How to arrive at the update terms of fuzzy clustering with the objective function? Can someone provide the mathematical proof?

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Panchatcharam Mariappan

The proof is available in this book. Cited in the same page, which you have posted.

J. C. Bezdek (1981), Pattern Recognition with Fuzzy Objective Function Algorithms

Please have a look at the 5th chapter, page 170-190.

James F Peters

This is a good question.

In addition to @Panchatcharam Mariappan's helpful answer, there is a bit more to add, thanks to many good articles on fuzzy clustering available on RG. For example, consider

https://www.researchgate.net/publication/28246684_Fuzzy_clustering_Insights_and_a_new_approach

Article Fuzzy clustering: Insights and a new approach