The positive rank of a square matrix is defined in Theorem $3$ of Expressing Combinatorial Optimization Problems by Linear Programs by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, the positive rank $rank_{\Bbb R}^+(A)$ is the smallest $m$ such that $A=LR$ for a non-negative $n\times m$ matrix $L$, and non-negative $m\times n$ matrix $R$.This concept is valuable in communication complexity, since it was shown that if $rank_{\Bbb R}^+(A)$ and $rank_{\Bbb R}(A)$ could be quasi-polynomially related for a $0/1$ matrix $A$, then the log-rank conjecture holds.Is there an example of a $0/1$ matrix $A$ whose positive rank is strictly smaller than the dimension $m$ of any positive decomposition of $A$ into $0/1$ matrices $L,R$? I checked the references in Jukna's book and Lovasz's 1989 survey, but I am still unable to clarify above question.
On the notion of positive rank
co.combinatorics;communication complexity