__ Summary and Contributions__: Update:
I have read the rebuttal and am glad to see that the authors are willing to update the paper to include more discussions about prior work.
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The paper discusses a variational EM approach to learning latent variable models, where the prior distribution is a learnable energy-based model and the posterior inference model is obtained by finite MCMC sampling over the true posterior (available via the energy of the joint). The paper demonstrates superior performance compared to existing VAE approaches that aims to use learnable prior in reconstruction and generation.

__ Strengths__: ### Technical Novelty
The paper proposes to consider a learnable energy model as the latent prior model. While there are various existing work on learning a latent prior model, framing it as an energy based model seems relatively interesting. The paper is conscious about the difference of "short-run" MCMC chains and the actual posterior, and provides a discussion in section 3.5 about it.
### Empirical evaluation
Empirically the proposed methods appears to have better performance than existing methods that uses learnable prior and and more explicit inference model (such as two-stage VAE).

__ Weaknesses__: ### Positioning and comparison with existing work
It seems that the authors are keen to describe the method as "maximum likelihood", and the short run Markov chain approach as an approximation to the method. However, one could also interpret the short run Markov chain sampler as an approximation to the true posterior p_\theta(z | x), and interpret the entire approach as EM / coordinate ascent between the inference sampling procedure and the generative model. Therefore, I personally feel like the claims in lines 191-194 about the approach being more general than exact MLE is slightly misleading.
If we use the EM perspective, there exists prior work that utilizes MCMC to derive a better posterior inference estimate; see Hoffman's 2017 paper "Learning Deep Latent Gaussian Models with Markov Chain Monte Carlo", where an inference model is augmented with MCMC steps over the true posterior. It seems to be a highly relevant work to this paper.
### Evaluation
Surprisingly, for the image generation task, there is no test log-likelihood metric (or lower bounds) that is observed on most VAE models (since the paper is mostly motivated from an MLE perspective). The lower bound seems to be relatively difficult to estimate due to the partition function being not easy to compute, but I think it is not impossible with annealed importance sampling?

__ Correctness__: I have not found any strict incorrectness with regards to the claims and method.

__ Clarity__: The paper is well written and detailed, but perhaps could benefit from bolding the variables that are assumed to be vectors. Moreover, a discussion of the computational efficiency as a variable of K in the main paper could help practitioners in understanding the computational trade-offs of this approach.

__ Relation to Prior Work__: As mentioned in the weakness section, the work misses one important related work [Hoffman et al, 2017] on short run MCMC utilized to improve VAE inference modeling. If we ignore the energy based latent variable, the proposed MLE method is a special case to the aforementioned paper with a trivial initial inference distribution.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper introduces a method to train generative models based on latent vectors while learning a flexible latent distribution alongside the generator neural network. This latent distribution is based on an energy-based correction of a standard Gaussian distribution, the energy value being defined as the output of a secondary neural network. The training of both networks is performed concurently, using short-run MCMC Langevin dynamics to approximate the latent prior p(z) and the latent posterior p(z|x).
The paper then provides experimental comparison of their model to several other approaches from the literature both in the context of image generation and text synthesis, showing their model is competitive or performs better.

__ Strengths__: The authors provide a detailed theoretical analysis and justification of their training algorithm. The model is compared to several other known methods on various datasets, and an ablation study is provided in appendix. The computing cost induced by the use of MCMC (as opposed to a standard VAE) is discussed and appears sufficiently low for the approach to be worth considering as a way to improve the quality of generative models.
The use of short-run MCMC as an affordable yet efficient approximation for sampling the variational posterior, and the empirical validation that it gives good results even for relatively short run-lenghts is an interesting result.

__ Weaknesses__: The main uncertainty I have is regarding the computing cost of this approach. The use of MCMC Langevin dynamics to estimate sampling the variational posterior implies a significant increase in the number of backpropagation passes through the generator neural network (K+1 passes in total for each mini-batch). On larger generator models this cost can become quite significant.
In particular, the authors only provide a numerical comparison of the training times in appendix for their SVHN example, which is the smallest of the models in their comparison. The computational cost induced on text-based auto-regressive models is not presented at all, which I find a little unsatisfying.

__ Correctness__: The paper provides a well-justified model, and compare it to the literature using standard evaluation scores.

__ Clarity__: The paper is clearly written.

__ Relation to Prior Work__: The prior work regarding MCMC energy based models and VAEs with flexible priors is clearly discussed.
The main contribution of the paper seems to be the use of MCMC Langevin dynamics to build an Energy-Based model in the latent space of a generator model, rather than directly on data space, and a joint procedure to train both the EBM and the generator network simultaneously.

__ Reproducibility__: Yes

__ Additional Feedback__: l.61 has a typo : "soild" instead of "solid".
**Post-rebuttal response**
I'll thank the authors for the clarifications regarding the computing cost of their approach.
I think this paper presents an interesting approach and could be a good addition to NeurIPS, however I agree with the concerns raised by the other reviewers regarding relation to prior work and comparison with relevant baselines. As such, I'm changing my score to 6.

__ Summary and Contributions__: Authors propose energy based generative model that augments a top down model(DAG) with a latent space learnt via EBM that potentially corrects standard noise prior. They modify MLE formulation by using short run MCMC for learning their model. In addition to providing theoretical basis, they also showcase qualitative samples and quantitative results on their setup.

