Tensors of Nonnegative Rank Two

TL;DR

This paper characterizes nonnegative tensors of rank two through supermodularity and flattening rank, and explores their geometric properties within phylogenetic models, with discussions on extending to higher ranks.

Contribution

It provides a complete characterization of nonnegative tensors of rank two and investigates their geometric structure in the context of phylogenetic models.

Findings

Nonnegative rank two tensors are supermodular with flattening rank at most 2.

The paper explores the semialgebraic geometry of the general Markov model on phylogenetic trees.

Discussion on potential extensions to higher-rank tensors.

Abstract

A nonnegative tensor has nonnegative rank at most 2 if and only if it is supermodular and has flattening rank at most 2. We prove this result, then explore the semialgebraic geometry of the general Markov model on phylogenetic trees with binary states, and comment on possible extensions to tensors of higher rank.