The data that support the findings of this study are openly available at https://www.robots.ox.ac.uk/~+vgg/data/flowers/, https://mkl.ucsd.edu/dataset/protein-fold-prediction/, https://ss.sysu.edu.cn/~+py/, and https://files.is.tue.mpg.de/pgehler/projects/iccv09/. Disclosure

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. As per the Indian Constitution, Public Servants are employed under Union Government, State Governments, Union Territories and Local Bodies, their subordinate organisations, Public Sector Undertakings, Public Sector Corporations, autonomous organisations, commercial and non-commercial organisations, Industrial-Non Industrial Establishments, organized and unorganised sectors etc.

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Definition: A meter, or metre (symbol: m), is the base unit of length and distance in the International System of Units (SI). The meter is defined as the distance traveled by light in 1/299 792 458 of a second. This definition was slightly modified in 2019 to reflect changes in the definition of the second.

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. 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 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: 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.

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Definition: A kilometer (symbol: km) is a unit of length in the International System of Units (SI). One kilometer is equivalent to 0.6214 miles. As ( 2) shows that only depends on and . However, the interactions between different kernel matrices are not considered. Liu et al. [ 14] defined a criterion to measure the correlation between and . A larger means high correlation between and , and a smaller one implies that their correlation is low. By introducing the criterion term in ( 2), we can obtain the following objective function: where is a hyperparameter to balance clustering loss and regularization term. 2.3. Localized SimpleMKKM