Nonnegative Rank vs. Binary Rank

Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a 0-1 matrix: its nonnegative rank and its binary rank (the log of the latter being the unambiguous nondeterministic communication complexity). We prove that for partial 0-1 matrices, there can be an exponential separation. For total 0-1 matrices, we show that if the nonnegative rank is at most 3 then the two ranks are equal, and we show a separation by exhibiting a matrix with nonnegative rank 4 and binary rank 5, as well as a family of matrices for which the binary rank is 4/3 times the nonnegative rank.