtain models that fit the data more closely. It can also be seen from Figure 4 that in all cases, the ACC values of the proposed SSCAC algorithm are higher than those of the comparison algorithms, and this advantage of SSCAC is more evident in the cases when the initially labeled data ratios are relatively low. This is because that in such cases, the newly labeled data occupies a larger proportion of the labeled dataset and therefore plays a dominant role in the self-training process. Thus, the adaptive adjustment strategy of label confidence of SSCAC can reduce the influence of mislabeled data to a greater extent. As the proportion of initially labeled data increases, the dominant role of the newly labeled data in the self-training process decreases, and the semi-supervised learning performance of each algorithm tends to be similar. The above results and analysis indicate that the SSCAC algorithm proposed in this paper is more suitable for solving the semi-supervised clustering problem with less initially labeled data.

In this section, we discuss the contribution of each part of the proposed model. The SSCAC model described by Eq. 9 consists of two parts: t r F T L F and tr( F − μY ) ^T U ( F − μY ), which together make the model have high clustering accuracy. t r F T L F is the manifold smoothness term of the objective function, the LRRADP low-rank representation matrix Z adopted in SSCAC can effectively enhance the sparsity of the similarity matrix W and improve the discriminative property of gene expression data, which can then improve the clustering accuracy through the graph Laplacian matrix L . To illustrate the advantage of the LRRADP low-rank representation, visualization of the original data matrix and the low-rank representation matrixs of LRR and LRRADP are plotted on the Gal dataset, as shown in Figure 5, and the data in each subplot are sorted according to their cluster labels in an ascending order. As seen from Figure 5, the low-rank representation matrix Z in both Figures 5B, C has a block-diagonal structure, i.e., the four high pixel rectangles along the diagonal of Z correspond to the four Gal gene clusters, respectively. It is obvious that compared with the original data matrix X , the low-rank representation matrix Z can better reveal the subspace structure of gene expression data, i.e., the block-diagonal structure. Comparing Figures 5B, C, it can be seen that since LRRADP considers the locality of gene expression data while focusing on the global low-rank constraint, the resulting low-rank representation matrix Z is more sparse and the diagonal-block structure is more obvious, and thus can provide more discriminative information for SSCAC.

On the other hand, the second term of the SSCAC model, tr( F − μY ) ^T U ( F − μY ), incorporates the label confidence μ with the aim to reduce the label confidences of mislabeled data through the proposed adaptive adjustment strategy of label confidence, and mitigate their negative impact during the self-training iterations. In order to verify the effectiveness of the adaptive adjustment strategy of label confidence in the SSCAC model, we focus on the newly labeled data, as well as their real labels, predicted labels, and label confidences during the self-training process. In our experiments, all algorithms are run 10 times with randomly selected initial labeled data. Thus, the newly labeled data selected during the iteration of SSCAC are different for different initial labeled data. Here, we take one case of random selection of initial labeled data on GAL as an example, where SSCAC achieves convergence in nine iterations. The newly labeled data selected in the last three iterations and their label confidences are reported in Table 4, and similar results can be obtained for other iterations.

As seen from Table 4, the adaptive adjustment strategy proposed in this paper can effectively reduce the label confidences of the mislabeled data, such as x ₉₂ and x ₁₀₈ in the 7-th iteration, and assign large confidences to the correctly labeled data. From Table 4, we can also observe that the label confidence of the correctly labeled datum x ₂₃ is rather small. As pointed out in the literature (Chen et al., 2011), even though some datum has correct label, it may have less impact on supervised learning due to its low partition uncertainty. Therefore, it is reasonable to assign a lower confidence to such correctly labeled datum. Compared with the existing self-training methods that do not consider the label confidence of newly labeled data, SSCAC can adaptively adjust the strength of supervisory guidance for different newly labeled data in the self-training process and effectively mitigate the negative impact of mislabeled data, which helps to significantly improve the clustering accuracy on gene expression data.