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arXiv · 2012.06471

Approximate Completely Positive Semidefinite Factorizations and their Ranks

Paria Abbasi·Andreas Klingler·Tim Netzer
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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.

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Paria Abbasi, Andreas Klingler, Tim Netzer. 2020-12-11. Approximate Completely Positive Semidefinite Factorizations and their Ranks. https://doi.org/10.1016/j.laa.2023.08.005

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