Given a non-negative matrix $M$, is there a nice method to characterize collections of non-negative matrices $P,Q$ such that $$\mathsf{rank}(M+P)\leq c\cdot\mathsf{rank}(P)$$ $$\mathsf{rank}(M+Q)\leq d\cdot\mathsf{rank}(M)$$with some fixed $c,d\geq1$?

Nice method implying convex characterizations or characterizations applicable to linear/non-linear programming.