An $O(n\log(n))$ Algorithm for Projecting Onto the Ordered Weighted $\ell_1$ Norm Ball

TL;DR

This paper introduces an efficient $O(n\,log(n))$ algorithm for projecting vectors onto the OWL norm ball, facilitating clustering and regression tasks with improved computational performance.

Contribution

The paper presents the first $O(n\,log(n))$ algorithm for projecting onto the OWL norm ball, advancing computational methods for this generalized norm.

Findings

Algorithm runs in $O(n\,log(n))$ time

Demonstrates effectiveness on synthetic regression data

Enables scalable clustering and regression with OWL norm

Abstract

The ordered weighted $\ell_1$ (OWL) norm is a newly developed generalization of the Octogonal Shrinkage and Clustering Algorithm for Regression (OSCAR) norm. This norm has desirable statistical properties and can be used to perform simultaneous clustering and regression. In this paper, we show how to compute the projection of an $n$-dimensional vector onto the OWL norm ball in $O(n\log(n))$ operations. In addition, we illustrate the performance of our algorithm on a synthetic regression test.