Hello I am looking for help on understanding the maths of Fuzzy C Means as explained here: Fuzzy C Means I was hoping for a broken down explantion of the actual math. I have tryed googling for tutorials but it has came up empty. I understand clustering and fuzzy c means and I know how to implement it but I still lack the understanding of the math.
I believe it is a version of an EM algorithm like a usual $k$-means algorithm; only that in this case each $x_i$ is allowed to belong to many clusters at the same time. If my intuition is right, you essentially try to maximize the log-likelihood function $\ell(c) = \log P(x|c)$. The EM proof should carry over to show that you can instead optimize $\ell(u, c) = \log P(x|u, c)$ in $u$, then in $c$, and repeat.