Abstract:Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of tha… Show more
“…Given a convex cone C ⊆ S n , its dual cone is defined as see [15]. The difference of this upper bound to the best known lower bound has recently been improved to O(n log log n), see [13]. The set of all n × n completely positive matrices also forms a proper cone, denoted by CP n .…”
In this paper we provide an approximation for completely positive semidefinite (cpsd) matrices with 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 Carathéodory 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 cpsdrank on the size.
“…Given a convex cone C ⊆ S n , its dual cone is defined as see [15]. The difference of this upper bound to the best known lower bound has recently been improved to O(n log log n), see [13]. The set of all n × n completely positive matrices also forms a proper cone, denoted by CP n .…”
In this paper we provide an approximation for completely positive semidefinite (cpsd) matrices with 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 Carathéodory 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 cpsdrank on the size.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.