Why be adversarial? Let's cooperate!: Cooperative Dataset Alignment via JSD Upper Bound
Authors: Wonwoong Cho
Ziyu Gong
David I. Inouye
Venue: ICLR 2022 Submitted (2022)

Abstract: Unsupervised dataset alignment estimates a transformation that maps two or more source domains to a shared aligned domain given only the domain datasets.
This task has many applications including generative modeling, unsupervised domain adaptation, and socially aware learning.
Most prior works use adversarial learning (i.e., min-max optimization), which can be challenging to optimize and evaluate.
A few recent works explore non-adversarial flow-based (i.e., invertible) approaches, but they lack a unified perspective.
Therefore, we propose to unify and generalize previous flow-based approaches under a single non-adversarial framework, which we prove is equivalent to minimizing an upper bound on the Jensen-Shannon Divergence (JSD).
Importantly, our problem reduces to a min-min, i.e., cooperative, problem and can provide a natural evaluation metric for unsupervised dataset alignment.
We present empirical results of our framework on both simulated and real-world datasets to demonstrate the benefits of our approach.

Under review as a conference paper at ICLR 2022
WHY BE ADVERSARIAL? LET'S COOPERATE!: COOPERATIVE DATASET ALIGNMENT VIA JSD UPPER BOUND
Anonymous authors Paper under double-bling review
ABSTRACT
Unsupervised dataset alignment estimates a transformation that maps two or more source domains to shared aligned domain given only the domain datasets. This task has many applications including generative modeling, unsupervised domain adaptation, and socially aware learning. Most prior works adversarial Iearn- ing (i.c., min-max optimization), which can be challenging to optimize and eval- uate. few recent works explore non-adversarial flow-based (i.e., invertible) ap- proaches, but they lack a unified perspective. Therefore, we propose to unify and generalize previous flow-based approaches under single non-adversarial frame- work, which we prove is equivalent to minimizing an upper bound on the Jensen- Shannon Divergence (ISD). Importantly, our problem reduces min-min, i.e., cooperative, problem and can provide natural evaluation metric for unsupervised dataset alignment. We present empirical results of our framework on both simu- lated and real-world datasets to demonstrate the benefits of our approach.
INTRODUCTION
In many cases, a practitioner has access to multiple related but distinct datasets such agricultural measurements from two farms, experimental data collected in different months, or sales data before and after a major event. Unsupervised dataset alignment (UDA) the ML task aimed at aligning these related but distinct datasets in a shared space, which may be latent space, without any pairing information between the two domains (i.e., unsupervised). This task has many applications such as generative modeling (e.g., (Zhu et al. (017)), unsupervised domain adaptation (e.g.. (Grover cct al. 2020. Hu et al. 2018 batch effect mitigation in biology (e.g., (Haghverdi et al. 2018)), and Tarrness-wware learning (e.g., (Zemel et al. 2013)).
The most common approach for obtaining such alignment transformations stems from Generative Adversarial Networks (GAN)GGoddieloow et al. 2014), which can be viewed minimizing lower bound on the Sensen-Shannol Divergence (JSD) between real and generated distributions. The lower bound is tight if and only if the inner maximization is solved perfectly. CycleGAN (Zhu al., 2017) maps between two datasets via two GAN objectives between the two domains and a cycle consistency loss, which encourages approximate invertibility of the transformations. However, ad- versarial learning can be challenging to optimize in practice (see e.g. (Lucic et al.] 2018 Kurach al. 2019h) in part because of the competitive nature of the min-max optimization problem. Per- haps more importantly. the research community only has reasonable GAN evaluation metrics for certain data types. Specifically, the commonly accepted Frechet Inception Distance (FID) (Heusel
et al. 2017) is only applicable to image auditory data, which have standard powerful pretrained classifiers, and even the implementation of FID can have evaluation issues (Parmar al. 2021). No
clear metrics exist for tabular data non non-perceptual data.
Recently, flow-based methods that leverage invertible models have been proposed for the UDA task (rroerr et al.) 2020 Usman et al. 2020). AlignFlow (Grover al., 2020) leverages invertible models to make the model cycle-consistens (i.e., invertible) by construction and introduce exact log-like likeliil loss terms derived from standard flow-based generative models complementary loss terms to the adversarial loss terms. Yet.-Alignnt AlignFlow still leverages adversarial learning- Log- likelihood ratio minimizing flows (LRMF) (Usman et al. 2020) invertible flow models and

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Z
T
T2
P
Q
Pxa
(x))
Pro(xa)
Pxa
(a)
(b)
Figure 1: Overview of our proposed method. In our framework, domain specific transformation functions {- and a density model Q are cooperatively trained to make the transformed representations be indistinguishable in the shared latent space. (a) I-D example. By minimizing our proposed AUB loss, the transformation functions Ti and T3 are trained to map the corresponding distributions Px, and Px, latent distributions (x,( and (X²) that have higher likelihood with respect to base distribution Q. The density model Q, on the other hand, is trained to fit the mixture of the latent distributions PI, and (1)() (b) Intuitively, the optimization process of our method can be seen to make the Q distribution tight around the mixture of latent distributions to increase the likelihood (i.e., MLE) while the transformation functions T₃ and T2 are encouraged to expand to fill the latent space defined by Q. Eventually, the latent distributions and Q are converge to be the same distribution, which means that they are aligned.
density estimation for distribution alignment without adversarial learning and define a new metric based on the log-likelihood ratio. However, LRMF depends heavily on the density model class and can only partially align datasets if the target distribution is not in the chosen density model class. Additionally, the LRMF metric depends this density model class and is only defined for two datasets.
Therefore, to provide alternative to adversarial learning and generalize previous flow-based ap- proaches, we propose a unified con-adversarial UDA framework, which we prove equivalent to minimizing upper bound on the JSD. Importantly, our problem reduces min-min, i.e., coop- erative, problem, and the JSD upper bound can provide natural evaluation metric for UDA that be applied in any domain. Our framework requires two parts, the outer minimization requires an invertible model and the inner minimization requires a density model (c.g., Gaussian mixture models or normalizing flows (Dinh al. 2017)). We summarize our contributions as follows:
We prove that a minimization problem over density models is an upper bound on a gen- eralized version of JSD that allows for more than two distributions. Importantly, we also theoretically quantify the bound gap and show that it can made tight if the density model class is flexible enough.
We use this JSD upper bound to derive novel regularized loss function for UDA and explain its relationship to prior methods.
