Hardness of some optimization problems over correlation polyhedra

We prove the $\mathbf{\text{NP}}$-hardness, using Karp reductions, of some problems related to the correlation polytope and its corresponding cone, spanned by all of the $n\times n$ rank-one matrices over $\{0,1\}$. The problems are: membership, rank of the decomposition, and a ``relaxed rank’’ obtained from relaxing the zero-norm expression for the rank to an ${\mathrm{\ell }}_1$ norm. While membership and rank are natural problems for any matrix cone, the relaxed rank problem occurs in some signal processing and statistical applications.