The lost honour of l2-based regularization

In the past two decades, regularization methods based on the l1 norm, including sparse wavelet representations and total variation, have become immensely popular. So much so, that there have even been recent suggestions that l1-based techniques ought to altogether replace the simpler, faster and better known l2-based alternatives as the default approach to regularization techniques. The occasionally tremendous advances of l1-based techniques are not in doubt. However, such techniques also have their limitations, and this article explores some of these. Taking into account the considerable added hardship in calculating solutions of the resulting computational problems, l1-based techniques must offer substantial advantages to be worthwhile. In this light our results suggest that in many applications, l2-based recovery may still be preferred.