Using the Eigenvalue Relaxation for Binary Least-Squares Estimation Problems

TL;DR

This paper surveys the eigenvalue relaxation method for binary least-squares problems, highlighting its convexity, efficiency, and applications in image reconstruction and CDMA detection.

Contribution

It demonstrates the properties and practical advantages of eigenvalue relaxation over SDP, including polynomial complexity and suitability for large-scale problems.

Findings

Eigenvalue relaxation is convex and solvable in polynomial time.

Efficient bundle methods enable large-scale problem solving.

Applications include binary image reconstruction and multiuser detection.

Abstract

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 pratical 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.