An $O(n\log(n))$ Algorithm for Projecting Onto the Ordered Weighted $\ell_1$ Norm Ball
This work provides an efficient algorithm for a specific norm used in statistical learning, which is incremental as it builds on existing norms like OSCAR.
The paper tackles the problem of efficiently projecting vectors onto the ordered weighted ℓ1 norm ball, achieving a result of O(n log(n)) computational complexity and demonstrating performance on a synthetic regression test.
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.