Speaker
Description
Matrix or tensor data often has structured rows, columns, or more generally modes. In particular, a mode may have a natural ordering that can be leveraged to obtain parsimonious representations of the data. To this end, the concept of the nondecreasing (ND) rank is introduced in this talk. 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 a variant of the hierarchical alternating least squares algorithm is introduced.