Approximate Completely Positive Semidefinite Factorizations and their Ranks

TL;DR

This paper demonstrates that approximate completely positive semidefinite factorizations can have bounded cpsd-rank independently of the original matrix's cpsd-rank, using advanced mathematical theorems to achieve low-rank approximations.

Contribution

It introduces a method to construct approximate cpsd factorizations with cpsd-rank bounds independent of the initial matrix's cpsd-rank, leveraging the Approximate Caratheodory Theorem and Johnson-Lindenstrauss Lemma.

Findings

Existence of approximate cpsd factorizations with bounded cpsd-rank

Use of Approximate Caratheodory Theorem for low-rank approximation

Application of Johnson-Lindenstrauss Lemma for logarithmic rank dependence

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.