  In this paper we show the existence of approximate completely positive
semidefinite (cpsd) factorizations with a cpsd-rank bounded above (almost)
independently from the cpsd-rank of the initial matrix. This is particularly
relevant since the cpsd-rank of a matrix cannot, in general, be upper bounded
by a function only depending on its size. For this purpose, we make use of the
Approximate Caratheodory Theorem in order to construct an approximate matrix
with a low-rank Gram representation. We then employ the Johnson-Lindenstrauss
Lemma to improve to a logarithmic dependence of the cpsd-rank on the size.
