Algorithms for box constrained M-estimation: Fitting large data sets with application to geodynamics

Abstract

Because of occurrence of ill-conditioning and outliers, use of direct Least Squares fit is now in decline, while robust M-estimators are currently attracting attention. We present here new algorithms based on the Spingarn Partial Inverse proximal decomposition method for L1 and Huber-M estimation that take into account both primal and dual aspects of the underlying optimization problem. The result is a family of highly parallel algorithms. Globally convergent, they are attractive for large scale problems as encountered in geodesy, especially in the field of Earth Orientation data analysis. The method is extended to handle box constrained problems. To obtain an efficient implementation, remedies are introduced to ensure efficiency in the case of models with less than full rank. Numerical results are discussed. Robust data pre-conditioning is shown to induce faster algorithm convergence. Practical implementation aspects are presented with application to series describing the Earth Rotation.