  In a recent article (Proc. Natl. Acad. Sci., 110(36), 14557-14562), El Karoui
et al. study the distribution of robust regression estimators in the regime in
which the number of parameters p is of the same order as the number of samples
n. Using numerical simulations and `highly plausible' heuristic arguments, they
unveil a striking new phenomenon. Namely, the regression coefficients contain
an extra Gaussian noise component that is not explained by classical concepts
such as the Fisher information matrix. We show here that that this phenomenon
can be characterized rigorously techniques that were developed by the authors
to analyze the Lasso estimator under high-dimensional asymptotics. We introduce
an approximate message passing (AMP) algorithm to compute M-estimators and
deploy state evolution to evaluate the operating characteristics of AMP and so
also M-estimates. Our analysis clarifies that the `extra Gaussian noise'
encountered in this problem is fundamentally similar to phenomena already
studied for regularized least squares in the setting n<p.
