what does the notation $x^{(n)}$ mean?

Question

What does the notation $x^{(n)}$ where $x$ is a matrix and $n$ is an integer?

$$\|W^T\mathbf{x}^{(n)}+b-y^{(n)}\|_2^2$$

Answer 1

As I see the optimization tag, this probably means the matrix that is yielded at the $n$-th iteration step.

In simple words, this means that $x^{(n)}$ is the $x$ matrix of your method at the $n$-th step and $y^{(n)}$ is the $y$ matrix of your method at the $n$-th step.

Answer 2

It is just an index for a sequence $x^{(1)}, x^{(2)}, x^{(3)}, \ldots$, similar to $x_1, x_2, x_3, \ldots$.

The reason why they avoid subscripts $x_i$ is because presumably $x$ is a vector, so $x_i$ may be reserved for denoting "the $i$th component of the vector $x$." Using the above notation allows one to do things like $x^{(2)}_3$, which denotes the third component of the vector $x^{(2)}$ (which itself is the second element in a sequence of vectors).

Finally, the reason for the parentheses is to differentiate it from an exponent, since "$y^2$" might look like the square of $y$.