Discussion on “Independent component analysis by robust distance correlation” by Leyder et al. (2026)

Abstract Leyder et al. (2026) make an important contribution to robust independent component analysis by introducing PICARD, a sequential procedure that estimates independent sources through a robust dependence criterion based on distance correlation. A central role is played by the bowl transform, which maps far outlying observations close to the origin while remaining continuous and injective, thereby preserving the characterization of independence through vanishing distance correlation. In this discussion, we focus on the statistical role of this transformation within the overall robustness of PICARD. We argue that the bowl transform is not only a device for robustifying a dependence measure, but also induces a geometry that can be interpreted in terms of observation weighting. This leads to two related weighting schemes: one based on the norm of the bowl-transformed observations, and one based on the radial component of the transform. To isolate the effect of such weights, we examine them in a simple projection pursuit setting, which provides a natural projection-based counterpart to ICA. The resulting exploratory study suggests that bowl-induced weights can transfer part of the robustness mechanism of PICARD to projection-based procedures. More broadly, the discussion highlights the importance of disentangling the contributions of robust whitening, bowl-transform geometry, and robust distance correlation in explaining the empirical success of PICARD.