Learning to Match Unpaired Data with Minimum Entropy Coupling
Authors: Mustapha BOUNOUA
Giulio Franzese
Pietro Michiardi
Venue: ICML 2025 poster (2025)
Keywords: Minimum entropy coupling
Unsupervised learning
Diffusion models
Primary area: general_machine_learning->unsupervised_and_semisupervised_learning
TL;DR: Minimum entropy coupling to pair continuous distributions

Abstract: Multimodal data is a precious asset enabling a variety of downstream tasks in machine learning. However, real-world data collected across different modalities is often not paired, which is a significant challenge to learn a joint distribution. A prominent approach to address the modality coupling problem is Minimum Entropy Coupling (MEC), which seeks to minimize the joint Entropy, while satisfying constraints on the marginals. Existing approaches to the MEC problem focus on finite, discrete distributions, limiting their application for cases involving continuous data. In this work, we propose a novel method to solve the continuous MEC problem, using well-known generative diffusion models that learn to approximate and minimize the joint Entropy through a cooperative scheme, while satisfying a relaxed version of the marginal constraints.
We empirically demonstrate that our method, DDMEC, is general and can be easily used to address challenging tasks, including unsupervised single-cell multi-omics data alignment and unpaired image translation, outperforming specialized methods.

Learning to Match Unpaired Data with Minimum Entropy Coupling
Mustapha Bounoua" Giulio Franzese? Pietro Michiardi 2
Abstract
disciplines, such as astronomy and geoscience, employ mul- timodal datasets combining spatial, spectral, and temporal data to understand complex systems (Srivastava et al. 2019
Multimodal data is precious asset enabling a variety of downstream tasks in machine learn- ing. However, real-world data collected across different modalities is often not paired, which is a significant challenge to Iearn a joint distri- bution. A prominent approach to address the modality coupling problem is Minimum Entropy Coupling (MEC), which seeks to minimize the joint Entropy, while satisfying constraints on the marginals. Existing approaches to the MEC prob- lem focus on finite, discrete distributions, lim- iting their application for cases involving con- tinuous data. In this work, we propose a novel method to solve the continuous MEC problem, us- ing well-known generative diffusion models that learn to approximate and minimize the joint En- tropy through a cooperative scheme, while sat- isfying a relaxed version of the marginal con- straints. We empirically demonstrate that our method, DDMEC, is general and can be easily used to address challenging tasks, including unsu- pervised single-cell multi-omics data alignment and unpaired image translation, outperforming specialized methods.
Zhang et al. 2024a Sošić Klindžić et al. 2024).
Modeling multimodal data allows for a more comprehen- sive understanding, reflecting the inherently multi- nature of the real world. Recent works in representation
learning (Radford et al. 2021
Lu
2023 Manzoor et al.;
2023
Chen et al.) 2023). the study of multivariate sys-
tems McSharry et al. 2024 Liang et al.
2023 Bounoua
ct al. 2024b), generative modeling (Rombach et al. 2022
Tang et al., 2023bb al Bounoua et al.
2024a Esser et al.
2024), and multimodal conversational agents (Li et al. 2023
Liu al. [023] Shukor et al., 2023 Xue et al.) 2024| Wu
ct al. 2024), are few examples illustrate the fervent effort in the machine learning community to address and exploit multimodality. However, the intrinsic complexity of multi- modal data introduces several challenges that hinder their application in machine learning research. Modality hetero- geneity complicates and sometimes impedes geometric com- parisons, requiring for example learning a mapping from ambient to latent spaces (Rombach et al.
2022
Tang et al.)
2023b Liu et al. 2023 assumptions (Liang et al
Bounoua al.)
2024a
stringent
2022; Xia et al.
2023
Dong et al.,
2024 Ibrahimi et al. [0244] Zhang et al.
2024b). Alignment
across modalities at spatial, temporal or semantic levels is another challenge, which calls for costly pre-processing
1. Introduction
steps such as synchronization (Hanchate et al. 2024 Chen
Nowadays, multimodal data is pervasive thanks advances in data collection technologies and the crucial need for sys- tems that can learn from the diversity of real-world phenom- ena. Healthcare, for example, is domain where patient data often spans electronic health records, radiological im- ages, genetic data, and wearable sensor outputs (Kline et al. 2022, Acosta et al., 2022). Autonomous systems rely on a suite of sensors, including LiDAR, cameras, and ultra- sonic sensors, navigate environments effectively (Caesar et al. 2020 Gu et al. 2023 Franchi et al. 2024). Scientific
et 2024 Scire 2024 Martin-Turrees et al.
2024
The major roadblock address is that of paired multimodal data, which is an underlying assumption in many works in the literature (Radford et al.) 2021 Rombach et al. 2022
Liu et al., 2023 Li et al. 2023 Bounoua et al.] 2024a) Paired data- for a given sample, all its various modalities are available - is either expensive, difficult to obtain, or sometimes impossible. For example, in genetic research, data is inherently unpaired due to the nature of the data acquisition process, such as single-cell RNA sequencing data, where measurements destroy the original cells (Kester & van Oudenaarden] 20188] Chen et al. 2019 Schiebinger et al. 2019). Similarly, matching image data from different domains a challenging endeavor when paired data is miss-
'Ampere Software Technology. France 'Department of Data Science, EURECOM, France. Correspondence to: munustapha.bounoue www.ouow.uuu...........
Proceedings of the 4rnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
ing, which calls for specialized methods (Zhu al. 2017
Huang et al. 2018
Pang et al. 2021 Sasaki et al.
2021

