TY - GEN

T1 - Efficient Euclidean projections in linear time

AU - Liu, Jun

AU - Ye, Jieping

PY - 2009/12/9

Y1 - 2009/12/9

N2 - We consider the problem of computing the Euclidean projection of a vector of length n onto a closed convex set including the ℓ1 and the specialized polyhedra employed in (Shalev-Shwartz & Singer, 2006). These problems have played building block roles in solving several ℓ1-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.

AB - We consider the problem of computing the Euclidean projection of a vector of length n onto a closed convex set including the ℓ1 and the specialized polyhedra employed in (Shalev-Shwartz & Singer, 2006). These problems have played building block roles in solving several ℓ1-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.

UR - http://www.scopus.com/inward/record.url?scp=71149115443&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=71149115443&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:71149115443

SN - 9781605585161

T3 - Proceedings of the 26th International Conference On Machine Learning, ICML 2009

SP - 657

EP - 664

BT - Proceedings of the 26th International Conference On Machine Learning, ICML 2009

T2 - 26th International Conference On Machine Learning, ICML 2009

Y2 - 14 June 2009 through 18 June 2009

ER -