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Together, these factors make LI-SimpleMKKM-MR significantly improved over other algorithms on the same dataset. In addition, due to time complexity and memory constraints, the effect of LMKKM on some datasets has not been shown. 4.5. Parameter Sensitivity of LI-SimpleMKKM-MR Optimization of ( 2) can be divided into 2 steps: optimizing or and fixing the other one. (i) Optimizing with is fixed, the problem of optimizing in ( 2) can be represented as follows: The optimization of of ( 3) can be easily solved by taking the first k eigenvalues of the matrix . (ii) Optimizing with is fixed, with the soft label matrix is fixed, the problem of optimizing in ( 2) can be represented as follows: According to the constraints, it can be easily solved by the Lagrange multiplier method [ 10]. 2.2. MKKM with Matrix-Induced Regularization Although localized SimpleMKKM shows excellent performance on MKC problems, we find that the correlation between the given kernels is not sufficiently considered providing an opportunity for improvement based on the listed problem statement. (i) The original method [ 21] makes the data stable by setting a larger weight in the gradient descent step and maintaining the summation and nonnegativity of the weights through the association with other weights. However, this idea only enhances the correlation between different view weights and and does not consider the relationship between view kernel matrices, especially between pairs. (ii) The original method is possible to select multikernel kernels with high correlation for clustering simultaneously. Repeated selection of similar information sources makes the algorithm redundant and has low diversity, leading to the low ratio of different kernel matrices’ effectiveness, ultimately affecting the accuracy of the clustering results. Compared with the original multiple kernel clustering, the proposed method optimizes kernel matrix weights by using gradient descent rather than coordinate descent, combined with localized sample alignment and kernel matrix induced regularization. This reduces the negative effects of forced alignment of long-distance samples and high redundancy and low complementarity of multiple kernel matrices.

We designed comparative experiments to study the influence of the setting of two hyperparameters, localized alignment, and matrix-induced regularization, on the clustering effect. According to equation ( 7), LI-SimpleMKKM-MR tunes the clustering performance by setting two hyperparameters and , referring to the regularization balance factor and the nearest neighbor ratio. Unlike the existing paradigm, SimpleMKKM adopts optimization [ 20]. However, it is extended to make full use of the information between local sample neighbors and optimization to enhance the clustering effect with a fusion algorithm called localized SimpleMKKM. The objective value of LI-SimpleMKKM can be represented as follows: where and with are the ith sample’s neighborhood mask matrices; that is, only the samples closest to the target sample will be aligned. This new method is hard to solve with a simple two-step alternating optimization convergence method. To solve this problem, LI-SimpleMKKM first optimizes by a method similar to MKKM and then converts the problem into a problem of finding the minimum with respect to . With proving the differentiability of the minimized formula, the gradient descent method can be used to optimize [ 21]. 3. Localized Simple Multiple Kernel K-Means with Matrix-Induced Regularization Here, means one soft label matrix, which is used to solve NP-hard problems caused by the direct use of hard allocation, which is also called the partition matrix. Moreover, means an identity matrix which is in size. Also, our proposed LI-SimpleMKKM-MR significantly outperforms the MKKM-MR algorithms by 3.6%, 3.8%, 4.7%, 7.5%, 3.3%, and 6.3% in terms of ACC on Flower17, Flower102, ProteinFold, DIGIT, Caltech-25 views, and Caltech-7 classes datasets. This result proves that utilizing the data’s local structure and optimization improves the clustering effect very well.

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Although the performance of clustering can be improved to some extent by aligning samples with closer samples, there is still room for further improvement of that algorithm. We experimented with the algorithm on 6 benchmark datasets and compared it with the other nine baseline algorithms that solve similar problems through four indicators: clustering accuracy (ACC), normalized mutual information (NMI), purity, and rand index. We find that LI-SimpleMKKM-MR outperforms other methods. This is the first work to fully consider and solve the correlation problem between the base kernels to the best of our knowledge. As for the latter term with the similar thought, it is also decreasing monotonically because is a PSD matrix, is not negative, and is bigger than 0; the second derivative of can be easy to be calculated bigger than 0 (since each element of is bigger than 0), so the latter term has the lower bound 0 and convex. At the same time, the whole equation ( 13) is monotonically decreasing and lower-bounded. 3.3. Computational Complexity Analysis On top of optimization, the clustering performance improves when the parameters are appropriately set by combining matrix-induced regularization and local alignment. 4.6. Convergence of LI-SimpleMKKM-MR We theoretically analyze the time complexity of the algorithm LI-SimpleMKKM-MR. We assume that n and m denote the number of samples and the number of base kernels. LI-SimpleMKKM-MR based on Algorithm 1 first computes a neighborhood mask matrix with computational complexity and then computes the regularization term with computational complexity . Therefore, the time complexity of LI-SimpleMKKM-MR is per iteration.

