An O(n2) Algorithm for Isotonic Regression

1999

Abstract. The isotonic regression problem has applications in statistics, operations research, and image processing. In this paper a general framework for the isotonic regression algorithm is proposed. Under this framework, we discuss the isotonic regression problem in the case where the directed graph specifying the order restriction is a directed tree with n vertices. A new algorithm is presented for this case, which can be regarded as a generalization of the PAV algorithm of Ayer et al.

2019

This note is a status report on the fastest known isotonic regression algorithms for various Lp metrics and partial orderings. The metrics considered are unweighted and weighted L0, L1, L2, and L∞. The partial orderings considered are linear, tree, d-dimensional grids, points in d-dimensional space with componentwise ordering, and arbitrary orderings (posets). Throughout, “fastest” means for the worst case in O-notation, not in any measurements of implementations. This note will occasionally be updated as better algorithms are developed. Citations are to the first paper to give a correct algorithm with the given time bound, though in some cases two are cited if they appeared nearly contemporaneously.

2010

A new algorithm for isotonic regression is presented based on recursively partitioning the solution space. We develop efficient methods for each partitioning subproblem through an equivalent representation as a network flow problem, and prove that this sequence of partitions converges to the global solution. These network flow problems can further be decomposed in order to solve very large problems. Success of isotonic regression in prediction and our algorithm's favorable computational properties are demonstrated through simulated examples as large as 2 x 105 variables and 107 constraints.

arXiv (Cornell University), 2021

2009

This paper gives an approach for determining isotonic regre ssions for data at points in multidimensional space, with the ordering given by domination. Recent algori thmic advances for 2-dimensional isotonic regressions have made them useful for significantly larger d ata sets, and here we provide an advance for dimensions 3 and larger. Given a set V of n d-dimensional points, it is shown that an isotonic regressio n on V can be determined iñ Θ(n2), Θ̃(n3), andΘ̃(n) time for theL1, L2, andL∞ metrics, respectively. This improves upon previous results by a factor of Θ̃(n). The core of the approach is in extending the regression to a set of pointsV ′ ⊃ V where the domination ordering on V ′ can be represented with relatively few edges.

Applied Mathematics and Parallel Computing, 1996

Algorithmica, 2013

Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm - SODA '06, 2006