We empirically demonstrate the benefits of our method compared to prior flow-based mod- els on both simulated and real-world datasets.
Notation We will denote distributions Px (x) where X is the corresponding random variable. Invertible functions will be denoted by T(-). We will use X, Px, to denote the observed random variable from the j-th distribution. We will use Zj A (X(,) Pz, PI,(X,) ) to denote the latent random variable of the j-th distribution after applying T₁ to X, (and note that X, = Tí (Z₂)). We will denote the mixtures of these observed latent distributions Px... $ E, Px, and Pz... A Wy where w is a probability vector. We denote KL divergence, entropy, and cross entropy (L(.,)) H(-). and Ho(:-)), respectively, where KL(P,, Q) H)(P,Q) H(P).
REGULARIZED ALIGNMENT UPPER BOUND Loss
In this section, we will introduce our main theoretical result proving an upper bound on the gener- alized JSD divergence, deriving our loss function based on this, and then showing that minimizing this upper bound results in aligned distributions assuming large enough capacity of the model com- ponents.
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Background: Normalizing Flows and Invertible Models Normalizing flows generative models that have tractable distributions where exact density evaluations and efficient samplings ensured (Kobyzev et al., 2021). Such models leverage the change of variables formula to cre- ate an invertible mapping 7 such that Px (x)
where Pz is a known latent distribution and [r)] is the absolute value of determinant of the Jacobian of the invertible map T. For sampling in distribution Px, one need to first sample from the latent distribution P'z and then apply the inverse transform T Therefore the key challenge in designing invertible models is to have computationally efficient inverse evaluation and Jacobian determinant calculation. Many approaches have been proposed by parameterizing mapping function T as deep neural networks cluding autoregressive structures (Kingma et al. 2016; Papamakarios et al.) (202). coupling layers (rattonl al.)
Pz
(Dinh et al. 2017 (inggma and Dhariwal] 2018), oridinary differential
2018), and invertible residual networks(Chen et al., 2019) |Behrmann et al. 2019). Flow models can be then learned efficiently by maximizing the likelihood for the given data.
Background: Generalized JSD We remind the reader of the generalized Jensen-Shannon diver- gence for more than two distributions, where the standard JSD is recovered if w1 W2 0.5.
Definition 1 (Generalized Lonsen-Shanon Divergence (GISD) (Lin 1991)), ) Given l distributions { and corresponding probability weight vector w, the generalized -aasennShannon di- vergence is defined as (proof of equivalence in appendix):
(Px. Px.)) = L,
= =H{E, - - H(Px,)
(I)
2.1 GJSD UPPER BOUND
The goal of distribution alignment is to find a set of transformations (-) (which will be in- vertible in our case) such that the latent distributions align, i.e., Pr,(x,) Pr,,(X,))) or equivalently Pz, = -P,, for all j + j'. Given the properties of divergences, this alignment will happen if and only if GJSD(PZ. ... .zz)) O. Thus, ideally, we would minimize GJSD directly with respect Ty.ie..
min Th TuET
GJSD( Pr(())) = min H ((, - TKET
-,, wy H(Pr)(x,),
where T is class of invertible functions. However, cannot evaluate the entropy terms in Eqn. 2 because we do not know the density of Px,: we only have samples from Px,- Therefore, we will upper bound the first entropy term in Eqn. 2 (H (E, "ey,,,x Px2) )) using an auxiliary density model and decompose the other entropy terms by leveraging the change of variables formula for invertible functions.
Theorem 1 (GJSD Upper Bound). Given an auxiliary density model class Q, we form a GJSD upper bound:
GJSD.s ,P2,) min He Q)- u, H(P2))
QEQ
where the bound gap is exactly minqeo KI.(Pz.. Q).
Proof of Theorem I
For any Q E Q, we have the following upper bound:
GJSD (P2,. P2,)) = HP 2.m.. Q) He 2mn: Q) + H((2...) -E H(P2,)
=0
=Ho(P HHomp Zmn Q) KL(( Zmme Q) -) H(P,,)
< Ha Q) E, H H(Pz,),
where the inequality is by the fact that KL divergence is non-negative and the bound gap is equal to (L,(Pznn Q). The Q that achieves the minimum in the upper bound is equivalent to the Q that
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minimizes the bound gap, i...
Q = arg min (P Q) Eun H(p),
(2)
QEQ
Constant W.LS. Q
arg min Ho (P Q)
H (an
(3)
QEO
Constant w.r.L. Q
arg min KT,(Pz Q) QEQ
(4)
The tightness of the bound depends how well the class of density models (e.g.. mixture models, normalizing flows, or autoregressive densities) can approximate Pz.. notably, the bound can be made tight if Pz.. E Q. Also, one key feature of this upper bound is that the cross entropy term can be evaluated using only samples from Px, and the transformations Tj. i.e., He(Pz Zmox Q E, L,n,Ers, Erx, [- log QT; (z;))). However, we still cannot evaluate the other entropy terms Hz,)) since we do not know the densities of P2, (or Px,). Thus, we leverage the fact that the T₃ functions invertible define entropy change of variables.
Lemma 2 (Entropy Change of Variables). Let X ~ Px and Z & T(X) ~ Pz. where T is an invertible transformation. The entropy of Z can be decomposed as follows: H(Pz) = H(Px) + Epx (log (r)))]]
(5)
where Jr(()) is the determinant of the Jacobian of
The key insight from this lemma that H(Px) is constant with respect T and can thus be ignored when optimizing T, while Epx [log JJr (x))] can be approximated using only samples from Px.
2.2 ALIGNMENT UPPER BOUND (AUB)
Combining Theorem I and Lemma 2 we can arrive at our final objective function which is equiva- lent to minimizing an upper bound on the GISD:
GJSD (P Pz.) < min Ho Zum. Q) - E, w, H H(P2,)
(6)
QEQ
min QEQ
E, |- log Q(T, (a)) ",
where the last term E, H(Px,) is constant with respect to T, functions so they can be ignored. We formally define this loss function as follows.
Definition 2 (Alignment Upper Bound Loss), Given ki continuous distributions (Px, Y class of continuous distributions 2, and a probability weight vector w, the alignment upper 1 loss is defined as follows:
CAUB(TI, Te: (Px,) Q, 2)
min E E, QED
(IT)
(8)
where Tý are invertible and JJr, (3)) is the absolute value of the Jacobian determinant.
