This paper proposes Dirichlet Variational Autoencoder (DirVAE) using a Dirichlet prior for a continuous latent variable that exhibits the characteristic of the categorical probabilities. To infer the parameters of DirVAE, we utilize the stochastic gradient method by approximating the Gamma distribution, which is a component of the Dirichlet distribution, with the inverse Gamma CDF approximation. Additionally, we reshape the component collapsing issue by investigating two problem sources, which are decoder weight collapsing and latent value collapsing, and we show that DirVAE has no component collapsing; while Gaussian VAE exhibits the decoder weight collapsing and Stick-Breaking VAE shows the latent value collapsing. The experimental results show that 1) DirVAE models the latent representation result with the best log-likelihood compared to the baselines; and 2) DirVAE produces more interpretable latent values with no collapsing issues which the baseline models suffer from. Also, we show that the learned latent representation from the DirVAE achieves the best classification accuracy in the semi-supervised and the supervised classification tasks on MNIST, OMNIGLOT, and SVHN compared to the baseline VAEs. Finally, we demonstrated that the DirVAE augmented topic models show better performances in most cases.
In this paper, authors proposes an algorithm to use Dirichlet prior on the variational auto-encoder (VAE). They used this prior as natural conjugate to likelihood distributtion of multinomial (categorical). The paper proposes a way to use scalability power of VAE for data distributed by categorical distribution. In order to apply reparametrization trick, authors have used iid Gamma random variable to construct draw from Dirichlet distribution and have used approximation with inverse gamma CDF,  it is discussed how this method has better performance than other approximations method for gamma distribution such as Weibull and logistic Gaussian.
Authors pointed out, one of the weak points in competing models such as  Guassian softmax prior or Griffith -Engen-McCloskey prior which has been used for Stick breaking VAE is to not encouraging of having multi-modal posteriori, while this prior empower having multi-modal posteriori distribution which give them advantage over previous papers. 
 In experimental results, paper has used different datasets of MNIST, MNIST+rotation , OMNIGLOT , 20newsgroup and RCVI and used different measures to compare the existing method with the baselines. 
To summarize the contribution of this paper, following three points can be named as main contribution of this paper:
- proposed a Dirichlet prior, for categorical likelihood which encourages having multi-modal posteriori. paper demonstrates couple of techniques  to apply the reparametrization trick on Dirichlet distribution, by using sum of iid Gamma random variables.  
- used method of moments estimator to update the hyper parameter of the Dirichlet distribution which helps to have closer approximation of log likelihood. They update hyper-parameters after every few updates of VAE parameters.
-discussed how to overcome  Stick-breaking VAE “component collapse” issue. Experiments show superior results on supervised and semi supervised, and authors claimed the main reason of this superiority being due to not having disadvantage of component collapse which happens in SBVAE.
Quality and Novelty:
claims in paper are supported by proofs and/or experimental results and there does not exist significant technical issues with the details of claims made in this paper and proofs provided. There are following issues with novelty and quality of paper that I would like discuss them under following three points:
- Authors need to be clear about the motivation of the paper, if the motivation of the paper is to encourage the multi-modality in posteriori distribution, using Gaussian prior and methods like normalizing flow Rezende, Danilo Jimenez, and Shakir Mohamed. "Variational inference with normalizing flows." arXiv preprint arXiv:1505.05770 (2015) or similar may be able to do the same work in which case paper should compare its results to those ideas which has not been done in this paper.
- second appealing point that this paper can make is to use Dirichlet prior for the purposes like community detection, topic modeling and LDA  etc etc. In this case, I did not find significant difference between the proposed method and what is found in Srivastava, Akash, and Charles Sutton. "Autoencoding variational inference for topic models." arXiv preprint arXiv:1703.01488 (2017), but due to the encourages of multi-modality authors show in average DirVAE performs better in measures like perplexity and NPMI. Under this condition, my main concern is interpretablity of posteriori. That will be discussed under next point
- Main motivation behind using Dirichlet prior, is to have posteriori with a few significant related topic and many unrelated topic for every word. By changing the concentration parameter in stick-breaking, it is possible that performance of stick-breaking method increase in perplexity and NPMI scores in cost of loosing interpretability of the model. So having higher concentration parameter can show better performance in the cost of interpretablity that put second point of the paper at risk
Clarity: 
The paper is well written and previous relevant methods have been reviewed well. The organization of paper is good, experiments well explained and proofs and mathematical reasoning are clear.
Significance of experiments:  
As discussed,in previous sections, the results show superior performance and compared to other methods on semi-supervised and supervised classification on different datasets. Also it has shown in average better perplexity and NPMI score for topic modeling, the only issue can be these scores come as cost of interpretablity of the model. Also it is possible that other competing models can be matching to this results if they do not aim for sparse posteriori.