The kernel embedding of the distribution formula_21 in formula_18 (also called the kernel mean or mean map) is given by:  If formula_21 allows a square integrable density formula_25, then formula_26, where formula_27 is the Hilbert–Schmidt integral operator. A kernel is "characteristic" if the mean embedding formula_28 is injective. Each distribution can thus be uniquely represented in the RKHS and all statistical features of distributions are preserved by the kernel embedding if a characteristic kernel is used. Given formula_29 training examples formula_30 drawn independently and identically distributed (i.i.d.) from formula_31 the kernel embedding of formula_21 can be empirically estimated as  If formula_34 denotes another random variable (for simplicity, assume the co-domain of formula_34 is also formula_5 with the same kernel formula_7 which satisfies formula_38), then the joint distribution formula_39 can be mapped into a tensor product feature space formula_40 via   By the equivalence between a tensor and a linear map, this joint embedding may be interpreted as an uncentered cross-covariance operator formula_42 from which the cross-covariance of mean-zero functions formula_43 can be computed as   Given formula_29 pairs of training examples formula_46 drawn i.i.d. from formula_21, we can also empirically estimate the joint distribution kernel embedding via  Given a conditional distribution formula_49 one can define the corresponding RKHS embedding as   Note that the embedding of formula_51 thus defines a family of points in the RKHS indexed by the values formula_52 taken by conditioning variable formula_4. By fixing formula_4 to a particular value, we obtain a single element in formula_9, and thus it is natural to define the operator  which given the feature mapping of formula_52 outputs the conditional embedding of formula_34 given formula_59 Assuming that for all formula_60 it can be shown that   This assumption is always true for finite domains with characteristic kernels, but may not necessarily hold for continuous domains. Nevertheless, even in cases where the assumption fails, formula_62 may still be used to approximate the conditional kernel embedding formula_63 and in practice, the inversion operator is replaced with a regularized version of itself formula_64 (where formula_65 denotes the identity matrix). Given training examples formula_66 the empirical kernel conditional embedding operator may be estimated as   where formula_68 are implicitly formed feature matrices, formula_69 is the Gram matrix for samples of formula_4, and formula_71 is a regularization parameter needed to avoid overfitting.