The Nondecreasing Rank

2025-08-29

AI Summary

This work addresses matrix/tensor decomposition under monotonicity constraints. We introduce the novel concept of “non-decreasing rank” (ND-rank), the first theoretical framework characterizing typical rank, maximum rank, and border rank for monotonic decompositions; we further prove that ND-rank is equivalent to nonnegative rank under partial order transformations. Methodologically, we propose an outer-product decomposition framework incorporating explicit monotonicity constraints and develop an enhanced hierarchical alternating least squares algorithm for low-ND-rank approximation. Experiments demonstrate that our approach effectively extracts interpretable, monotonically evolving patterns in real-world applications—specifically, pig weight growth modeling and pandemic-related mental health data analysis—thereby significantly improving model transparency and domain adaptability. This work establishes a new theoretical foundation and practical algorithmic toolkit for ordinal-constrained tensor learning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

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.

Problem

• Introducing nondecreasing rank concept for matrices
• Developing theory for ND rank properties and limitations
• Proposing algorithm for low ND rank approximations

Innovation

• Nondecreasing rank factorization with monotonic constraints
• Hierarchical alternating least squares algorithm variant
• Transformed tensor nonnegative rank equivalence