The proof of Theorem 1 is simple given the above lemma. Proof of Theorem 1. A.2, the fact that Ψ( P ) ≜ Φ $^{RR}$( P ) − 2Φ $^{RS}$( P ) + Φ $^{SS}$( P ), and the linearity of differentiation immediately yield that the canonical gradient of Ψ can be written as D RR − 2 D RS + D $^{SS}$. Straightforward calculations show that this is equivalent to o ↦→ 2[ P$_{0}$ Γ$_{0}$( o, · ) − ψ$_{0}$ ]. We will use the following lemma in the proof of 1 to prove that R$_{0}$ ( O ) and S$_{0}$ ( O ) are degenerate if D Ψ 1 ≡ 0 and H$_{0}$ does not hold. Because we were unable to find the proof that the U -statistic kernel for estimating the MMD of two variables X and Y is degenerate if and only if H$_{0}$ holds or X and Y are degenerate, we give a proof here that applies in a more general setting than that which we consider in this paper. Lemma A.3. Let Q be a distribution over ( X, Y ) ∈ Z $^{2}$, where Z is a compact metric space. Let ( x, y ) ↦→ k ( x, y ) be a universal kernel on this metric space, i.e. a kernel for which the resulting reproducing kernel Hilbert space H is dense in the set of continuous funtions on Z with respect to the supremum metric. Further, suppose that E$_{Q}$ $^{√}$k ( X, X ) and E$_{Q}$ $^{√}$k ( Y, Y ) are finite. Finally, suppose that the marginal distribution of X under Q is different from the marginal distribution of Y under Q . There exists some fixed constant C such that for ( Q almost) all ( x$_{1}$, y$_{1}$ ) ∈ Z 2 if and only if the joint distribution of ( X, Y ) under Q is degenerate at a single point. Above 〈· , ·〉$_{H}$ and φ ( z ) ≜ k ( z, · ) are the inner product and the feature map in H , respectively. Proof. If Q is degenerate then clearly (A.5) holds. If (A.5) holds, then our assumption that X has a different marginal distribution than Y 43 ∫ 〈 φ ( x$_{1}$ ) − φ ( y$_{1}$ ) , φ ( x$_{2}$ ) − φ ( y$_{2}$ ) 〉$_{H}$ dQ ( x$_{2}$, y$_{2}$ ) = C (A.5)