Approximate Completely Positive Semidefinite Factorizations and their Ranks
arXiv:2012.06471 · doi:10.1016/j.laa.2023.08.005
Abstract
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.
v2: clarified and corrected some citations, v3: new title, close to published version