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

Approximate Completely Positive Semidefinite Factorizations and their Ranks