Clarify the relationship between geometric median embeddings and KQE-based 1D projected quantiles

Determine the precise relationship between the geometric median embedding of a distribution in a reproducing kernel Hilbert space and kernel quantile embeddings constructed from one-dimensional projected quantiles, including the specific case of the 1D-projected median, by formally characterizing any equivalences, differences, or conditions under which they coincide or diverge.

Background

The paper reviews kernel median embeddings (the geometric median of k(X,·) in the RKHS) and contrasts them with the newly proposed kernel quantile embeddings, which are based on directional quantiles of RKHS projections. While KQEs admit clear statistical and computational properties, the theoretical status of median-characteristic kernels and formal connections between geometric medians and KQEs are not established.

The authors explicitly state that the connection between geometric median embeddings and 1D-projected quantiles used in KQEs is unclear, motivating a precise characterization to understand how these approaches relate and whether one can recover or approximate the other.