Deep generative models have advanced the state-of-the-art in semi-supervised classification, however their capacity for deriving useful discriminative features in a completely unsupervised fashion for classification in difficult real-world data sets, where adequate manifold separation is required has not been adequately explored. Most methods rely on defining a pipeline of deriving features via generative modeling and then applying clustering algorithms, separating the modeling and discriminative processes. We propose a deep hierarchical generative model which uses a mixture of discrete and continuous distributions to learn to effectively separate the different data manifolds and is trainable end-to-end. We show that by specifying the form of the discrete variable distribution we are imposing a specific structure on the model's latent representations. We test our model's discriminative performance on the task of chronic lymphocytic leukemia (CLL) diagnosis against baselines from the field of computational flow cytometry (FC), as well as the Variational Autoencoder literature.
Summary
The authors propose a hierarchical generative model with both continuous and discrete latent variables. The authors empirically demonstrate that the latent space of their model separates well healthy vs pathological cells in a dataset for Chronic lymphocytic leukemia (CLL) diagnostics. 
Main
Overall the paper is reasonably well written. There are a few clarity issues detailed below.
The results seem very promising as the model clearly separates the two types of cells. But more baseline experiments are needed to assess the robustness of the results. 
Novelty
The model introduced is a variant of a deep latent Gaussian model, where the top-most layer is a discrete random variable. Furthermore, the authors employ the Gumbel-trick to avoid having to explicitly marginalize the discrete latent variables.
Given the extensive literature on combining discrete and continuous latent variables in VAEs, the novelty factor of the proposed model is quite weak. 
The authors use the Gumbel-trick in order to avoid explicit marginalization over the discrete variables. However, the number of categories in their problem is small (n=2), so the computational overhead of an explicit marginalization would be negligible. The result would be equivalent to replacing the top of the model p(y) p(z_L|y) by a GMM p_{GMM}(z_L) with two Gaussian components only.
Give these observations, it seems that this is an unnecessary complication added to the model as an effort to increase novelty. 
It would be very informative to compare both approaches.
I would perhaps recommend this paper for an applied workshop, but not for publication in a main conference.
Details:
1) Variable h was not defined before it appeared in Eq. (5). From the text/equations we can deduce h = (y, z_1, …, z_L), but this should be more clearly stated.
2) It is counter-intuitive to define the inference model before having defined the generative model structure, perhaps the authors should consider changing the presentation order.
3) Was the VAE in VAE+SVM also trained with lambda-annealing?
4) How does a simple MLP classifier compares to the models on Table 1 and 2?
5) It seems that, what is called beta-VAE here is the same model HCDVAE but trained with a lambda that anneals to a value different than one (the value of beta). In this case what is the value it terminates? How was that value chosen?
6) The authors used 3 stochastic layers, how was that decided? Is there a substantial difference in performance compared to 1 and 2 stochastic layers?
7) How do the different models behave in terms train vs test set likelihoods. Was there overfitting detected for some settings? How does the choice of the MCC threshold affects train/test likelihoods?  
8) Have the authors compared explicit marginalizing y with using the Gumbel-trick?
Other related work:
A few other papers that have explored discrete latent variables as a way to build more structured VAEs are worth mentioning/referring to:
[1] Dilokthanakul N, Mediano PA, Garnelo M, Lee MC, Salimbeni H, Arulkumaran K, Shanahan M. Deep unsupervised clustering with gaussian mixture variational autoencoders. arXiv preprint arXiv:1611.02648. 2016 Nov 8.
[2] Goyal P, Hu Z, Liang X, Wang C, Xing E. Nonparametric Variational Auto-encoders for Hierarchical Representation Learning. arXiv preprint arXiv:1703.07027. 2017 Mar 21.