Efficient euclidean projections in linear time

Jun Liu, Jieping Ye

Research output: Chapter in Book/Report/Conference proceedingConference contribution

14 Scopus citations


We consider the problem of computing the Euclidean projection of a vector of length n onto a closed convex set including the l1 ball and the specialized polyhedra employed in (Shalev-Shwartz & Singer, 2006). These problems have played building block roles in solving several l 1- norm based sparse learning problems. Existing methods have a worst-case time complexity of O(n log n). In this paper, we propose to cast both Euclidean projections as root finding problems associated with specific auxiliary functions, which can be solved in linear time via bisection. We further make use of the special structure of the auxiliary functions, and propose an improved bisection algorithm. Empirical studies demonstrate that the proposed algorithms are much more efficient than the competing ones for computing the projections.

Original languageEnglish (US)
Title of host publicationProceedings of the 26th Annual International Conference on Machine Learning, ICML'09
StatePublished - 2009
Event26th Annual International Conference on Machine Learning, ICML'09 - Montreal, QC, Canada
Duration: Jun 14 2009Jun 18 2009

Publication series

NameACM International Conference Proceeding Series


Other26th Annual International Conference on Machine Learning, ICML'09
CityMontreal, QC

ASJC Scopus subject areas

  • Software
  • Human-Computer Interaction
  • Computer Vision and Pattern Recognition
  • Computer Networks and Communications

Cite this