# The Nondecreasing Rank

## TL;DR

This paper introduces the concept of nondecreasing (ND) rank for matrices and tensors, explores its properties, and develops algorithms for low ND rank approximations, with applications to real-world datasets.

## Contribution

It defines ND rank for tensors, analyzes its theoretical properties, and proposes a hierarchical ALS algorithm for low ND rank tensor approximation.

## Key findings

- ND rank relates to nonnegative rank under certain conditions.
- Not all monotonic tensors have finite ND rank.
- Low ND rank factorizations reveal meaningful data insights.

## Abstract

In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity constraint. It is shown that for certain poset orderings finding an ND factorization of rank $r$ is equivalent to finding a nonnegative rank-r factorization of a transformed tensor. However, not every tensor that is monotonic has a finite ND rank. Theory is developed describing the properties of the ND rank, including typical, maximum, and border ND ranks. Highlighted also are the special settings where a matrix or tensor has an ND rank of one or two. As a means of finding low ND rank approximations to a data tensor we introduce a variant of the hierarchical alternating least squares algorithm. Low ND rank factorizations are found and interpreted for two datasets concerning the weight of pigs and a mental health survey during the COVID-19 pandemic.