Let $A$ be a $n \times p$ matrix and $B$ be $p \times m$ with values in $\mathbb{C}$.
Let $C$ be a $n \times p$ matrix whose entries are a permutation of the entries of A and let $D$ be a $p \times m$ matrix whose entries are a permutation of $B$ such that $AB = CD$.
Suppose I perform an SVD of A and B and truncate the number of singular values to obtain new matrices $A'$ and $B'$. And suppose I do the same for $C$ and $D$ to obatin $C'$ and $D'$.
How small can I make $||A'B'-C'D'||_{2}$ ?