A Gaussian Tree Approximation for Integer Least-Squares

Abstract

This paper proposes a new algorithm for the linear least squares problem where the unknown variables are constrained to be in a finite set. The factor graph that corresponds to this problem is very loopy; in fact, it is a complete graph. Hence, applying the Belief Propagation (BP) algorithm yields very poor results. The algorithm described here is based on an optimal tree approximation of the Gaussian density of the unconstrained linear system. It is shown that even though the approximation is not directly applied to the exact discrete distribution, applying the BP algorithm to the modified factor graph outperforms current methods in terms of both performance and complexity. The improved performance of the proposed algorithm is demonstrated on the problem of MIMO detection.