# 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.)

$$\arg\min\arg\min$$ is argument of the minimum.

The simplest example is

$$\arg\min _{x} f(x)\arg\min _{x} f(x)$$ is the value of $$xx$$ for which $$f(x)f(x)$$ attains its minimum.

$$x_nx_n$$ is known and depends on $$\pi_{nk}\pi_{nk}$$
and $$kk$$ equals to $$jj$$ such that $$\begin{Vmatrix} x_n-\mu_j \end{Vmatrix}^2\begin{Vmatrix} x_n-\mu_j \end{Vmatrix}^2$$ attains minimum among all values of $$\mu_j\mu_j$$ and given $$x_nx_n$$.