(...) By combining these matrices, i.e. $X= M_r^{'}||M_r$ and $Y=M_G^{'}||M_G$, a coupled matrix factorization is applied to $X,Y$ and $Z$, with the objective functions as follows (...)
I stumbled upon this notation and i'm having some difficulties in undertanding its meaning
On
Without more information, if $A$ is a $k\times m$ matrix and $B$ is a $k\times n$ matrix I would assume that $A\|B$ is a notation for the $k\times(m+n)$ matrix, the first $m$ columns of which are the columns of $A$ and the last $n$ columns of which are the columns of $B$. That is, that this is a notation for the block matrix construction $$ A\|B = (A : B).$$
A soft check of this assumption would be that in your case, the number of rows in $M_r$ should match the number of rows in $M_r'$, and so on.
It means matrix concatenation.