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I want to find the minimum value of the objective function if we set K equal to the number of samples.

I know the objective function is $J=\Sigma_{n=1}^{N}\Sigma_{k=1}^{K}r_{nk}||x_n-\mu_k||^2$ And we take the derivative to get $\mu_k=\frac{\Sigma_{n=1}^{N}r_{nk}x_n}{\Sigma_{n=1}^{N}r_{nk}}$

However, when plugging that back in and set K=N, it seems unable to be simplified: $J=\Sigma_{n=1}^{N}\Sigma_{k=1}^{N}r_{nk}||x_n-\frac{\Sigma_{n=1}^{N}r_{nk}x_n}{\Sigma_{n=1}^{N}r_{nk}}||^2$

Any help?

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Well, when k = n, the obvious global minimum occurs where $x_n = \mu_k$ for all cluster centers. This falls out from your derivate because $r_{nk} = 1$ iff $n = k$.

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    $\begingroup$ So would the minimum value then be 0? $\endgroup$ – IrCa May 21 '19 at 22:38
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    $\begingroup$ Yes exactly. This is when every point is its own cluster center $\endgroup$ – Sean Owen May 23 '19 at 2:26

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