$L_0$ Isotonic Regression With Secondary Objectives

TL;DR

This paper introduces efficient algorithms for $L_0$ isotonic regression with secondary objectives, applicable to ordinal and real-valued labels, improving computational complexity and addressing multidimensional cases.

Contribution

It presents new algorithms for $L_0$ isotonic regression with secondary criteria, reducing computational complexity and extending to multidimensional orderings.

Findings

Algorithms for arbitrary ordinal labels optimize label changes.

Methods for real-valued labels minimize combined $L_p$ and weighted $L_0$ errors.

Reduced algorithm complexity from $ heta(n^3)$ to $o(n^{3/2})$.

Abstract

We provide algorithms for isotonic regression minimizing $L_0$ error (Hamming distance). This is also known as monotonic relabeling, and is applicable when labels have a linear ordering but not necessarily a metric. There may be exponentially many optimal relabelings, so we look at secondary criteria to determine which are best. For arbitrary ordinal labels the criterion is maximizing the number of labels which are only changed to an adjacent label (and recursively apply this). For real-valued labels we minimize the $L_p$ error. For linearly ordered sets we also give algorithms which minimize the sum of the $L_p$ and weighted $L_0$ errors, a form of penalized (regularized) regression. We also examine $L_0$ isotonic regression on multidimensional coordinate-wise orderings. Previous algorithms took $\Theta(n^3)$ time, but we reduce this to $o(n^{3/2})$.