What is Huber loss, median absolute deviation?

The Huber loss with threshold $c$ is $\rho_c(r) = r^2/2$ for $|r| \le c$ and $c|r| - c^2/2$ otherwise; the Huber M-estimator of location minimises $\sum_i\rho_c((x_i - \mu)/s)$ for a robust scale $s$, so its estimating function $\psi_c(r) = \max(-c, \min(c, r))$ is bounded (Huber, 1964). The median absolute deviation is $\mathrm{MAD} = 1.4826\,\mathrm{med}_i|x_i - \mathrm{med}_jx_j|$, the constant making it consistent for the standard deviation of a normal law.