$L_p$ Isotonic Regression Algorithms Using an $L_0$ Approach

TL;DR

This paper introduces a unified plug-in approach for $L_p$ isotonic regression algorithms that leverages advances in flow algorithms and violator dags, achieving faster exact and approximate solutions for various graph classes.

Contribution

It presents a simple, systematic method combining $L_0$ and $L_1$ isotonic regression techniques, improving computational efficiency for $L_p$ isotonic regression.

Findings

Algorithms are faster than previous methods for key graph classes.

Achieves near-optimal complexity bounds for certain cases.

Significantly outperforms existing implementations in statistical packages.

Abstract

Significant advances in flow algorithms have changed the relative performance of various approaches to algorithms for $L_p$ isotonic regression. We show a simple plug-in method to systematically incorporate such advances, and advances in determining violator dags, with no assumptions about the algorithms' structures. The method is based on the standard algorithm for $L_0$ (Hamming distance) isotonic regression (by finding anti-chains in a violator dag), coupled with partitioning based on binary $L_1$ isotonic regression. For several important classes of graphs the algorithms are already faster (in O-notation) than previously published ones, close to or at the lower bound, and significantly faster than those implemented in statistical packages. We consider exact and approximate results for $L_p$ regressions, $p=0$ and $1 \leq p < \infty$, and a variety of orderings.