Generative adversarial nets (GANs) have become a preferred tool for tasks involving complicated distributions. To stabilise the training and reduce the mode collapse of GANs, one of their main variants employs the integral probability metric (IPM) as the loss function. This provides extensive IPM-GANs with theoretical support for basically comparing moments in an embedded domain of the \textit{critic}.

Generative adversarial nets (GANs) have become a preferred tool for tasks involving complicated distributions. To stabilise the training and reduce the mode collapse of GANs, one of their main variants employs the integral probability metric (IPM) as the loss function. This provides extensive IPM-GANs with theoretical support for basically comparing moments in an embedded domain of the \textit{critic}. For rigour, we first establish the physical meaning of the phase and amplitude in CF, and show that this provides a feasible way of balancing the accuracy and diversity of generation. We then develop an efficient sampling strategy to calculate the CFs.

Weaknesses: My primary concern is that: 0. The paper seems to propose two ideas: 1) measuring distance between distributions as an expected squared difference between empirical characteristic functions evaluated at points sampled according to some adversarially learned distribution T; 2) the reciprocal training of adversarial autoencoders, i.e. adversarially aligning embeddings of X and Y, while making sure that these embeddings follow the Gaussian distribution and minimize the reconstruction loss. I wonder whether the impact of these two design choices can be evaluated independently: 1) seeing how direct minimization of C_T(X, g(Z)) wrt g performs compared to the model with a dedicated encoder/critic; 2) replacing C_T in Algorithm 1 with MMD / Sliced Wasserstein Distance or another statistical distance (moreover, distance to a Gaussian can often be estimated in closed form); does Lemma 4 hold for other statistical distances? And there are some things that I must have misunderstood. In general, authors discuss in great details possible interpretations of phase and amplitude components of CFs, but cram a lot of content critical to proper understanding of the final model on the first half of page 6. For example, in lines 214-215: "we further re-design the critic loss by finding an anchor as C(f(Y),Z) C(f(X),Z)" - it is still not clear to me what "anchors" authors are referring to.

The reviewers have reached a consensus that this work constitutes a novel and interesting extension of previous work on learning implicit models using GANs and integral probability metrics. The problems identified in the first round of reviewing were largely addressed in the author response, and I therefore can comfortably recommend accepting this paper.

Generative adversarial nets (GANs) have become a preferred tool for tasks involving complicated distributions. To stabilise the training and reduce the mode collapse of GANs, one of their main variants employs the integral probability metric (IPM) as the loss function. This provides extensive IPM-GANs with theoretical support for basically comparing moments in an embedded domain of the \textit{critic}. For rigour, we first establish the physical meaning of the phase and amplitude in CF, and show that this provides a feasible way of balancing the accuracy and diversity of generation. We then develop an efficient sampling strategy to calculate the CFs.