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.

翻译：本文引入了矩阵或张量的非递减（ND）秩概念。若一个张量可表示为r个向量外积之和，且每个向量满足单调性约束，则称其具有非递减秩r。研究表明，对于某些偏序集序，寻找秩为$r$的ND分解等价于寻找变换后张量的非负秩r分解。然而，并非所有单调张量都具有有限的ND秩。本文建立了描述ND秩性质的理论体系，包括典型ND秩、最大ND秩和边界ND秩。特别探讨了矩阵或张量具有一阶或二阶ND秩的特殊情形。为寻找数据张量的低ND秩近似，我们引入了一种分层交替最小二乘算法的变体。通过对猪体重数据集和COVID-19疫情期间心理健康调查数据集的分析，我们获得并解释了低ND秩分解结果。