Let $J$ be all one matrix and $A$ be a non-negative matrix such that $J-A$ is also non-negative. Then what can we say about the non-negative rank of $J-A$ in terms of the non-negative rank of $A$. We know that for rank, we have
$$\mbox{rank}(A)-1 \le \mbox{rank}(J-A) \le \mbox{rank}(A)+1$$
by subadditive property of rank. Can we bound the non-negative rank of $J-A$ similarly in terms of the non-negative rank of $A$? We can apply subadditivity for non-negative rank but we would have to ensure that matrices are non-negative and will give us following
$$\mbox{rank}_{+}(J) - \mbox{rank}_{+}(A) \le \mbox{rank}_{+}(J-A)$$
which is useless. Can we say more? How far can the non-negative rank of $J-A$ be from $A$?
Any help, comments, hints are greatly appreciated. Thanks.