Appendices A Proofs for theorems in Section 4.1 A.1 Proof for Theorem 4.1

We recall Theorem 4.1: Theorem 4.1 The optimal generator of Eq. 4 induces a distribution The optimization problem in Eq. 4 is: min The proof proceeds as follows: We first simplify the objective function into two terms. Strong duality also holds because Slater's condition holds. Similar to the proof for Theorem 4.1, we can rewrite the objective function in Eq. 31 as [16]: V (G,G We claim that the solution to the optimization problem above is as follows. We show the proof by contradiction. Figure 2: Left: We use a single generator. Right: We use dual generator technique.