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The Nondecreasing Rank

Andrew McCormack

TL;DR

The paper introduces nondecreasing rank, denoted $NDrank(T)$, as a monotone generalization of nonnegative tensor rank and develops the geometry of order cones, as well as conditions for the existence of finite factorizations. It shows how ND rank can be reduced to nonnegative rank via invertible maps when order cones are simplicial, and it provides bounds and typicality results for the maximum ND rank, along with a matrix tri-factorization formulation that links ND factorizations to fixed dictionaries. An ND HALS–like algorithm is proposed to compute low ND rank approximations, with theoretical assurances of existence and convergence to stationary points under common likelihood models. The methodology is demonstrated on pig weight data and a mental health tensor, yielding interpretable, monotone components and highlighting subpopulation structure and temporal interactions, thereby offering a principled approach for structure discovery in ordered data.

Abstract

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.

The Nondecreasing Rank

TL;DR

The paper introduces nondecreasing rank, denoted , as a monotone generalization of nonnegative tensor rank and develops the geometry of order cones, as well as conditions for the existence of finite factorizations. It shows how ND rank can be reduced to nonnegative rank via invertible maps when order cones are simplicial, and it provides bounds and typicality results for the maximum ND rank, along with a matrix tri-factorization formulation that links ND factorizations to fixed dictionaries. An ND HALS–like algorithm is proposed to compute low ND rank approximations, with theoretical assurances of existence and convergence to stationary points under common likelihood models. The methodology is demonstrated on pig weight data and a mental health tensor, yielding interpretable, monotone components and highlighting subpopulation structure and temporal interactions, thereby offering a principled approach for structure discovery in ordered data.

Abstract

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 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.

Paper Structure

This paper contains 17 sections, 26 theorems, 38 equations, 6 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on Low-rank Tensors and Convex Geometry
  4. The Nondecreasing Rank
  5. Geometry of the Order Cone and the Existence of Nondecreasing Factorizations
  6. Reduction of the ND rank to Nonnegative Rank
  7. Maximum and Typical ND Ranks
  8. The Maximum ND Rank
  9. A Matrix Tri-Factorization Formulation
  10. Typical ND Ranks
  11. ND Rank One and Two
  12. ND Border Rank Equals ND Rank
  13. Finding Low ND Rank Approximations
  14. Applications of Low ND Rank Decompositions
  15. Pig Weight Data
  16. ...and 2 more sections

Key Result

Lemma 1

The extremal rays of $\otimes_{j = 1}^k \mathcal{C}_j$ consist exactly of the rank-one tensors $\otimes_{j = 1}^k {\mathbf v}^{(j)}$ where every ${\mathbf v}^{(j)} \in \mathcal{C}_j$ is extremal.

Figures (6)

  • Figure 1: The respective Hasse diagrams of $\mathcal{P}_1 = \{\text{Type I,\, Type II,\, Type III}\}$ and $\mathcal{P}_2 = \{0,100,200,400\}$.
  • Figure 2: Hasse Diagram of the product poset $\{\text{Type I,\, Type II,\, Type III}\} \times \{0,100,200,400\}$.
  • Figure 3: A Hasse diagram with a tree shape that has no colliders.
  • Figure 4: Order polytopes for two different orders. The order cone is found by extending the rays connecting the origin to the top face.
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (52)

  • Definition 1: Order cone and order polytope of functions
  • Definition 2: Nondecreasing (ND) rank
  • Definition 3: Projective Tensor Product of Cones
  • Lemma 1
  • Theorem 1
  • Lemma 2: OrderConeFacesgarcia2020order
  • Definition 4
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 42 more