Behavior of matrix rank under thresholding of its elements Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is there anything known about by how much the rank of $A$ can increase by such a deformation? 
I am typically thinking of the situation when $r \ll n$ and I am wondering if there are conditions known which if true then the new matrix also continues to have a rank far below $n$. 

An analogy can be drawn to how under the Hadamard product rank is sub-multiplicative. Hadamard product of two low rank matrices can't be of rank arbitrarily high. 
 A: The deformation of a matrix of rank two can have full rank, e.g., $$\pmatrix{1&1&-1&-2&-3\cr1&2&0&-1&-2\cr1&3&1&0&-1\cr1&4&2&1&0\cr1&5&3&2&1\cr}$$
A: In some cases, the rank is preserved under thresholding. For example, let
$$\rm A := \begin{bmatrix} 1\\ 0\\-1\end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix}^\top = \begin{bmatrix} 1 & 1 & 1\\ 0 & 0 & 0\\-1 & -1 & -1\end{bmatrix}$$
Note that $\rm A$ is a rank-$1$ matrix. Thresholding $\rm A$, we obtain another rank-$1$ matrix
$$\max \{ \mathrm O_3, \mathrm A \} = \begin{bmatrix} 1 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{bmatrix} = \begin{bmatrix} 1\\ 0\\ 0\end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix}^\top$$
Thinking of the thresholding operation in terms of the Hadamard product
$$\max \{ \mathrm O_3, \mathrm A \} = \underbrace{\begin{bmatrix} 1 & 1 & 1\\ 0 & 0 & 0\\-1 & -1 & -1\end{bmatrix}}_{= \mathrm A} \circ \underbrace{\begin{bmatrix} 1 & 1 & 1\\ 1 & 1 & 1\\ 0 & 0 & 0\end{bmatrix}}_{=: \mathrm B}$$
where $\rm B$ is a binary matrix that contains information pertaining to the signs of the entries of $\rm A$. In this case, $\rm B$ is also a rank-$1$ matrix. Since
$$\mbox{rank} (\mathrm A \circ \mathrm B) \leq \mbox{rank} (\mathrm A) \cdot \mbox{rank} (\mathrm B)$$
and $\mbox{rank} (\mathrm A) = \mbox{rank} (\mathrm B) = 1$, the rank does not increase under thresholding. Since $\mathrm A \circ \mathrm B \neq \mathrm O_3$, we conclude that the rank is actually preserved.