Notice that this alignment loss can be seen as learning the best base distribution given fixed flow models Ty. We now consider the theoretical optimum if we optimize over all invertible functions. Theorem (Alignment Global Minimum of LAUB). If LAUB is minimized over the class of all functions, invertiblefunnctons,, global minimum of LAUB implies that the latent distributions are aligned, i.e., Pri(x,) = Pry, y for all j & j'. Notably, this result holds regardless of Q.
invertible
Informally. this can be proved by showing that the problem decouples into separate normalizing flow losses where Q is the base distribution and the optimum is achieved only Pry(x,) = Q for all Tç (formal proof in the appendix). This alignment of the latent distributions also implies the translation between any of the observed component distributions. The proof follows directly from Theorem and the change of variables formula.
Corollary (Translation at Global Minimum of LAUB). Similar to Theorem 3 global minimum of CAUB implies translation between any component distributions using the inverses of Tý, i.e Przz (T,(5))())) = Px,, for all j +
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Regularization via Transportation Cost While the alignment objective is the most challenging part of UDA, we suggest that regularization may also useful for practical and stable alignment (or translation) between datasets because there many optimal alignment solutions even many in most cases (see appendix for two examples). We alleviate this issue by adding expected transportation cost (usually squared Euclidean distance) regularization by optimal transport (OT) concepts,
even infinitely
objective inspired
Definition 3 (Regularized Alignment Upper Bound Loss, RAUB). Given similar setup as in Def. and a transportation cost function c(a, b) z foror transporting a point from a to b, the regularized alignment upper bound loss is defined as:
CRAUB(TI, Tk: Pxx, 1=== Q w. 2,
(9)
- min w, log Jr, (x)) Q(T)(x)) + Ac(a, (x)))
QEo
Pseudo-code for training our model with RAUB objective can be found in the appendix Section C.
3 RELATIONSHIE TO PRIOR WORKS
AlignFlow without adversarial terms is a special case AlignFlow (Grover al. 2020) 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., singleton class) and no regularization is used (i.e., À = 0). Thus, AlignFlow can be viewed as initially optimizing poor upper bound on ISD; however, the JSD bound becomes tighter as training progresses because the latent distributions independently move towards the same normal distribution.
LRMF is special case with only (LRMF) (Usman et al. 2020) also a special case of our method for only two distributions, where transformation
transformation Log-likelihood ratio minimizing flows
fixed at the identity (i.e., T2 Id) and regularization is applied (i.e., = 0). While the final practical LRMF objective is a special case of ours, the theory is developed assump- from different but complementary perspective. The LRMF metric developed requires tion about a given density model class, which enables zero point (or absolute value) of the metric estimated but requires fitting extra domain density models. Usman et al. (2020) also do not uncover the connection of the objective as upper bound on JSD regardless of the density model class. Additionally, to ensure alignment, LRMF requires that the density model class includes the true target distribution because only one invertible transform is used, while our approach can the- oretically align even if the shared density model class is weak (see [heorem 3 and our simulated experiments). See Fig. 2 for a comparison of our approach to prior works.
fixed Q
learnable
T
T2
T
T
T2
X1
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X2
X1
Z
X2
X1
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X2
Getting:
doosdd
Getting 'close
Getting dooeed
AlignFlow (MLE only)
LRMF
Ours
Invertible function
Fixed dist.
Changed dist.
Q dist.
Figure 2: High-level comparison with the baseline models. AlignFlow, LRMF, and our setup are illustrated in from left. Transformation functions in AlignFlow are independently trained to be fitted to the fixed standard gaussian distribution. Tj in LRME aimed to directly map the given X, another image distribution X2. The density model Q in LRMF is fixed and learned fit to (Z, X2). In our setup, Ti and 72 are trained to obtain the high likelihood from the learnable Q distribution which is fitted to the shared latent distributions, In every setup, the latent distributions Section
Z1 and Zi are getting closer to the target distribution as training goes by. Details are provided in
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Cooperative networks versus adversarial networks Analogous to the generator G and the dis- criminator D in adversarial learning, our framework has two main networks, Tú and Q2. We can use any invertible function for I (e.g.. coupling- based flows (Dinh et al. 2017), neural ODE flows Grathwohl et al. 2 or residual flows (Chen et al. 2019)) and any (approximate) density models for Q= (e.g., kernel densities (in low dimensions), mixture models, autoregressive densities (Sali- mans et al. 2017). normalizing flows (Kingma and Dhariwal 2018), or even VAEs (Kingma and Welling 2019)). Thus, our framework has similar modularity compared to adversarial approaches. In contrast, we have min-min, i.e., cooperative, optimization problem, but our transformations must be invertible. As another difference, the inner optimization problem may be more challenging (i.e., fitting density model Q2 may be more difficult than fitting an auxiliary discriminator D), but the overall min-min likely to be more stable than the min-max croblem-Specificall our problem can align even if the density model Q2 is not optimal (see SeeeThheremt 3). We expect some of these limitations to be alleviated as new invertible models and density models are continually be- ing developed. Therefore, the proposed approach provides a fundamental alternative to adversarial learning with different strengths and weaknesses.
note that
4 EXPERIMENT
4.1 2D DATASET COMPARISON WITH RELATED WORKS
We compare our method with related works in order to illustrate the limitations of prior methods and how our method overcomes these limitations. All implementation details for the 2D dataset available in the appendix.
Single T vs. Double T (LRMF vs. Ours) We first compare our method with LRMF (Usman et al. 2020) method. We construct the experiment to have the task: Translation between the two half-cireled distributions: X1 and X₂ in the moons dataset. In this example, we made two models, one with LRMF setup and one with our AUB setup. As illustrated in Fig. 3] the LRMF method fails to transform between X₁ and X2. Even though Q can model well enough for Ti (X,). Q can only model the mean and variance of X2 which is obviously not informative enough. Therefore, the LRMF fails to transform between two datasets. While in the AUB setup, both 7 and Tx2X2) modeled to the same distribution which Q can be learned to fit with high likelihood which leads to better translation results. In conclusion, the performance of the LRMF model is limited by the power of the density model Q which means if Q fails model one of the domain distribution in high likelihood, data alignment cannot be achieved with good performances.
