Independent component analysis by robust distance correlation

Independent component analysis (ICA) is a powerful tool that attempts to decompose a multivariate signal or distribution into fully independent sources, not just uncorrelated ones like PCA does. ICA is harder to do, but it has many important applications. Unfortunately, most approaches to ICA are not robust against outliers. Here we propose a robust ICA method called PICARD, which estimates the components by minimizing a robust measure of dependence between multivariate random variables. The dependence measure used is the distance correlation (dCor). In order to make it more robust we first apply a new transformation called the bowl transform, which is bounded, continuous, injective, and maps far outliers to points close to the origin. This preserves the crucial property that a zero dCor implies independence. PICARD estimates the independent sources sequentially, by looking for the component that has the smallest dCor with the remainder. We prove that PICARD is strongly consistent. Its robustness is investigated by a simulation study, in which it generally outperforms its competitors. The method is illustrated on three applications, including the well-known cocktail party problem.