Self-tuning M-estimators

M-estimators are the de-facto standard method of robust estimation in robotics. They are easily incorporated into iterative non-linear least-squares estimation and provide seamless and effective handling of outliers in data. However, every M-estimator's robust loss function has one or more tuning parameters that control the influence of different data. The choice of M-estimator and the manual tuning of these parameters is always a source of uncertainty when applying the technique to new data or a new problem. In this paper we develop the concept of self-tuning M-estimators. We first make the connection between many common M-estimators and elliptical probability distributions. This connection shows that the choice of M-estimator is an assumption that the residuals belong to a well-defined elliptical distribution. We exploit this implication in two ways. First, we develop an algorithm for tuning the M-estimator parameters during iterative optimization. Second, we show how to choose the correct M-estimator for your data by examining the likelihood of the data given the model. We fully derive these algorithms and show their behavior on a representative example of visual simultaneous localization and mapping.