A More General Robust Loss Function

We present a two-parameter loss function which can be viewed as a generalization of many popular loss functions used in robust statistics: the Cauchy/Lorentzian, Geman-McClure, Welsch/Leclerc, and generalized Charbonnier loss functions (and by transitivity the L2, L1, L1-L2, and pseudo-Huber/Charbonnier loss functions). If this penalty is viewed as a negative log-likelihood, it yields a general probability distribution that includes normal and Cauchy distributions as special cases. We describe and visualize this loss and its corresponding distribution, and document several of their useful properties. Many problems in statistics [8] and optimization [6] require robustness -- that a model be insensitive to outliers. This idea is often used in parameter estimation tasks, where a non-robust loss function such as the L2 norm is replaced with some most robust alternative in the face of non-Gaussian noise.