  We give an algorithm for computing approximate PSD factorizations of
nonnegative matrices. The running time of the algorithm is polynomial in the
dimensions of the input matrix, but exponential in the PSD rank and the
approximation error. The main ingredient is an exact factorization algorithm
when the rows and columns of the factors are constrained to lie in a general
polyhedron. This strictly generalizes nonnegative matrix factorizations which
can be captured by letting this polyhedron to be the nonnegative orthant.
