Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.
review_content: summary: This manuscript proposed a framework for flow-based unsupervised dataset/distribution alignment, where a variational objective was derived and is shown to upper bound the JSD. Based on the analysis, the authors derive results linking deep techniques for UDA and motivated their approach. Unlike the adversarial training-based UDA methods, this work adopted the flows-based methods to minimize the JSD upper bound and has been shown tighter than other methods. The paper is generally well written and easy to follow the main idea.
strengths_and_weaknesses: Strengths:

1. This paper investigates an interesting flow-based distribution alignment method. It provides good insights into the flow-based approaches, which is a promising research direction.

2. The proposed work is well-motivated and the analysis seems to be technically sound.

Weakness:
1. The cooperative training, which looks like a min-min optimization, seems to lead the Q model to be difficult to train since the proposed methods require the transformations must be invertible. In this respect, I think adversarial training might be better. I would expect the authors to provide further explanations/ discussions for this point.  

2. The empirical evaluations are not sufficient, I would expect to see more comparisons with recent strong baselines in DA (especially those with adversarial training objectives) with more challenging datasets (e.g. Office-31, Office-Home or DomainNet etc.)

questions: 1. The proposed methodology has cooperative training (min-min optimization). Will this optimization be easy to solve, considering that the method requires the transformations should be invertible?

2. Since this work claims to opt for the cooperative objective rather than adversarial training ones, the benefits of this choice should be well-explained.

3. I would expect to see more comparisons with recent strong baselines in DA (especially those with adversarial training objectives) with more challenging datasets (e.g. Office-31, Office-Home or DomainNet etc.)

Minor questions:
What is the exact meaning of $Z_{mix}$ in Theorem 2.2?
limitations: The current evaluations are only conducted for some simple datasets, making it hard to check the empirical effectiveness.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: Authors propose a novel non-adversarial Alignment Upper Bound (AUB) objective that uses normalizing flows to learn invertible mappings from respective domains to a shared latent space modeled explicitly by another parametric density model Q. The objective, up to a constant, equals the weighted log-likelihood of multiple flow densities each fitted to samples from the respective domain using a shared prior Q that is also optimized. Authors show that the proposed objective upper bounds the JSD and leads to a minimization alignment problem. Authors show that the objective is a generalization of LRMF and AlignFlow, and illustrate specific toy configurations in which equivalent LRMF and AlignFlow models fail. Authors show that their method achieves better alignment on tabular data, and achieves better image realism when transforming across MNIST digit classes compared to both LRMF and AlignFlow.
strengths_and_weaknesses: 1. The main idea is well-motivated and is presented very well. 
2. The relation to prior work is established.
3. The derivations are sound.
4. Toy experiments highlight specific cases in which prior work fails.

Nevertheless, I still have some concerns:

1. L168 “AlignFlow without adversarial loss terms is a special case of our method for two distributions where the density model class Q only contains the standard normal distribution (i.e., a singleton class).”, In the “Maximum Likelihood Estimation” section of the AlignFlow paper, Grover et al. discuss that parameter sharing across early layers might be helpful ("Instead, we can share parameters between the two mappings"). In my understanding, an MLE-based AlignFlow with shared layers would be equivalent to an AUB with flow-based Q - is that correct?

2. L178 “[LRMF does] not uncover the connection of the objective as an upper bound on JSD”. Looking at the LRMF paper, the sub-section “Relation to Jensen-Shannon divergence and GANs” seems to establish some relationship between the introduced objective the JSD?

3. Both tabular and digit experiments tell very little about the semantic quality of the alignment. The MNIST digit class transformation task is somewhat contrived. Is there a chance you could include at least an experiment with USPS-to-MNIST adaptation with post-adaptation classification scores?

Some minor comments:

The equation introduced in L119-120 is critical for going from (3) to (4), but can be easily missed. I would have appreciated it if the eq in L119-120 had its own number and was referenced around L130.

There seems to be a pair of parentheses missing in eq (4-5) - the log() should be applied to the product of J and Q, but log Q |J| reads as if log Q is weighed by |J|.
questions: See concerns above.
limitations: Authors discuss that the performance of their approach is limited by the quality of the underlying density model.
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper addresses the problem of alignment of multiple distributions. The authors propose a new method for distribution alignment based on invertible flow models and a new training objective which gives an upper bound on the generalized Jensen-Shannon divergence (GJSD) between the distributions. The method learns invertible transformations (one per each distribution) from the data space to a shared latent space. The invertible mappings are optimized so that the induced latent space distributions are as close as possible to a variational distribution Q which is chosen from an appropriate family of distributions. The parameters of the variational distribution Q are optimized (via maximum likelihood) so that Q approaches the mixture of the latent space distributions. An important feature of the approach is that the invertible transformations and the variational distribution are trained cooperatively meaning that the training is cast as  a min-min game in contrast to min-max games arising in adversarial alignment approaches.

The authors derive a variational upper bound on the GJSD and convert this upper bound to a loss function which can be computed (and optimized) using only samples from the original distributions. The authors prove that global minima of the objective correspond to invertible mappings that align the distributions. The authors also explain that previous flow based methods (AlignFlow with MLE objective only and LRMF) can be seen as special cases of the proposed approach. Finally, the authors suggest that the proposed training objective can be used as a relative metric for evaluation of alignment methods (similar to how ELBO is used as a metric for density estimation).