According to Liu et al. [ 21], the relative value of is only dependent on , , and , where u is the largest component of . Only the weights of different kernels are linked, indicating that the LI-SimpleMKKM algorithm is not fully considered the interaction of the kernels when optimizing the kernel weights. This motivates us to derive a regularization term which can measure the correlation between the base kernels to improve this shortcoming. 3.1. Formulation These employees are divided into departments, services, cadres, branches, units, Technical-Non Technical, Scientific Non-Scientific, Gazetted Non-Gazetted, Permanent-Temporary, Regular-casual/Contractual, recruited or outsourced, Agricultural Non-Agricultural, semi-self employed or Home Servants etc. The Private Sector employees are also divided in organized and unorganized sectors. The total workforce in this country is more than 40 Crores out of which 90% are belonging to SC/ST/OBC and Converted Religious Minorities.With the hyperparameter defined, we can regard as a whole, which is global kernel alignment and PSD [ 21]. For convenience, we let . We first prove the differentiability of ( 9), then calculate the gradient, and optimize by the gradient descent method. The first part of the objective function in ( 9) is as follows: Current use: Being the SI unit of length, the meter is used worldwide in many applications such as measuring distance, height, length, width, etc. The United States is one notable exception in that it largely uses US customary units such as yards, inches, feet, and miles instead of meters in everyday use. In addition to the localized SimpleMKKM with matrix-induced regularization, we tested nine other comparative algorithms from the other MKKM algorithms, including, average kernel k-means ( Avg-KKM), multiple kernel k-means ( MKKM) [ 10], localized multiple kernel k-mean ( LMKKM) [ 12], optimal neighborhood kernel clustering ( ONKC) [ 24], multiple kernel k-mean with matrix-induced regularization ( MKKM-MR) [ 14], multiple kernel clustering with local alignment maximization ( LKAM) [ 22], multiview clustering via late fusion alignment maximization ( LF-MVC) [ 25], simple multiple kernel k-means ( SimpleMKKM) [ 20], and localized SimpleMKKM ( LI-SimpleMKKM) [ 21].

For all the datasets, we set the number of clusters k according to the actual number of categories in the dataset. We engage 4 indicators: clustering accuracy (ACC), normalized mutual information (NMI), purity, and rand index to measure the effect of clustering. To reduce the harmful effects of randomness, we initialized and executed all algorithms fifty times (50×) to obtain the mean and variance of the experimental indicators. 4.4. Experimental Results The proposed localized SimpleMKKM with matrix-induced regularization significantly outperforms localized SimpleMKKM. For example, it outperforms the LI-SimpleMKKM algorithm by 1.8%, 0.1%, 3.1%, 0.3%, 0.6%, and 3.4% in terms of ACC on Flower17, Flower102, ProteinFold, DIGIT, Caltech-25 views, and Caltech-7 classes datasets. These results validate the effectiveness of enhancing the correlation between matrices. The implementations of the comparison algorithms are publicly available in the corresponding papers, and we directly apply them to our experiments without tuning. Among the previous algorithms, ONKC, MKKM-MR, LKAM, LF-MVC, and LI-SimpleMKKM need to adjust hyperparameters. Based on the published papers and actual experimental results, we show the best clustering results of the previous methods by tuning the hyperparameters on each dataset. 4.3. Experimental Settings The incorporation of use of the basic kernel better, thus improving clustering performance. Moreover, we can clearly see that if we set , equation ( 7) is a special case of LI-SimpleMKKM.

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Let be a set of n samples, and means mapping the features of the sample of the th view into a high-dimensional Hilbert space . According to this theory, each sample can be represented by , where means the weights of m prespecified base kernels . The kernel weights will be changed according to the algorithm optimizing in the kernel learning step. According to the definition of and the definition of kernel function, the kernel function can be defined as follows: Motivated by these, we propose a localized SimpleMKKM with matrix-induced regularization (LI-SimpleMKKM-MR) to improve upon the LI-SMKKM algorithm by adding an entry containing a matrix to measure the correlation between each two basis kernel matrices. LI-SimpleMKKM-MR algorithm can reduce the probability and simultaneously select high-correlation kernels, thereby enhancing the diversity of synthetic kernels and enhancing the complementarity of low-correlation kernels. Moreover, it adopts the advantage of localized SimpleMKKM, which has a better optimization effect that can be achieved by clustering the neighbor index matrix formed by the sample and the nearest k neighbors, and uses the optimization strategy instead of .