Help with mathematic notation

Question

I am looking at a mathematical expression and could use some help in understanding these notations and also If you could please provide a reference where I could read more about these notations so that I can learn this order of operation and learn how to solve similar problems on my own. This is a computer algorithm for machine learning (in this particular instance):

$\|x\|_1$ is solved by: $$x_1 = \arg\min \|x\|_1 \text{ s.t. } y = Ax$$

I have never seen the large "||" symbols before, so It would be awesome if someone could tell me what they are called, how they work, and where I can get more details on its function.

Thanks everyone!

Answer 1

They are called norm.

$$\|x\|_1 = \sum_{i=1}^n |x_i|$$

The problem that you mentioned can be solved by a linear program.

Answer 2

$\lVert \cdot \rVert_1$ is called the 1-norm. For a vector $x \in F^n$, where $F$ is a field, the 1-norm of $x$ is defined as $$\|x\|_1 = \sum_{i=1}^n |x_i|.$$

Answer 3

Here, $x$ is a vector, presumably in $\mathbb{R}^n$, meaning you can write it as a $1\times n$ column matrix, $\begin{bmatrix}x_1\\x_2\\ \vdots \\ x_n\end{bmatrix}$. The norm $||x||_p = \left(\sum_{i=1}^n |x_i|^p\right)^{1/p}$. When $p=1$, it is the sum of the absolute values of the components, $$||x||_1 = |x_1|+|x_2|+\dots+|x_n|$$

The problem given here is an optimization problem, specifically a convex optimization problem called $\ell_1$ minimization.