提出单调秩概念，用于分析具有单调性的张量数据。
The Nondecreasing Rank
- 引入单调秩定义：向量满足单调性约束的外积和
- 某些偏序下，单调秩等价于非负秩的变换形式
- 适用于猪体重与新冠心理调查等真实数据建模
本文提出矩阵或张量的非递减（ND）秩概念。若张量可表示为满足单调性约束的向量外积之和，则其ND秩为r。对于特定偏序结构，寻找ND秩为r的分解等价于寻找变换后张量的非负秩-r分解。但并非所有单调张量都有有限的ND秩。本文建立了ND秩的理论，包括典型、最大及边界ND秩性质。特别分析了矩阵或张量的ND秩为1或2的特殊情况。为求低ND秩近似，提出一种改进的分层交替最小二乘算法。在猪体重数据集和新冠疫情期间心理健康调查数据上实现了低秩分解并加以解释。
原文摘要 · Abstract (English)
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.
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