Paper:
Identifiability, or recovery of the true latent representations from which the observed data originates, is a fundamental goal of representation learning. However, most deep generative models do not address the question of identifiability, and cannot recover the true latent sources that generate the observations. Recent work proposed identifiable generative modelling using variational autoencoders (iVAE) with a theory of identifiability. However, due to the intractablity of KL divergence between variational approximate posterior and the true posterior, iVAE has to maximize the evidence lower bound of the marginal likelihood, leading to suboptimal solutions in both theory and practice. In contrast, we propose an identifiable framework for estimating latent representations using a flow-based model (iFlow). Our approach directly maximizes the marginal likelihood, allowing for theoretical guarantees on identifiability, without the need for variational approximations. We derive its learning objective in analytical form, making it possible to train iFlow in an end-to-end manner. Simulations on synthetic data validate the correctness and effectiveness of our proposed method and demonstrate its practical advantages over other existing methods.

Review:
This paper is about learning an identifiable generative model, iFlow, that builds upon a recent result on nonlinear ICA. The key idea is providing side information to identify the latent representation, i.e., essentially a prior conditioned on extra information such as labels and restricting the mapping to flows for being able to compute the likelihood. As the loglikelihood of a flow model is readily available, a direct approach can be used for learning that optimizes both the prior and the observation model.	
The paper is very clear and very easy to follow. The idea is quite clear, and the direct approach is really attractive. Unfortunately, the experimental section is quite limited and does not fully study representational aspects. There is only an illustrative simulation on synthetic data, that in a sense verifies the theory. I was also not able to see the additional insight that the identifiability theory in 4.2 provides additional to Khemakhem et al. (2019). Please clarify.
My main concern is that this model is actually a supervised model that learns a mapping from u to x via z. Hence, the theoretical ‘identifiability’ guarantee needs to be stated with some care as this depends on the choice of an arbitrary u. For example if we set x = u, will learning take place? Please comment.  
Overall, I like the approach but I am uncertain about the level of novelty. For such a paper, one should expect a much more involved computational study. In this respect, I feel that the paper could be accepted but it certainly feels as if it needs more computational results as otherwise the original contribution would be too incremental for ICLR standards. I am giving a provisional reject in the hope that the authors will provide convincing arguments about their original contributions for clarification.