The authors conduct experiments on toy data, UCI tabular datasets, and MNIST image dataset. The proposed approach is shown to outperform the AlignFlow and LRMF in terms of alignment, parameter efficiency, translation quality, and training stability.

strengths_and_weaknesses: S: strength W: weakness

S1: The paper contributes to the line of work on unsupervised distribution alignment via flow models and proposes a novel (to the best of my knowledge) approach which generalizes some of the existing techniques.

S2: The paper is technically sound. The theoretical results and the proofs are correct. The chosen experimental setups and metrics are adequate. 

S3: The paper is clearly-written and well-organized. The presentation of the theoretical motivation behind the method is precise. The experiments are described in detail and the results are reported clearly. 

W1: Paper is missing references to a related line of work (references [1] and [2] below) on non-adversarial alignment methods via iterative flow models based on sliced optimal transport.

[1] Dai, Biwei, and Uros Seljak. "Sliced Iterative Normalizing Flows." International Conference on Machine Learning, 2021

[2] Zhou, Zeyu, et al. "Iterative Alignment Flows." International Conference on Artificial Intelligence and Statistics, 2022.

questions: Q1: I believe that Theorem 2.5 can be formulated as “global minima of L_AUB is attained if and only if the distributions are aligned”. So, the statement of the theorem can be stronger (implication both ways rather than only “global minima implies alignment”). The current proof already confirms the implication in the opposite direction.

Q2: (line 125) “|J_T(x)| is the determinant of the Jacobian of T” -> “|J_T(x)| is the absolute value of the determinant of the Jacobian of T”

Q3: legends in Figure 3 are barely readable. What do the green ellipses represent?

Q4: (line 227) what does “identity-initialized” Q mean?

Q5: The role of the variational distribution Q is still not completely clear to me after reading the paper. In principle the alignment of P_1, …, P_k can be achieved by optimizing the AUB loss with any fixed Q (as AUB is equivalent to the sum of KL(P_j, Q)). I understand that when Q minimizes the AUB loss (cooperatively with P_1, … P_k) it is supposed to help optimization over P_1, …, P_k. The optimal Q fits the mixture of P_1, … P_k exactly, and when Q reaches that optimum, the AUB becomes exactly equal to the GJSD between the distributions. Given this interaction between P_1, …, P_k and Q, I have two questions:

Q5.1: Is it really important that AUB loss gives an upper bound on GJSD? It seems that this fact does not affect the ability of the method based on AUB minimization to align the distributions. I understand that having a limited family of variational distributions Q might be problematic (as the results of AlignFlow on synthetic data suggest). However, it is still not clear what is the best choice of Q (since any Q should work in principle).

Q5.2: It seems that optimization of L_AUB w.r.t. Q could lead to a problematic optimization landscape for P_1, … P_k. Assuming that Q reaches its goal and fits the mixture of P_1, …, P_k, such Q most likely will be multimodal (especially if distributions P_i have disjoint supports at the initialization) with each mode corresponding to one of P_i. Then, the optimization objective w.r.t. P_i is equivalent to reverse-KL (KL(P_i, Q) and we are optimizing over P_i). The reverse-KL optimization is known to have a bias towards locally fitting modes of the target distribution, so P_i being a mode of the mixture might be stuck in a local optima when Q fits the mixture. 

limitations: I do not foresee any direct negative societal impact of this work.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper proposes a new objective function to calculate the Generalized Jensen-Shannon Divergence, which can be used to the field of the unsupervised distribution alignment. The experimental results on different tasks show the effectiveness of the proposed objective.
strengths_and_weaknesses: Strengths:
1. The theoretical analysis provided is interesting and make sense.
2. This paper is easy to read.
3. The experimental results are good and adequate.

Weaknesses:
1. The motivation of this paper is vague and unconvincing.
2. The novel of this paper is limited.
questions: 1. The impact of different values of $w$ on the final performance of the model is not clear.
2. The main part of the article does not refer to Figure 2.
3. As for the objective 6 and Algorithm 1, we can see that the update of $Q$ and $T$ is in a step by step manner, why not update the $Q$ and $T$ in and end-to-end manner, 
4. How to guarantee ${P_{{Z_{mix}}}} \in Q$ ？
5. From Figure 5, we can see that the training process of the proposed method in this paper is obviously more stable than that of AlignFlow, I hope the author can give a theoretical analysis.
6. I hope the author can provide the results when using this objective function in the domain adaptation and image generation.
limitations: The authors have adequately addressed the limitations and potential negative societal impact of their work.
review_title: null

Meta-review: metareview: This paper addresses the problem of the alignment of multiple distributions. The authors propose a new method for distribution alignment based on invertible flow models and a new training objective that gives an upper bound on the generalized Jensen-Shannon divergence (GJSD) between the distributions.   The authors derive a variational upper bound on the GJSD and convert this upper bound to a loss function.  They prove that global minima of the objective correspond to invertible mappings that align the distributions and conduct experiments on a few toy data.   Most of the reviewers recognize the motivation, and effort of the paper.  During the discussion, the authors also successfully addressed some of the reviewers' questions.  However, concerns about the novelty as well as the lack of in-the-wild evaluation remain.   Reviewers acknowledge the problem of domain alignment using normalizing flows is hard, and this paper does not make any significant breakthroughs in this matter.  But, it seems it has made a small measurable step towards solving it, which is good enough to be recommended to accept.  

confidence: Certain
Decision: Accept