The Nondecreasing Rank
This work addresses the problem of constrained tensor factorization for researchers in applied mathematics and data analysis, but it is incremental as it builds on existing factorization methods with new monotonicity constraints.
The paper introduces the nondecreasing (ND) rank for matrices and tensors, defined by monotonicity constraints in factorizations, and develops theoretical properties like typical and maximum ranks, with applications to pig weight and mental health survey datasets.
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.