What does do double vertical bars with numerical parameters on the right hand side mean?

Question

In the following equation, $$\large R(Y) = \lambda \left\| \frac {x^\top Y} {\|x\|} \right\|^2_2$$ What do the double bars with range $2$ to $2$ on the right hand side mean? I need to work with this equation, but I don't understand what it is doing. Note, $x$ and $Y$ are both matrices, and $\lambda$ is just a constant.

Answer 1

In the notation $$\left\|x\right\|_\color{blue}{2}^\color{red}{2}$$

• the top (red) $\color{red}{2}$ simply means squaring, as in $x^2$;
• the bottom (blue) $\color{blue}{2}$ refers to the fact that it's the "2-norm", the standard Euclidean norm.

See here for the more general $p$-norm, of which this is a special case: $$\left\| x \right\| _p = \left( |x_1|^p + |x_2|^p + \dotsb + |x_n|^p \right) ^{1/p}$$