Solution of Robust Linear Regression Problems by Preconditioned Conjugate Gradient Type Methods

Venansius Baryamureeba


Abstract

In this paper, we consider solving the robust linear regression problem by an inexact method. We show that iteratively reweighted least squares method and Newton method can each be combined with an iterative method to solve large, sparse, rectangular systems of linear, algebraic equations efficiently . We suggest and discuss preconditioners based on a new convex weighting function. One of these preconditioners is a constant preconditioner which is computed once at the beginning of the iterative process. For these preconditioners, we show that the upper bound on the spectral condition number of the preconditioned matrix is determined by a predefined constant, and is independent of the conditioning of the data matrix. Finally we give two sets of numerical results. One set is on demonstrating the properties of the preconditioners based on the new convex weighting function. The second set is a comparison with other convex weighting function on solving some given test problems. Both sets of numerical testing clearly show that preconditioners based on the new function are very effective in solving robust linear regression problems.

Key words: Robust linear regression, Iteratively reweighted least squares method, Newton's method, New weighting function, Conjugate gradient least squares method, Preconditioner

AMS subject classification: 62J05, 65D10, 65F10, 65F20.