Measuring Dependence Beyond Linearity: Distance Correlation, Robust Extensions, and Applications – Sarah Leyder (University of Antwerp)

Distance correlation, introduced by Székely, Rizzo, and Bakirov (2007), provides a powerful measure of statistical dependence between random vectors of arbitrary dimension. Unlike classical correlation coefficients, distance correlation is zero if and only if the variables are independent, making it a fundamental tool for independence testing. This talk begins with an overview of the definition of distance correlation, along with its main properties and intuition, and contrasts it with more classical correlation measures which are often limited to linear dependencies.

Although distance correlation is very powerful, it can be sensitive to outliers and heavy-tailed data. The second part of the talk presents a robust version based on the tailored biloop transform, which limits the influence of extreme observations while preserving the underlying dependence structure (Leyder, Raymaekers, and Rousseeuw, 2025). We explain how this transformation leads to a more stable dependence measure and discuss its robustness properties in theory and practice.

In the final part, we look at an application of distance correlation to Independent Component Analysis (ICA). It builds on the framework of Matteson and Tsay (2017), who minimize the distance correlation between the components to extract the independent sources. However, we robustify their approach by utilizing a more robust version using a new transform, the bowl transformation. This leads to an ICA method that performs better in the presence of outliers or model deviations. Extensive simulation results and real data examples demonstrate the good performance of this new ICA method and its resistance towards contamination.

Székely, Gábor J., Maria L. Rizzo, and Nail K. Bakirov. “Measuring and testing dependence by correlation of distances.” (2007): 2769-2794.
Matteson, David S., and Ruey S. Tsay. “Independent component analysis via distance covariance.” Journal of the American Statistical Association 112.518 (2017): 623-637.
Leyder, Sarah, Jakob Raymaekers, and Peter J. Rousseeuw. “Robust Distance Covariance.” International Statistical Review (2025).
Leyder, Sarah, et al. “Independent Component Analysis by Robust Distance Correlation.” arXiv preprint arXiv:2505.09425 (2025).