# Explanation on arg min

would someone be so kind to explain this to me:

Especially the arg min part.

(it's from the k-means algorithm)

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arg min is argument of the minimum.

The simplest example is

$arg min _{x} f(x)$ is the value of $x$ for which $f(x)$ attains it's minimum.

$x_n$ is known and depends on $\pi_{nk}$ and $k$ equals to $j$ such that $\begin{Vmatrix} x_n-\mu_j \end{Vmatrix}^2$ attains minimum among all values of $\mu_j$ and given $x_n$.
$\operatorname{argmin}(f(x))$ simply returns the value of $x$ which minimizes $f(x)$ over the set of candidates for $x$ as opposed to the minimum value itself. This arises, of course, in all kinds of statistical estimates of parameters when building models (like the LS situation alluded to in your example).