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Proof. For any given , the maximum of optimization problem is unique [ 21], with the corresponding maximizer. According to theorem in [ 23], the former part of is differentiable. By defining other elements in except for as s and the latter part of the as , the differential of can be expressed as follows: where means one of the components of and s means all of the other components so that , and the whole in ( 12) is differentiable.

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 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

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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.

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.Current use: It is currently the official unit of measurement for expressing distances between geographical places on land in most of the world. However, there still remain a number of countries that primarily use the mile instead of the kilometer including the United States and the United Kingdom (UK). Unlike the United States, the UK has adopted the metric system; while the metric system is widely used in government, commerce, and industry, remnants of the imperial system can still be seen in the UK's use of miles in its road systems. Meter In our proposed algorithm, we set for each element in to measure the correlation between and . Choosing this method makes the calculation not too complicated and adopts the Hilbert–Schmidt independent criterion in disguise, which can reflect the correlation between different base kernels to a certain extent. 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.

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-MRCurrent 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].