Completely Positive Binary Tensors

TL;DR

This paper provides a complete characterization of binary tensors that are completely positive, along with algorithms for decomposition, cp-rank computation, and conditions for uniqueness, applicable to both odd and even order tensors.

Contribution

It introduces linear matrix inequalities to characterize binary CP tensors and offers algorithms for cp-rank computation and decomposition, including uniqueness conditions.

Findings

Binary CP tensors are characterized by two linear matrix inequalities.

Algorithms for cp-rank computation and tensor decomposition are provided.

Uniqueness of cp-rank decomposition is established for odd and even order tensors.

Abstract

A symmetric tensor is completely positive (CP) if it is a sum of tensor powers of nonnegative vectors. This paper characterizes completely positive binary tensors. We show that a binary tensor is completely positive if and only if it satisfies two linear matrix inequalities. This result can be used to determine whether a binary tensor is completely positive or not. When it is, we give an algorithm for computing its cp-rank and the decomposition. When the order is odd, we show that the cp-rank decomposition is unique. When the order is even, we completely characterize when the cp-rank decomposition is unique. We also discuss how to compute the nearest cp-approximation when a binary tensor is not completely positive.