What do the double bars and the 2 2 mean in this equation? I was trying to understand this equation which is used to find the Rotation and translation between a set of 3D points x, and y that brings them closest together. It says you look for the solution where it is minimized.

R I think is the 3x3 rotation matrix and t is a 3x1 translation matrix. I think it starts by saying its taking the average or 1/n of the sum of each matched point from the x and y list. And because it is y -x the minimum value would be when the position of x and y were made equal.
I don't follow the bars or the 2 over 2 at the end.  Bars mean absolute value to me and the 22 I don't know what that means.
 A: The bars stand for norm. The subscript $2$ indicates that this is the 2-norm, or euclidean norm, which is the one defining the usual euclidean distance between vectors or euclidean length of a vector. Mathematically, given a vectors $x=(x_1,\dots, x_n)$ the euclidean norm of $x$ is defined to be
$||x||_2=\sqrt{x_1^2+\cdots+x_n^2}.$
The superscript 2 is just a power, meaning that the norm is squared, so
$||x||^2_2=x_1^2+\cdots+x_n^2.$
This is done because square roots are problematic in optimizations problems due to non-differentiability at the origin and because squaring is a monotone function for non-negative values, so the minimum is attained at the same point as the original function.
A: First of all, note that the object between the $\| \cdots\|_2^2$ is a vector, so it does not make sense to take its "absolute value."
It is likely that $\|v\|_2$ refers to the 2-norm, which is the usual notion of the "length" of a vector. The upper $2$ is an exponent, which is to say that
$$
\|v\|_2^2  = (\|v\|_2)^2 = v_1^2 + v_2^2 + v_3^2.
$$
A: The $\Vert.\Vert_2^2$ is the notation for the squared 2-norm.
The 2-norm is also widely known as the euclidean norm.
\begin{align}
\Vert.\Vert_2^2 = \sum_{i=1}^{n}x_{i}^2
\end{align}
The general form of the p-norm is
\begin{align}
\Vert.\Vert_p = \left(\sum_{i=1}^{n}x_{i}^p\right)^{1/p}.
\end{align}
The 2-norm is commonly present in regression problems as the loss function between regressors $x$ and the response variable $y$. The following technique to find parameters is known as ordinary least squares:
\begin{align}
\hat{\beta} = \arg\min_{\beta\in\mathbb{R}^{n_x}}\frac{1}{n}\Vert Y - \beta X\Vert_2^2
\end{align}
where $\hat{\beta}$ is the estimated parameter vector and $n_x$ is the number of regressors.
