A Proof of Prop

The upper bound of gradient's Frobenius norm for spectrally-normalized discriminators follows directly. From Theorem 1.1 in [44] we know) that if w Proposition 3 (Upper bound of Hessian's spectral norm). Consider the discriminator defined in Eq. (1). The proof is in App. Corollary 1 (Upper bound of Hessian's spectral norm for spectral normalization). Moreover, if the activation for the last layer is sigmoid (e.g., for vanilla GAN [14]), we have H The desired inequalities then follow by induction. Now let's come back to the proof for Prop. Applying Eq. (5) and Lemma 1 we get We use the following lemma to lower bound (10). The following lemma upper bounds the square of the infinity norm of this vector.