Simple Fixed Q vs. Learnable Q (AlignFlow vs. Ours) Next we compare our method with AlignFlow(Grover et al.) [020 Hu et al., 2018)) setup. We construct the experiment to have the task: Transform between the two random patterns X, and X₂ from the randomly generated datasets. Again, we made two models with AlignFlow and our AUB setups respectively. As illustrated in Fig. 3 the AlignFlow method fails to transform between X1 and X2, because the transformed dataset (X1)) and (22)2) failed to reach the normal distribution Q. While in the AUB setup, the density model Q is learned to help fit the transformed distributions Ti (X₂) and T2 (X2). which allows them to be aligned with each other easier. In conclusion, the performance of the AlignFlow model is limited by the performances of the invertible functions.
Regularized vs. Un-regularized (Some prior works vs. Ours) We finally show the importance of the regularization term. We construct the experiment to have the task: Transform between two concentric circles with the same mean but slighted different radius. In this example, we make two models with our AUB approach, but one with regularization and one without. As illustrated in the Figure 4 both models are able to transform between two distributions. However, the transformation pattern is not natural in terms of the movement of each point. Each pair created by the unregularized model has bigger transportation cost compared to the pairs created by the regularized model. There- fore, suggest that by adding transportation cost, the resulting transformation between samples will be closer to the identity function and therefore more stable.
4.2 REAL-WORLD DATASETS
In this subsection, we use two metrics to measure the overall model performances: Frechet Inception Distance (FID) (Heusel al.) 2017) implemented by (Seitzer 2020) and our metric Alignment Upper Bound (AUB).
Metrics

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(a) Original
(b) LRMF
(c) RAUB(Ours)
(d) Original
(c) AlignFlow
(f) RAUB(ours)
Figure 3: Top row is latent space and bottom the data translated into the other space. (a-c) LRMF, which only has one transformation T may not be able to align the datasets if the density model class Q is not expressive enough (in this case Gaussian distributions) while using two transformations as in our framework can align them. (d-f) AlignFlow (without adversarial terms) may not align because Q= is fixed at standard normal, while our approach with learnable mixture of Gaussians for Q2 is able to learn an alignment (both use the same T, models).
EN. DE EN
19
10
IN IN
(a)
(b)
(c)
(d)
Figure 4: An unregularized aligmment loss (figure (a) and (b)) can lead to excessive and unexpected movement of points in the latent representation (lines connect transported points), while regular-
ized alignment loss ((figure (c) and (d))) yields a unique and regularized solution that moves points significantly less and is closer to the identity function.
Unsupervised domain translation We first perform three image translation tasks on MNIST dataset; more specifically, we are flipping digit images in "0/1", "0/2" and "1/2" pairs shown in Fig. 5 We use RealNVP invertible models for all translation maps T,, as well the density model Q. We evaluate our model performances in Table I along with baseline models: LRMF and MLE version of AlignFlow. Note that all methods are flow-based models and thus images translated back to original domain are exactly the same, which implies exact cycle consistency.
As represented in Table. T both of our approaches outperform the baseline models terms of FID and AUB. The lower score in AUB indicates our model has the tighter bound than baseline model. This implies the statistical distance between the shared distribution 1 and our density model Q is smaller than baselines, meaning believe this result comes from our model setup, i.c., a Icarnable density model and domain specific invertible transformation functions with shared space. Specifically, AlignFlow with fixed standard normal distribution as their Q obtains worse AUB because the Q is not powerful enough to model the complex shared space trained from the real world dataset. On the other hand, LRMF shows the lack of stability when trained with relatively simple models that we are using, i.c., RealNVP T and Real NVP Q. This is because the transformation function and the density model should be able to directly model the complex real distribution in their setup. By comparing our approach with and without transportation cost, we observe introducing transportation cost yields insignificant decrease in AUB and an increase in FID. This indicates the transportation cost regularizes our method well without degeneration, so that our method can find better solution (which theoretically unique).
model has the better dataset alignment performance. We
The impressive performance of our method in AUB shows a consistent pattern in FID score as well. Across entire translation cases, our method shows the better performance in terms of FID. This result can be intuitively understood by Fig. 5 AlignFlow shows less stable translation results than our method especially for translating to digit 22 from digits "0" and '1'. We believe this phenomenon comes from the lack of expressivity of their Q model. On the other hand, our model shows stable results across all translation cases, which are also quantitatively verified via the lower FID score. Also, from the comparison the latent representations (second column of each macro column) of
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ours with and without transportation cost, observe that the latent representation of the À = setup contains numerous excessive values. This implies adding transportation cost (lambda = 1) prevents the excessive movement in the latent space, thus, model can find simpler and smoother solution during the training.
LRMF
0
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translation from 2
Figure 5: All pair-wise translation results among MNIST digits 0-2 where each block has first image as the original digit followed by the latent image, and three corresponding translated digits. Please note that second column of LRMF is set to be black because it does not have latent representation, and LRMF fails to translate in this situation which is why the numbers are all the same.
Table I: FID and AUB score for three images translation tasks in MNIST (for both metrics, lower the better). FID score for each translation task is calculated by averaging scores from each direction and AUB score is shown in nats. This table shows that our model has overall better performances than all baselines models in terms of both metrics.
FID
AUB
0c+1
0c+2
1c+2
Avg.
0+>1
0++2
1++2
Avg.
AlignFlow (MLE)
38.90
71.17
61.33
57.13
4799.27
-450.200
-4834.30
-4711.92
LRMF
224.02
141.70
182.31
182.68
-713.54
-592.90
-1323.75
-876.73
Ours, À
31.26
43.29
41.55
38.70
-4824.69
-4555.30
-4862.84
-4747.61
Ours, À =1
29.30
45.21
39.95
38.15
4819.73
-4547766
-4857.67
-4741.72
Multi-domain translation To illustrate that our method can be easily scaled to more domain distributions, we present qualitative examples of translating between every digit and every other digit for MNIST in Fig. 6 with performances in Table 2] To show the effect of transportation cost proposed in our paper, we also compare our model with and without transportation cost. Note that we omit LRMF in this experiment because the multi-domain situation is hard to deal with for LRMF setup due to LRMF's two distribution setup and assymetric model structure.
As shown in Table 2 our approach with and without transportation cost shows better performance in terms of FID and AUB than AlignFlow since our learnable density estimator can model more complex distribution than fixed simple density model in AlignFlow. In other words, the shared space of AlignFlow is limited because of the fixed simple density model. Note that the gap in the AUB score between AlignFow and our model is larger in this multi-domain setting (Table 2 than in the two domain case (Table []. We hypothesize that this is because the shared density model across many domains needs be more complex than for only two domains and thus, our learnable Q can create better upper bound to optimize than AlignFlow. In this experiment, while ours with transportation cost performs slightly worse, the results are comparable and it does provide a natural regularization that ensures a theoretically identifiable solution.
