Are there any methods for obtaining a low-rank approximation of a matrix that preserves the diagonal? For example, given a square matrix $A$, determine if there exists $A'$ such that $$ \|A-A'\| <= \delta, \\ \|diag(A)-diag(A')\| <= \epsilon, \\rank(A') <= \rho. $$ Other variations of this problem can be considered.
I'm particularly interested in the case of null diagonals, but I reckon the general case is interesting as well.