Exploring the bounds on the positive semidefinite rank

The nonnegative and positive semidefinite (PSD-) ranks are closely connected to the nonnegative and positive semidefinite extension complexities of a polytope, which are the minimal dimensions of linear and SDP programs which represent this polytope. Though some exponential lower bounds on the nonnegative [FMP12] and PSD[LRS15] ranks has recently been proved for the slack matrices of some particular polytopes, there are still no tight bounds for these quantities. We explore some existing bounds on the PSD-rank and prove that they cannot give exponential lower bounds on the extension complexity. Our approach consists in proving that the existing bounds are upper bounded by the polynomials of the regular rank of the matrix, which is equal to the dimension of the polytope (up to an additive constant). As one of the implications, we also retrieve an upper bound on the mutual information of an arbitrary matrix of a joint distribution, based on its regular rank.