The superiority of our method compared to AlignFlow in multiple-domain translation can also be verifed through qualitative comparisons in Fig. 6 The lefumost column is an input image and the second and third macro columns are the results from ours and AlignFlow. by forwarding a given k-th latent T& ( into 10 inverse transformation functions, respectively. It is easy to observe that ours has more clear results in most cases than baseline. Moreover, our model shows better performance in maintaining the original identity (e.g., width and type of a stroke) than the baseline, as seen in third, fourth and ninth rows. This is because we jointly train our transformation functions with a leamaale density model, while AlignFlow independently train their transformation functions. This benefit of our approach may be crucial for other datasets such as human faces (Choi et al., 2018) where maintaining the original identity is important.
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Table 2: FID and AUB score for do-
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main alignment task 10 domains. FID score is calculated by average across all paired translations and AUB score is shown in nats
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AlignFlow (MLE)
49.82
4661.05
Ours, Ours, X ==1
43.25
4715.02
Figure 6: Qualitative comparison on translation results across 10 classes with AlignFlow and ours with transportation cost.
44.06
4697.42
DISCUSSION AND CONCLUSION
Transfer Learning Capability The performance of our model can be improved by adapting pre- trained density models. Fig. aa) shows preliminary results. Specifically, MLE version of Align- Flow model may still struggle to translate between USPS and MNIST dataset after learning for 200 epochs while ours can show qualitative results after only I epoch if we are using pretrained powerful density model for Q Salimans et al. (2017).
Generative Tasks to sample from the density model Q to have a latent image z first, and then forward to the inverse function 777 get the image sampled in jth domain. Examples of generated images for each domain in MNIST data are shown in Fig. 7b). The quality of the generative result also reflects the tightness of the bound between the latent space and the latent density model as illustrated in Eqn. 4.
Our model is also capable of generating samples in each domain. One needs
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(a) Figure 7: Figure (a) is translation result between USPS and MNIST dataset. Top two images are the training and testing result if using pretrained PixeICNN++(Saliman et al. (2017) models after only I epochs, and the bottom images are the results of MLE version of AlignFlow models after 200 epochs. Figure (b) shows the generated images of our model for all domains. Model used in this task adapted from multi-domain translation tasks in subsection 4.2
(b)
In this paper, we proposed novel upper bound the generalized JSD that leads to theoretically grounded alignment loss function. We then show that this framework unifies previous flow-based approaches dataset alignment and demonstrate the benefits of our approach compared prior flow-based methods. Additionally, we show that our framework could enable the use of pre-trained density models which would enable type of transfer learning for distribution alignment. More broadly, we expect that our AUB metric can be useful dataset alignment methods beyond images. An alignment metric that is not tied to a particular pretrained model (as for FID) or to particular data type will be critical for systematic progress, in unsupervised dataset alignment. We hope this paper provides one step in that direction.
comain-agnostic metric for comparing

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Under review as a conference paper at ICLR 2022
REFERENCES
Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei Efros. Unpaired image- sogge-to-mmne image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223-2232, 2017.
Aditya Grover, Christopher Chute, Rui Shu, Zhangjie Cao, and Stefano Ermon. Alignflow: Cycle consistent learning from multiple domains via normalizing flows. In AAAI, 2020.
Lanqing Hu, Mcina Kan, Shiguang Shan, and Xilin Chen. Duplex generative adversarial network for un- supervised domain adaptation. In Proceedings of the IEEE Conference Computer Vision and Pottern Recognition (CVPR), June 2018.
Laleh Haghverdi, Aaron TL Lun, Michael D Morgan, and John C Marioni. Batch effects in single-cell ma- sequencing data are corrected by matching mutual nearest neighbors. Nature biotechnology. 36(5):421
2018.
Rich Zemel, Yu Wu, Kevin Swersky, Toni Pitassi, and Cynthia Dwork. Learning fair representations. Sanjoy Dasgupta and David McAllester, editors, International Conference on Machine Learning, volume 28 of Proceedings of Machine Learning Research, pages 325-333, Atlanta, Georgia, USA, 17-19 Jun 2013.
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair. Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances Neurul Information Process- ing Systems, 2014.
Mario Lucic, Karol Kurach, Marein Michalski, Sylvain Gelly, and Olivier Bousquet. Are gans created equal? large-scale study. In Advances in neural information processing systems, pages 700-709, 2018.
Karol Kurach, Mario Lucic, Xiaohua Zhai, Marcin Michalski, Sylvain Gelly, The GAN landscape: Losses, architoctures, regularization, and normalization, 2019. URL httpp:r//peeneeeeewwnett///e/eee
1d= rkGG650GKQ
Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Berhhard Nessler, and Sepp Hochreiter. Gans trained by two time-scale update rule converge a local nash equilibrium. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc.. 2017. URL
neurips ca/ Paperddapppppppppp pdf
Gaurav Parmar, Richard Zhang, and Jun- Yan Zhu, On buggy resizing libraries and surprising subtletics in fid calculation. arXiv preprint arXiv:2104. 1222, 2021.
Ben Usman, Avneesh Sud, Nick Dufour, and Kate Saenko. Log-likelihood ratio minimizing flows: Towards robust and quantifiable neural distribution aligmment. In H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin, editors, Advances Neural Information Processing Systems, volume 33, pages 21118 21129. Curran Associates, Inc., 2020. URL https: /ptoc:eed///occiiigggg nourips wttp://pppppp////eeeeppp
[1
1191777111111100110000 www550050-0--000 Faper. pdf
Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. In International Conference on Learning Representations, 2017.
Ivan Kobyzev, Simon J.D. Prince, and Marcus A. Brubaker. Normalizing flows: An introduction and review of
3979, Nov current methods. IEEE Transactions on Pattern Analysis and Machine Intelligence, dol. http://///oooo11.oo/000011101190 2021. ISSN 1939-3539, doi: 10.1 TPAM 2000 2992933
URL http:
Durk P Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Suiskever, and Max Welling. Improved variational inference with inverse autoregressive flow. In Advances in neural information processing systems, pages 4743-4751, 2016.
