Completely Positive Tensors and Multi-Hypergraphs

TL;DR

This paper introduces the concept of completely positive multi-hypergraphs to extend the study of zero patterns from matrices to tensors, providing conditions for their complete positivity.

Contribution

It defines completely positive multi-hypergraphs and characterizes conditions for (0,1) tensors and uniform multi-hypergraphs to be completely positive.

Findings

Necessary and sufficient conditions for (0,1) tensors to be completely positive.

Characterization of completely positive multi-hypergraphs.

Conditions for uniform multi-hypergraphs to be completely positive.

Abstract

Completely positive graphs have been employed to associate with completely positive matrices for characterizing the intrinsic zero patterns. As tensors have been widely recognized as a higher-order extension of matrices, the multi-hypergraph, regarded as a generalization of graphs, is then introduced to associate with tensors for the study of complete positivity. To describe the dependence of the corresponding zero pattern for a special type of completely positive tensors--the $\{0,1\}$ completely positive tensors, the completely positive multi-hypergraph is defined. By characterizing properties of the associated multi-hypergraph, we provide necessary and sufficient conditions for any $(0,1)$ associated tensor to be $\{0,1\}$ completely positive. Furthermore, a necessary and sufficient condition for a uniform multi-hypergraph to be completely positive multi-hypergraph is proposed as well.