We have matrices $A$ and $B$ of the same dimension. Generally, we use the Frobenius norm of the difference, $\|A-B\|_F^2$, to compare these two matrices and measure how close they are to each other.
Here we assume the $\mbox{col}_i$ of matrix $A$ corresponds to $\mbox{col}_i$ of matrix $B$ and so on and, hence, the subtraction makes sense. If matrices $A$ and $B$ are equal, the Frobenius norm is zero. When matrices $A$ and $B$ deviate, the Frobenius norm is positive.
In our case, $\mbox{col}_i$ of matrix $A$ may corresponds to $\mbox{col}_j$ of matrix $B$. Is there any measure to compare these two matrices where the columns are permuted in one matrix?
One can say that, we can find all the permuted matrix $B$ and evaluate $\|A -\operatorname{perm}(B)\|_F^2$ and choose the best $B$. However, this gets problematic when the number of columns increases, as we need to evaluate $n!$ permutations.
Is there any other measure (not necessarily using the Frobenius norm) to evaluate the closeness of matrices $A$ and $B$ where $B$ is a version of $A$ in which the columns have been permuted?