George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshmi- narayaman. Normalizing flows for probabilistic modeling and inference. Journal of Machine Learning Research, 22(57): 2021. URL. httt::://mmr.org///
v22/19 1 1288 html
Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1xl convolations. In Advances in neural information processing systems, pages 10215-1000 10224, 2018
Will Grathwohl, Ricky TQ Chen, Jesse Bettencourt, Ilya Sutskever, and David Duvenaud. Ffjord: Free-form continuous dynamies for scalable reversible generative models. arXiv preprint 1810.000 2018
10

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Under review as a conference paper at ICLR 2022
Ricky T. Q. Chen, Jens Behrmann, David K Duvenaud, and Joem--Henrik Jacobsen. Residual flows Wallach, H. Larochelle, A. Beygelzimer, d'Alché- Buché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32, Cur- ran Associates, Inc., 2019. URL - Paper.
for invertible generative modeling.
F0 FF83d10
Jens Behrmann, Will Grathwohl, Ricky TQ Chen, David Duvenaud, and Jörn-Henrik Jacobsen. Invertible residual networks. In International Conference Machine Learning, pages 573-582. PMLR, 2019.
Jianhua Lin. Divergence measures based on the shannon entropy. IEEE Transactions on Information theory, 37(1):145-151, 1991.
Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelen+++: Improving the pixelenn with discretized logistic mixture likelihood and other modifications. arXiv preprint 17Xiv:170..0517, 2017.
Diederik P Kingma and Max Welling.
An introduction to variational autoencoders.
arXiv preprint
arXiv: arXiv:/906.00 /9069996669 2019.
Maximilian Seitzer, pytorch-fid: FID Score for PyTorch. bytorch-fid August 2020. Version 0.1.1.
Yunjey Choi, Minje Choi. Munyoung Kim. Jung- Woo Ha, Sunghun Kim, and Jaegul Choo. Stargan: Unified generative adversarial networks for multi-domain image-to-image translation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1789-8797, 2018.
Gabriel Peyré and Marco Cuturi. Computational optimal transport: With applications data science. Foun- dations and Trends Machine Learning, 1165-6):355-677, 2019. ISSN 1935-8237. doi: 10.1561/ 2200000073. URL htRt:htt:://o /x doi wor. 10. 1561022000110222000000000
Dheeru Dua and Casey Graff. UCI machine learning repository, 2017. UBLLEttpp//arcch.e..co..
m
George Papamakarios, Thco Pavlakou, and Jain Murray. Masked autoregressive flow for density timation. In I. Guyon, U. Luxburg, Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Cur- Associates, Inc., 2017. URL http:///pppoeeeee...r.rroo.ooppp
Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceed- ings International Conference Computer Vision (ICCV), December 2015.

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A PROOFS
Proof of Equivalence in Def. 1 While the proof of the equivalence is well-known, we reproduce
here for completeness. As reminder, the KL divergence is defined as:
P(a) Q)) = Ep|log Q(a). = Epl-log (=)) Epl- log P(z)] = -L(P
(10)
where -) denotes the cross entropy and H(-) denotes entropy. Given this, we can now easily
derive the equivalence:
GJSD (P,,.... .xx.) = :x)==Ew, (L(Pxx Pxms)
(11)
= (Ho,)(((PX,, Pxmn)) -((xx)))
(12)
= mny Hc(Px,: Pxmna) - -E, un H(Px,)
(13)
EjwylEPs |- Pxmil - -- H(Px,)
(14)
= E, Sx -Px, (x)) log Pxm. (x) - E, u H(Px,)
(15)
= [x Ex (xr) log Pxm. (z) Eg H(Px,)
(16)
= -Sx -Pxmi (x)) log Px.. (x)dx mn H(Px,)
(17)
-((Xxm)) - -E H(Px,).
(18)
Proof of Lemma 2
First, note the following fact from the standard change of variables formula:
Px (x) = (z(
(19)
Px(x))]T<x)) (=)))
We can now derive our result using the change of variables for expectations (i.e., LOTUS) and the probability change of variables from above:
H(Pz) Ep - log Pz(z)] = Ep |- log (T(
Epx |- log Px (x) (x)
Epx |- log Px (x)) + Epx |o Jr |r
= -((x)) Fx [lo (x)])
Proof of Theorem 3 Given any fixed Q, minimizing CAUB decouples into minimizing separate nor- malizing flow losses where Q is the base distribution. For each normalizing flow, there exists an invertible T, such that (X,) Q. and this achieves the minimum value of LAUB. More formally,
min LAUB (Tun(Ti:- (Ti,.... ,Tx)
(20)
TI.", TT
min 1
Σ, WyyEPs - log Jr, (x: (x))
(21)
En min Epx, | log 12y (+)
H(Px,)
(22)
T
Ejwy min Epx, T3
log Jry (x) Q(T H(Px,) IE Exx |- log Px, (w)))
(23)
Px (z)) (w)
Ejw; H(Px,) + Σ, w minr; Exx, [log
(24)
(TT(
Pr. (x,) (X; (ce(
-Em,H((P)) H(Px,))))))))))))(()))) + Σ, wy minr, Exx, [log
(25)
(11)
Pr. (z)
= E,myH((x,) + Σ, Wy minr, E E flog
(26)
((2)
=E, H(Px,) E, w. minz; (L(Pr,(X,)))) Q)-
(27)
Given that KL(P,Q) 2 0 and equal to if and only Pf Q, the global minimum is achieved only if Pr,(X,) Q, vj and there exist such invertible functions (e.g.. the optimal Monge map between Px, and Q for squared Euclidean transportation cost (Peyré and Cuturi, 2019)). Additionally, the optimal value is E, H H(Px,), which constant with respect the 1 transtormations.
n
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B EXAMPLES OF NON-UNIQUE ALIGNMENT SOLUTIONS
B.1 GAUSSIAN EXAMPLE
Suppose the component distributions normal distributions, i... X1 (( 1) and X2 (1,2,, T), and for even greater simplicity, we assume T2 is the identity, i.e., 72(x) = z. Then, a global optimal solution could be Ti (æ) U( 141 (42) for any orthogonal matrix U, i.e., there are infinitely many invertible functions that align the distributions. Note that this lack of unique solutions is not restricted orthogonal rotations (see appendix for more complex example).
