How can I match higher-order moments with kernel tricks if I do not have the complete distribution but only the higher-order moments?

[bmmddzgj]

As you said in Appendix B.3 of ref, "It is difficult to match high-order moments, because we have to deal with high order tensors directly."
However, if I only know little information about the true distribution such as up to 4-order moment or cumulant, how can I match these information with kernel tricks?