Discussion of “Independent component analysis by robust distance correlation”: a note on the geometry of the bowl transform and on the reach of the consistency for robust ICA

Abstract This is a discussion of PICARD, the robust independent component analysis (ICA) estimator built on a transformed distance correlation and on the bowl transform.I develop two threads.The first reads the bowl transform as a multivariate redescending $\psi$-function, makes its exponential decay explicit, and describes how the transformed norm concentrates as the dimension grows.The second examines the reach of the strong-consistency result, relating its population target to classical ICA identifiability and to the behavior of robust whitening under non-elliptical sources.