What does this math notation mean? $\min_{i} \|x_{i}\|$

Question

$\qquad\min_{i} \|x_{i}\|$

I'm doing some machine learning problems and I ran into this notation.

I don't understand what "min_i" means in this case.

Is it the smallest element in the norm of $x$? Any help would be appreciated.

Answer 1

$\min_{i}\Vert x_{i}\Vert$ is a sloppy shorthand notation for the smallest norm (i assume || denotes a norm, otherwise some sort of absolute value) of each of the $x_i$. For example if you have $3$ elements $x_1, x_2$ and $x_3$ with norms $\Vert x_1\Vert = 5$, $\Vert x_2\Vert = 8$ and $\Vert x_3\Vert = 2$, then

\begin{align*} \min_{i}\Vert x_{i}\Vert = \Vert x_3\Vert = 2 \end{align*}

As has already been pointed out in the comments, more formally $i$ ranges over all elements in some index set $I$, i.e. \begin{align*} \min_{i}\Vert x_{i}\Vert = \min\{\Vert x_{i}\Vert, \, i\in I\}. \end{align*} In the example I gave, one could have had $I = \{1, 2, 3\}$ as index set, as there were $3$ elements of which the minimum norm was chosen.