# Meaning of “arg min”

Would someone be so kind to explain this to me:

$$\pi_nk=\left\{\begin{array}{cl}1&\textrm{if }k=\arg\min_j\left\Vert\mathbf x_n-\mu_j\right\Vert^2\\0&\textrm{otherwise}\end{array}\right..$$

Especially the $\arg\min$ part.

(It's from the $k$-means algorithm.)

Define $$\arg\min_x f(x)$$ as the set of values of $$x$$ for which the minimum of $$f(x)$$ is attained, so it is the set of values where the function attains the minimum. Thus, $$\arg\min_x f(x)$$ is a subset of the domain of $$f(x)$$.

For your example: $$x_n$$ is known and depends on $$\pi_{nk}$$; $$k$$ equals to the value of $$j$$ such that $$\begin{Vmatrix} x_n-\mu_j \end{Vmatrix}^2$$ attains a minimum among all values of $$\mu_j$$ and given $$x_n$$.

Hopefully that helps.

• If you define arg min $f$ as the set of values where the function attains its minimum, then one should write $x\in$arg min $f$ whereas $x=$ arg min $f$ is not what most authors intend to say. Sep 18, 2023 at 8:56

$$\arg \min$$ (or $$\arg \max$$) return the input(s) for which the output is minimum (or maximum).

For example:

The graph illustrates the function $$f(x)=2 \sin(x-0.5)+\cos(x)^2$$.

The global minimum of $$f(x)$$ is $$\min(f(x)) \approx -2$$, while $$\arg \min f(x) \approx 4.9$$.

• The arrow is approximately pointing to the point: (4.9, -2) Sep 23, 2021 at 4:24

$\operatorname{argmin}(f(x))$ 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).

• Practical! yes it means argument that returns least not the least passed argument! Apr 27, 2016 at 15:07