__ Strengths__: 1)At the outset, paper clearly describes their method and is well articulated.
2)Also appreciate the theoretical section that delves deep into the learning mechanism while providing required insights.
3)Supplementary material comprehensive and has all the required details

__ Weaknesses__: Results are not compelling enough to showcase what the model is capable of.
It could be important to showcase what the corrected prior/posterior is capable of than standard distributions
comparisons with up-to-date literature could be useful

__ Correctness__: yes

__ Clarity__: yes

__ Relation to Prior Work__: work includes relevant literature, but could include more recent research on inference models (both GAN and VAE based) other than just BiGAN

__ Reproducibility__: Yes

__ Additional Feedback__: I have one main concern that experiment section is not comprehensive enough
I would appreciate if there are additional experiments to include usefulness of learnt latent space such as semi-supervised or one-shot related results.
I was wondering if some details of theoretical results can go into additional material and possible include some intuitive connections with other topdown inference based generative models.

__ Summary and Contributions__: This paper proposes a generative model in the form of p(z, x) = p(z) p(x|z) where p(z) is an exponential tilting of a simple Gaussian prior (i.e., the prior is an energy-based model), the likelihood function p(x|z) is a decoding distribution similar to the one in VAEs.
This paper proposes to use short-run MCMC sampling to sample from both true posterior and EBM prior distributions during training. The experimental results show that the proposed model outperforms small VAE modes and different variants of VAEs in image generation, text generation, and anomaly detection tasks.

__ Strengths__: * Novelty:
Recently energy-based generative models have gained a lot of momentum in the community. The idea of using an EBM in the latent space of generative models, as proposed in this paper, is an interesting and impactful extension to the current efforts. EBMs for the prior distribution has been examined in the past for binary latent variables (in DVAEs, see below). However, this paper goes beyond simple energy functions and it shows how EBMs modeled by a neural network can be applied to the continuous latent variables.
* Theoretical grounding:
I found the theoretical discussion in Sec 3.5 very helpful for understanding the error induced by approximate sampling. It is very interesting to see that short-run MCMC can be thought of as a variational bound on log-likelihood where the "variational" distribution is the distribution of samples generated by the short-run MCMC samples.
* Empirical Evaluation:
The method is examined in several image and text datasets. It has been shown that the proposed model outperforms several prior works including VAEs that use simple Gaussian distribution.
* Relevance to the NeurIPS community:
The work is highly relevant to the community.

__ Weaknesses__: * Missing comparison against persistent sampling:
This paper proposes to use short-run MCMC to sample from both the prior and true posterior. In practice, since we have only one prior distribution, sampling from the prior can be also done using persistent sampling which often improves the performance of EBMs by a large margin. It's not clear why the proposed method uses short-run MCMC that can potentially mix slowly and can introduce sampling error. Moreover, Eq. 13 shows that sampling error turns the objective into an upper bound on the log-likelihood. This can be dangerous as the model may start increasing the gap between the distribution of approximate samples and the EBM prior by making the distribution harder to sample from.
* Practical limitations:
While we have a single prior distribution, we have true posterior distributions as many as training data points. This means that in every parameter update, we require sampling from the true posterior distribution per data point. The computational complexity of sampling from the true posterior is much more than the prior as the former requires evaluation of both prior and decoder networks. This paper limits the decoder to small models with a few layers, but in practice when decoders are deep, running 20 MCMC steps requires evaluating a very expensive network 20 times in each training iteration. It is not clear why the proposed model abandons amortized inference for approximating the true posterior. A variational distribution can be easily used to infer the latent variables in an amortized fashion (as done in VAEs and DVAEs).
* Claims on the stability of the algorithm:
In line 48, it is claimed that training EBMs unlike GANs doesn't suffer from instability. However, as observed by Du & Mordatch NeurIPS 2019 training energy-based models can be unstable when the sampling procedure in the negative phase cannot catch up with sharp energy functions. In my experience with EBMs, this problem can be a big barrier to training EBMs.

__ Correctness__: As long as I can see the derivations and the overall hypotheses seem correct.
Regarding the evaluation, it seems that all the comparisons are done internally by comparing the proposed model against re-implementation of the prior art by authors (please correct me if I am wrong). It would be nice if the authors could also compare the method against prior work that uses similar network architectures by reporting results from the corresponding papers. For example, 2-Stage VAE [12] have reported better FID scores for their method originally (FID score for 2-stage VAE on CIFAR 10, original 72.9, reported 109.77)

__ Clarity__: Yes, the paper is well written.

__ Relation to Prior Work__: * I found the main missing prior works that are not discussed in this paper are DVAEs (DVAE [1], DVAE++[2], and DVAE#[3]) that use energy-based Boltzmann machines in the prior of VAEs. This work differs from DVAEs in two aspects:
i) The proposed model goes beyond binary latent variables and simple RBM energy function (which were used in DVAEs), and it uses neural networks to define an energy function over continuous variables.
ii) The proposed work uses short-run MCMC to samples from true posterior which can have computational disadvantages as discussed in the weaknesses.
[1]: Discrete variational autoencoders, ICLR 2017.
[2]: Dvae++: Discrete variational autoencoders with overlapping transformations, ICML 2018.
[3]: DVAE#: Discrete Variational Autoencoders with Relaxed Boltzmann Priors, NeurIPS 2019.
* Recently, Ghosh et al. ICLR 2020 showed that regularized autoencoders can generate high-quality images. Although the proposed method outperforms Ghosh et al. on the CIFAR-10 and CelebA datasets, it would be nice to include this when referred to regularized autoencoders. Also, VQ-VAE can be considered as a 2-stage VAE and is missing in the discussion.

__ Reproducibility__: Yes

__ Additional Feedback__: **** After rebuttal ****
After discussing with all the reviewers, I agree that this paper has its own merits and it is a good addition to NeurIPS. However, given the current concerns regarding the discussion of the previous work, I'm going to stick to my original rating. I believe the stability issues and the efficiency of training for big models can be a good subject for future studies.
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