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.
Department of Biological Physics, Eötvös Loránd University
Abstract
Ranking athletes by their performance in competitions and tournaments is
common in every popular sport and has significant benefits that contribute to
both the organization and strategic aspects of competitions. Although rankings
are perhaps the most concise and most straightforward representation of the
relative strength among the competitors, beyond this one-dimensional
characterization, it is also possible to capture the relationships between
athletes in greater detail. Following this approach, our study examines the
networks between athletes in individual sports such as tennis and fencing,
where the nodes are associated with the contestants and the edges are directed
from the winner to the loser. We demonstrate that the connections formed
through matches arrange themselves into a time-evolving hierarchy, with the top
players positioned at its apex. The structure of the resulting networks
exhibits detectable differences depending on whether they are constructed
purely from round-robin data or from purely elimination-style tournaments. We
find that elimination tournaments lead to networks with a smaller level of
hierarchy and thus, importantly, to an increased probability of circular
win-loss situations (cycles). The position within the hierarchy, along with
other network metrics, can be used to predict match outcomes. In the systems
studied, these methods provide predictions with an accuracy comparable to that
of forecasts based on official sports ranking points or the Elo rating system.
A deeper understanding of the delicate aspects of the networks of pairwise
contests enhances our ability to model, predict, and optimize the behaviour of
many complex systems, whether in sports tournaments, social interactions, or
other competitive environments.