B.2 COMPLEX EXAMPLE
Consider the ID case where Q only contains the uniform distribution. Thus, Ti and 72 must map their distributions the uniform distribution for alignment. One solution would be that Ti and T2 F3 where Fi and F2 are the CDFs of Px, and Px.- Yet, there are infinitely many other possible solutions. Consider an invertible function that subdivides the unit interval into arbitrar- ily large number of equal length intervals and then shuffles these intervals with fixed arbitrary permutation. More formally, we could define this as:
m(1)
E [0, 1 if 1 2)
if x
I
Sm, (I)
m
m
(28)
((m)
if E
where x--) is permutation of the integers to m. Given this, then other optimal solutions could be Ti = Sm.st o Fi and T2 = F2 for any m > 1 and any permutation æ. This idea could be generalized to higher dimensions well by mapping to the multivariate uniform distribution and subdividing the unit hypercube similarly.
C ALGORITHM
We summarize computation of regularized alignment upper bound in Alg.
Algorithm Training algorithm for our model
Input: datasets { for & domains; normalizing flows (1); density model
Q(2;)) learning rate transportation cost factor À maximum epoch Emazi initial parameters
value {I, ( (())(()))
Output:
{0 Jy:
k.
for epoch =1 -1, Emaz do { do
for each batch
Update Q using data from all domains d --0+ x ++ Ag log (1;)
end for for each batch { do for domainj 1, do
0- 0 77 log Jr, (x) (z))())) Ac(x, (x()) Update T' independently across
all domains
end for end for end for
D DETAILED PARAMETERS USED IN 2D DATASET EXPERIMENT
D.1 LRMF VS. OURS EXPERIMENT
T' for LRMF setup: T: channel-wise mask for Real-NVP model with s and derived from 64 hidden channels of fully connected networks. T2: Identity function.
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T for RAUB setup: 7j and Ti: 8 channel-wise mask for Real-NVP model with and t derived from 64 hidden channels of fully connected networks. Regularization coefficient
À = 0
Q for both: A single Gaussian distribution with trainable mean and trainable variances.
D.2 ALIGNFLOW VS. OURS EXPERIMENT
T for both: channel-wise mask for RealNVP model with s and & derived from 8 hidden channels fully connected networks.
Q for Alignflow setup: single fixed normal distribution.
Q for RAUB setup: A Iearnable mixture of Gaussian with 3 components. Regularization coefficient =
D.3 REGULARIZED VS. UNREGULARIZED EXPERIMENT
1. 7' for both: 8 channel-wise mask for RealNVP model with and derived from 64 hidden channels of fully connected networks.
2. Q for both: A learnable mixture of Gaussian with 2 components.
3. À for unregularized Experiment: :
4. À for regularized Experiment À 1
E INTERPOLATION OVER LATENT SPACE
We use the same model used in the "Multi-domain translation" part in section 4.2 and perform interpolation in the latent space. We first randomly select two distinct real images in domain (in this case two Os). and do a linear interpolation of the selected two images in the latent space. Then we translate all the interpolated images (including the two selected images) to all of the domains to generate ceanslaaeddinterpolaaeedd images -i..,, the corresponding interpolations in each of the domains.
As shown in Fig. 8 all traanslaeddiinterolleee results can preserve the trend of the stroke width of the digits from the original interpolated domain. These results suggest that our approach aligns the domains so that some latent space directions have similar semantic meaning for all domains.
F TABULAR/STRUCTUREE DATA EXPERIMENT
We follow the same preprocessing for four UCI tabular datasets (uua and Graff 2017) from the MAF paper (Papamakarios et al., 2017), specifically MINIBOONE, GAS, HEPMASS, and POWER. In order obtain binary label for creating domains, choose the last input feature for each dataset and discretize it based on whether it is higher or lower than the median value, which ensures the datasets of equal size. We the MLE version AlignFlow baseline compare with our model and use the AUB score as the metric for comparison.
As shown the Table. 3 our method shows better performance with large gap across all com- parisons. This provides strong evidence that our proposed method (trained with learnable Q) better aligns the two domains than AlignFlow (trained with fixed Q).
It is worth emphasizing that there is no natural metric for evaluating GAN (Goodfellow et al. 2014) in tabular dataset. We believe this demonstrates one of the key benefits of our proposed method over GAN-based alignment methods.
G MULTI-TRANSILATION TASKS CELEBA DATASET
In order verify proposed method performs well in more complex setup than MNIST, conducted experiments on the CelebA (Liu et al. 2013) dataset. We first create three domains (Black Hair, Blond Hair and Brown Hair) by using the hair color attribute. We further center crop
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Figure 8: This figure shows the translation results of interpolated images. In the first row, the two
images selected by red rectangles are real images from the dataset; and all eight images between generated by linear interpolation in the latent space. Starting from the second row, each row contains translated images which are transformed from the same latent vector in the same column and the first row.
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Under review as a conference paper at ICLR 2022
Table AUB score for the domain alignment task in four tabular datasets. AUB score is shown in nats (the lower the better).
MINIBOONE
GAS
HEAPMASS
POWER
AlignFlow (MLE) Ours
13.26
-5.65
19.57
-0.89
-47.39
-137.91
-188.11
-91.17
the images to have the size 148 x 148 then resize them to 64 x 64. For the baseline model, we again uses MLE version of AlignFlow. We use both FID and AUB metric compare the performances.
As shown in the Table 4 our proposed method demonstrates better performance in almost all periments terms of both FID and AUB. This consistently validates that the flexibility of Q in our framework has clear advantages over the fixed Q AlignFlow for the domain alignment task. Intuitively, we think the tighter upper bound from our proposed method plays an important role in aligning distributions in the latent space.
We also provide qualitative results in Fig. Compared to the AlignFlow (MLE) baseline, our model performs significantly better in translating between selected attributes. In particular, while our approach maintains a valid face structure during translation, AlignFlow is unable to maintain a valid face structure. Note that absolute image quality is not focus of our paper as we use relatively simple flow models, but rather the comparison prior methods and show the feasibility of our approach in higher dimensions. We expect better qualitative results could be achieved using more advanced flow models such GLOW, Residual Flows, Flow++, etc. but given that the scope of our work which focuses on the theoretical and foundational approach, we leave this to future work.
Table This table provides evidence that our model has overall better performances than all base- lines models in terms of both metrics. FID and AUB score for three images translation tasks in CelebA (for both metrics, lower is better). For domain names, means Black Hair, means Blond Hair, and 2 means Brown Hair. FID score for each translation task is calculated by averaging scores from each direction and AUB score is shown in nats.
FID
AUB
0++1
0+>2
Avg.
0<>1
0++2
1++2
Avg.
