Before revision:  Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the {\em metric properties} of a kernel k , i.e., the relationship between the RKHS distance d_k and the diagram distances, are of central interest for understanding if the persistence diagram guarantees carry over to the embedding.
After revision:  Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the {\em metric properties} of the feature map , i.e., the relationship between the RKHS distance d_k and the diagram distances, are of central interest for understanding if the persistence diagram guarantees carry over to the embedding.