Variational inference is a fundamental problem in Variational Auto-Encoder (VAE). By virtue of high-dimensional geometry, we propose a very simple algorithm completely different from existing ones to solve the inference problem in VAE. We analyze the unique characteristics of random variables on spheres in high dimensions and prove that the Wasserstein distance between two arbitrary datasets randomly drawn from a sphere are nearly identical when the dimension is sufficiently large. Based on our theory, a novel algorithm for distribution-robust sampling is devised. Moreover, we reform the latent space of VAE by constraining latent random variables on the sphere, thus freeing VAE from the approximate optimization pertaining to the variational posterior probability. The new algorithm is named as Spherical Auto-Encoder (SAE), which is in essence the vanilla autoencoder with the spherical constraint on the latent space. The associated inference is called the spherical inference, which is geometrically deterministic but is much more robust to various probabilistic priors than the variational inference in VAE for sampling. The experiments on sampling and inference validate our theoretical analysis and the superiority of SAE.
Summary
This paper considers the L2 normalization of samples “z” from a given prior p(z) in Generative Adversarial Netowks (GAN) and autoencoders. The L2 normalization corresponds to projecting samples onto the surface of a unit-hypersphere. Hence, to attempt to justify this normalization, the authors rely on some already established results regarding high dimensional hyperspheres. In particular, the focus is on the fact that, the Euclidean distance between any given point on a hypersphere and another randomly sampled point on the hypersphere tends to a constant, when the number of dimensions goes to infinity. This result is then used to show that the Wasserstein distance between two arbitrary distributions on a hypersphere converges to a constant when the number of dimensions grows. Based on this result, the authors claim that projecting the latent samples onto the surface of a hypersphere would make GAN less sensitive to the choice of the prior distribution. Moreover, they claim that such normalization would also benefits inference, and that it addresses the issue of variational inference in VAE.
Main comments. 
This paper is hard to follow and requires substantial improvements in terms of writing, owing to several grammatical and semantic issues. Moreover, there is a lack rigor; some important claims are supported neither by experiments nor by theoretical analysis. Experiments in the main paper are also weak. I can therefore not recommend acceptance. My detailed comments are below.
- An important claim in this paper is that the proposed approach “alleviates variational inference in VAE”. However, this requires clarification as well as theoretical/empirical justifications.
- In the introduction, it is stated that generated samples from VAE may deviate from real data samples, because “the posterior q(z|x) cannot match  the prior p(z) perfectly”. However, in VAE we do not expect the posterior to match the prior perfectly, as this would result in useless data representations or inference. Generation issues in VAE may rather be explained by the fact that, in this context we optimize a lower bound on the KL-divergence between the empirical data distribution and the model distribution. The latter objective does not penalize the model distribution if it puts some of its mass in regions where the empirical data distribution is very low or even zero.
- Theorem 2 (on the convergence of the Wasserstein distance (W2) on high dimensional hyperspheres) does not seem to hold if, for instance, P and P’ are empirical distributions with overlapping supports. Further, even when the above Theorem holds, the W2 distance may be relatively high since it is proportional to the square root of the number of samples.
- Moreover, why and how would Theorem 2 justify improved inference when projecting latent samples onto a hypersphere?
- Please consider revising the following statement in the introduction: “The encoder f in VAE approximates the posterior q(z|x)”. The encoder “f” in VAE parametrizes the variational posterior.
- Some typos,
	- Abstract, “… by sampling and inference tasks”  -- “on sampling …”
	- Introduction second paragraph after eq 2. “… it also causes the new problems” – “ … causes new problems”
	- Section 2.1, “For convenient analysis …” – “For a convenient …” 
	- Second paragraph after Theorem 1. “… perform probabilistic optimizations … ” – “… optimization …” 
	- Section 5.2, second paragraph. Is it Figure 9?
The main recommendations I would make are as follows.
- Consider revising the paper to improve its writing.
- Provide rigorous theoretical analysis and discussions to support the main claims. 
- Improve experiments by including more datasets and baselines (e.g., hyperspherical VAE [1]), as well conduct more targeted experiments to give more insights regarding the effect of the L2 normalization on inference and generation. 
[1] Davidson, Tim R., et al. "Hyperspherical variational auto-encoders." UAI, 2018.