Introduction

The Huber norm Huber (1973) is an alternative to Iteratively Reweighted Least Square programs for solving the hybrid l²-l¹ problem. In this note, I detail a method for minimizing the Huber norm. Because the Huber norm gives rise to a non-linear problem with non-twice continuously differentiable objective functions, its use is quite challenging. Claerbout (1996) implemented a Huber regression based on conjugate-gradient descents. However, the final results were not satisfying. Here I propose to solve the Huber problem using a quasi-Newton update of the solution with the computation of an approximated Hessian (second derivative of the objective function). This strategy is innovative in seismic processing and merits some explanation.

In this paper I first provide general definitions plus sufficient conditions to solve the optimization problem. Then, I present the quasi-Newton method and the complete algorithm used to solve the Huber problem.