Lipschitz Unimodal and Isotonic Regression on Paths and Trees

TL;DR

This paper introduces efficient algorithms for Lipschitz and unimodal isotonic regression on sequences and trees, enabling constrained data fitting with near-linear computational complexity.

Contribution

It extends isotonic and unimodal regression algorithms from sequences to trees, providing near-linear time solutions for these constrained regression problems.

Findings

Algorithms operate in near-linear time.

Applicable to sequences and tree-structured data.

Supports both isotonic and unimodal constraints.

Abstract

We describe algorithms for finding the regression of t, a sequence of values, to the closest sequence s by mean squared error, so that s is always increasing (isotonicity) and so the values of two consecutive points do not increase by too much (Lipschitz). The isotonicity constraint can be replaced with a unimodular constraint, where there is exactly one local maximum in s. These algorithm are generalized from sequences of values to trees of values. For each scenario we describe near-linear time algorithms.