On {0,1} CP Tensors and CP Multi-hypergraphs

TL;DR

This paper introduces Gramian tensors, explores 2-dimensional completely positive tensors and {0,1} CP tensors, and characterizes complete positive multi-hypergraphs, extending concepts from matrix theory to tensors.

Contribution

It extends the concept of complete positivity from matrices to tensors, providing new characterizations and defining complete positive multi-hypergraphs.

Findings

Characterization of 2-dimensional completely positive tensors

Introduction of Gramian tensors and {0,1} CP tensors

Necessary and sufficient conditions for tensor complete positivity

Abstract

A doubly nonnegative matrix can be written as a Gramian matrix, and a completely positive matrix can therefore be written as a Gramian matrix of some nonnegative vectors. In this paper, we introduce Gramian tensors and study 2-dimension completely positive tensors and {0,1} CP tensors. Also investigated are the complete positive multi-hypergraph, a generalized form of a cp graph. We also provide a necessary and sufficient condition for a 2-dimensional tensor to be completely positive.