Using the eigenvalue relaxation for binary least-squares estimation problems

Stéphane Chrétien (1) , Franck Corset (2)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 SAGAG - Statistique Appliquée et de Géométrie Aléatoire de Grenoble

The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact constraint and as such, is a convex problem with polynomial time complexity. Moreover, as a main practical advantage of this relaxation over the standard semi-definite programming approach, several efficient bundle methods are available for this problem allowing to address problems of very large dimension. The necessary tools from convex analysis are recalled and shown at work for handling the problem of exactness of this relaxation. Two applications are described. The first one is the problem of binary image reconstruction and the second is the problem of multiuser detection in CDMA systems.

Mots clés: Binary least squares; Eigenvalue relaxation; Semi-definite relaxation; Image denoising; CDMA; Multiuser detection

Domaines: Optimisation et contrôle [math.OC]; Méthodologie [stat.ME]; Calcul [stat.CO]

Stéphane Chrétien, Franck Corset. Using the eigenvalue relaxation for binary least-squares estimation problems. Signal Processing, 2009, 89 (11), pp.2079-2091. ⟨10.1016/j.sigpro.2009.04.025⟩. ⟨hal-00385002⟩