论文标题
近似完全积极的半限定因素及其等级
Approximate Completely Positive Semidefinite Factorizations and their Ranks
论文作者
论文摘要
在本文中，我们表明存在与上面（几乎）（几乎）与初始矩阵的CPSD级别界定的CPSD级别的近似完全正半数（CPSD）因素化的存在。这尤其重要，因为矩阵的CPSD级别通常不能仅取决于其大小的函数。为此，我们利用近似caratheodory定理，以构建具有低级数克表示的近似矩阵。然后，我们采用Johnson-Lindenstrauss Lemma来改善CPSD级的对数依赖性。
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.
