Paper:
We introduce and demonstrate the variational autoencoder (VAE) for probabilistic non-negative matrix factorisation (PAE-NMF). We design a network which can perform non-negative matrix factorisation (NMF) and add in aspects of a VAE to make the coefficients of the latent space probabilistic. By restricting the weights in the final layer of the network to be non-negative and using the non-negative Weibull distribution we produce a probabilistic form of NMF which allows us to generate new data and find a probability distribution that effectively links the latent and input variables. We demonstrate the effectiveness of PAE-NMF on three heterogeneous datasets: images, financial time series and genomic.

Review:
TITLE
A VARIATIONAL AUTOENCODER FOR PROBABILISTIC NON-NEGATIVE MATRIX FACTORISATION
REVIEW SUMMARY
Well written, interesting new idea, modest technical contribution, limited demonstration.
PAPER SUMMARY
The paper presents an approach to NMF within a variational autoencoder framework. It uses a Weibull distribution in the latent space. 
QUALITY
The work appears technically sound except for minor typos. 
CLARITY
Overall the paper is a pleasure to read. Only the presentation of the standard vae could be more clear.
ORIGINALITY
The method is (to my knowledge) novel. 
SIGNIFICANCE
I think this paper is a significant contribution. I feel I have learned something from reading it, and am motivated to try out this approach. I believe there should be a wide general interest. The technical contribution is perhaps somewhat modest, as the paper fairly straightforwardly includes non-negativity in a vae setting, but I think this is a good idea. The demonstration of the algorithm is also quite limited - I would have enjoyed seeing this applied to some more reaslistic, practical problems, where perhaps the quantification of uncertaincy (which is one of the main benefits of a vae-based nmf) would come more directly into play. 
FURTHER COMMENTS
page 3
The presentation of the VAE objective is a bit oblique. The statement "they require a different objectiv function" is not wrong, but also not very precise. The equality in eq. (2) is incorrect (I assume this is meant to be a stochastic approximation, i.e. the expectation over q approximated by sampling?)
"with \hat v  the reconstructed vector" Not clear. I assume \hat v is reconstructed from a sample from q given v ?
There is a typo in eq. 3. The first factor in the second to last term should be (lambda_1/\lambda_2)^(k_2)