# The Nondecreasing Rank

arXiv: https://arxiv.org/abs/2509.00265 · HTML: https://mlsanity.com/paper/2509.00265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 60 references that are in the map.
- First posted on arXiv 29 Aug 2025.

## Builds on (most cited first, 5 of 5)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Selecting the number of principal components: estimation of the true rank of a noisy matrix](https://mlsanity.com/paper/1410.8260/) (2014)
- [Hierarchical Clustering of Hyperspectral Images using Rank-Two Nonnegative Matrix Factorization](https://mlsanity.com/paper/1310.7441/) (2013)
- [Fixed points of the EM algorithm and nonnegative rank boundaries](https://mlsanity.com/paper/1312.5634/) (2013)
- [Self-scaled bounds for atomic cone ranks: applications to nonnegative rank and cp-rank](https://mlsanity.com/paper/1404.3240/) (2014)

## Abstract (arXiv)

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.