AlignFlow (MLE)
243.27
99.11
127.10
156.49
-38109.25
-26437.60 6.6
-26288.99 88
-30278.61
Ours, = 0.01
101.20
104.72
109.99
117.86
-34041.39
-33884.92
-33902.84
-33943.05
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Ours, 2 = 0.01
AlignFlow, MLE
Figure 9: This figure shows the translation results for three attributes (Black Hair, Blonde Hair, Brown Hair) in CelebA dataset. The first column corresponds to real images in the test dataset. The next three columns are translated results into three different attrbbutes/dmmann (Each of the three columns represents translation to Black Hair, Blonde Hair. Brown Hair attribute respectively) from our model. The final three columns shows the translation results of MLE version of AlignFlow model with the same format the previous three columns.
17


Reviews:
[{"review_id": "a7L-mIhI_z-", "reviewer": "Reviewer_B4sF", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "* The major contribution of this paper is providing new domain alignment loss with theoretical guarantee and connections with JSD.\n\n* The paper is well written and easy to follow. Details of experiments can be easily located and confirmed.\n\n* A major concern of this paper is to which extend the current model can be applied to more general learning tasks. The current experiments (most of them) show the produced samples, compared with two SOTAs. It seems to me that the advantage is not very significant. See Fig. 6. Also, the quantitative results through FID and AUB are comparable to AlignFlow.  \n\n* Authors may be interested in showing more results of different learning tasks. As the approach is mainly for domain alignment, it would be interesting to show, e.g., visual domain adaptation results.\n\n* More exploration and discussions are expected for high-dimensional or structured data, which may affect the design of transformation T and its inverse. The current model is only evaluated on simple or low-dimensional data, and the model generality is unclear. \n\n* While GAN model is computationally expensive and hard to optimize, the proposed model did not prove itself to be efficient or more optimizable in conventional vision tasks where GAN was testified.\n", "summary_of_the_paper": "The paper presents a unified framework for domain alignment through non-adversarial approaches. Instead of solving a min-max adversarial problem, this paper aims to solve a min-min optimization equivalent to minimizing the upper bound of JSD. It proves the equivalence and offers a computational and feasible solution that can be inserted into flow-based non-adversarial approaches in a plug-and-play fashion. Preliminary results on generating digits from different classes or domains have demonstrated the correctness of the model.", "summary_of_the_review": "In brief, I was impressed by the theoretical results and connections of the new alignment loss and conventional JSD bound. Experiments seem to prove the model is workable on toy data and digits datasets. However, authors still need to explore and demonstrate its feasibility and value in more challenging learning tasks using high-dimensional or structured data.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "Y3EnbAGUYW4", "reviewer": "Reviewer_sZ2C", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "## Concerns:\n\nMy major concern is why adversarial learning is challenging for distribution alignment? Could you provide a concrete example like an experiment? Do the authors justify this only according to, \"However, adversarial learning can be quite challenging in practice\", with two references are both earlier than 2019. How about nowadays? As the author claimed, \"the auxiliary density model $Q_z$ may be more difficult to train than the auxiliary discriminator $D$\". Does this seem to be contradictory to your justification?\n\n1. The experiment design is trivial with MNIST. Using FID to measure generate data quality on MNIST is not well acknowledged. The authors should either choose a different dataset or report some other metrics.\n\n2. There exist lots of multi-domain translation models with GANs, e.g. MUNIT and StarGAN v2. Without this kind of comparison, adversarial learning models shouldn't be criticized.\n\n3. Since the proposed model introduces flow-based models, some ablation studies by changing the based model, e.g. Glow, NICE etc, would be better.\n\n4. Since the flow-based model is used, is there any way to visualize the latent space on real data?", "summary_of_the_paper": "The paper provides an interesting idea by unifying flow-based models under a non-adversarial framework to realize multiple-distribution alignment.", "summary_of_the_review": "It's an interesting idea of using non-adversarial learning to realize multi-distribution alignment. But lots of concerns remain as given above. Thus, the impact of this paper is weak.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "4: The contributions are significant, and do not exist in prior works."}, "extra_text": {}}, {"review_id": "Shf3SvQWrd", "reviewer": "Reviewer_fTCq", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Strengths:\n1. This paper theoretically gives an upper bound of GJSD with a learnable density function Q in a variational way, which extends prior method AlignFlow in multi-distribution alignment;\n2. The authors give extensive discussions about the relationship between prior methods (including LRMF that tackles two data set alignment) and a regularization term.  \n3.  The author provides extensive experiments to illustrate its effectiveness. Moreover, the author also experimentally verify the statement in the discussions about prior methods and a regularization term.\n\nWeaknesses:\n1. Compared to AlignFlow, this work provides a learnable Q rather than a fixed function. This design seems reasonable. Could the authors give more discussions about the learnable Q? The authors state that a fixed normal distribution Q is insufficient. However, VAE[1] can generates very realistic figures from a normal distribution in a reversible way? Under what cases is a fixed Q not enough? \n\n2. Compared to LRMF which tackles two data set alignment, this work can align multiple distributions. From Figure 3, the authors claim that if the density model class Q is not expressive enough, LRMF could fail to achieve the alignment. Could the authors explain the difference between the two methods (LRMF and yours) when optimizing the function Q?  Moreover, the number of the data sets to be aligned in the experiments is relatively small. When the number of distributions is large, is there some problems when the proposed method to align these distributions? This may be a common issue for existing methods, could the authors discuss it more?\n\n[1]Kingma, Diederik P., and Max Welling. \"Auto-encoding variational bayes.\" arXiv preprint arXiv:1312.6114 (2013)", "summary_of_the_paper": "This paper proposes a flow-based method for the unsupervised data set alignment problem. It firstly reveals that the minimization problem over density models can be addressed by optimizing the upper bound of generalized JSD. Based on this theoretical result, the authors propose a new regularizer for the multi-distribution alignment problem. Compared to prior work, e.g., AlignFlow and LRMF, the authors derive a more general framework for the unsupervised data set alignment problem. The authors also provide extensive experiments to verify its effectiveness and superiority.", "summary_of_the_review": "The framework advances the improvements in the problem of multi-distribution alignment, although it is an extension of prior works.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper offers flow-based alignment methods for alignment of distributions in a domain adaptation setting.  While there are many positive aspects of the submission, the experimental results only weakly support the results.  The AC agrees with the critical comments mentioned by reviewer sZ2C, and in particular observes that the experimentation is not state of the art with regard to current domain adaptation literature. Unfortunately the submission is not acceptable in present form.
Decision: Reject