Suppose $A$ is a matrix, and its rank-$r$ SVD approximation looks like $A \approx U \Sigma V^\top$. I want to apply some invertible point-wise non-linear function $f$ and apply it to $A$ to make a new SVD approximation $U_1 \Sigma_1 V_1^\top$ of $f(A)$ more precise. Obviously, the choice of $f$ depends on $A$, but probably there are some wide families to search in.
Have you ever heard of this problem? Unfortunately, I can't find anything.