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Learning to Match Unpaired Data with Minimum Entropy Coupling
Yang et al. 2023 Sun et al. 2023 Xie et al.
2023).
according reward functions, following an alternating opti- mization process. In summary, our contributions are:
In this work, we study the problem of unpaired multimodal data through the lens of coupling, a fundamental problem in probability theory, that aims at determining the optimal joins distribution of random variables given their marginal distributions, with early attempts at solving it dating back to the work by Fréchet [195T). The pairing problem belongs to broad class of methods (Den Hollander, 2012 Lin al.] 2014. Benes& & Stepán 2012 Yu & Tan 2018): some cast it quantities, where optimality is defined in terms of Entropy minimization or Mutual Information maximization, others focus on Opti- mal Transport (OT) (Villani 2009| Peyré& & Cuturi 2019), where optimality defined as minimizing the expected value of a transport cost over the joint distribution. Our focus is the Minimum Entropy Coupling (MEC) problem, which aims at finding the joint distribution with the smallest Entropy, given the marginal distribution of some random variables. Recent applications include entropic causal infer- ence (Kocaoglu et al. 2017; Javidian et al.] 2021: Comp ton 2022), communication systems (Sokota et al. 2022),
We propose an approximation of the MEC problem for arbitrary, continuous distributions, which is general, and that does not require stringent assumptions on the marginal distributions, nor the definition of geometric cost functions (Section 2
We present a practical implementation of our method (Section B. that relies on generative models, that inter- act through a cooperative scheme aiming at optimizing an information-theoreti cost function related to the Entropy of the joint distribution. Our training proce- dure overcomes numerical instabilities and degenerate solutions by relying on the application of soft marginal constraints, as well as the natural approximation stem- ming from capacity denoising model.
through the lens of information-theoreer
We illustrate the benefits and performance of our method on two important use cases (Section First, we solve the coupling problem between incomparable spaces with a single-cell multi-omic dataset, where we compare our method to state-of-the-ard alternatives that rely on OT. Second, we focus unsupervised im- age translation between uncoupled pairs, and compare against state of the art.
steganography (de Witt et al. 2022). random number gener-
ation (Li 2021), dimensionality reduction (Cicalese al.
2016 Vidyasagar 2012), lossy compression (Ebrahimi et al.
2024 and multimodal learning (Liang et al. 2024).
While the MEC problem is known to be NP- Kovačević et al., 2012). the
Hard (Vidyasagar 2012.
literature contains many approximation and greedy algorithms (Painsky et al.
2. Problem Formulation
2013; Kovacevic et al.) 2013 Cicalese et al. 2016 Li (021). and theoretical studies about the approximation qualities of such approaches (Cicalese eet al. 2017] 2019). Nevertheless, the vast majority of prior work on the MEC problem focus on discrete distributions: instead, we consider the continuous variant of MEC, and propose a flexible and general solution to the coupling problem for arbitrary, continuous distributions. The MEC problem for continuous random variables is much more complex than its discrete counterpart, and can be ill-defined in certain cases due to the properties of differential Entropy and the challenges inherent to continuous distributions living in an infinite dimensional space.
Given two random variables X E X' and Y E y with marginal probability distributions pxx px (a) and py respec- tively, we consider parametric space po = (pxx (a,w)) of joint distributions over the space X' x Y, with induced marginal distributions r'x (a), p& (y) (where p&x (x) a Sy (x, y) dy and similarly for (v)).. The MEC prob- lem between the two original distributions px px(x)) (I) and py (v) consists in finding a joint distribution pry- y) such that i) the induced marginal distributions px (x), py (v) match them either exactly or approximately and ii) the joint dis- tribution is the one with minimal entropy (Kovacevic et al., 2013 Cicalese et al. 2017 2019). The constraints over the search space pd are referred to as marginal contraints
The gist of our method is to consider a parametric class of joint distributions, which we reinterpret as conditional generative models, with additional terms to steer adherence to marginal constraints. Then, the MEC problem requires access to the conditional Entropy, which we rewrite as log- likelihood. Crucially, our method exploits two specular generative models, which cooperate to minimize the joint Entropy, while approximately satisfying the marginal con- straints. Our approach materializes as two denoising dif- fusion probabilistic models (Ho et al.] 2020),, which we first pre-train on marginal distributions, and then fine-tune
Definition 2.1. A joint distribution p&rr (x, y) from po is said to be an exact coupling iff
p& (x) = (0)
(1)
In general, exact coupling is not possible (nor wanted, to avoid overfitting) and the goodness of the solution in terms of marginal constraints is approximated through some distance function between the induced and original distributions, e.g. using the Kullback-Leibler divergence KL [p& || px] A Enapg
[log
P (rr). Then, we define px
2

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Learning to Match Unpaired Data with Minimum Entropy Coupling
the MEC problem with soft constraints follows
3. Methodology
Consider two random variables in continuous domains, X E X and Y E y. We begin by considering a parametric class for the joint distribution expressed as p'x (zz, y) ( (y), such that the joint entropy HI(p',y min- imization becomes equivalent to minimizing the conditional entropy H(p x Note that the marginal constraint on Y from Equation 22 verified by construction. To satisfy the marginal constraint on X we consider the IKL divergence. This leads to an alternative definition of the MEC problem with soft constraints, that reads as
min H(p Xx KLL [p& ll px] + KL
KL [pt. |l pr].
(2)
where the entropic term is defined H((x))) - -E Brarokk. [lo (a,w)] w)]
Previous work have mainly focused on the exact MEC in dis- crete settings, where px and py have finite or countably infinite number of outcomes. Exact solution in such set- tings is known to be NP-Hard (Vidyasagar 2011 Kovacevic et al. 2013). Under our assumption of continuous spaces the problem is more complex. Exact matching is not generally possible due to the finite complexity of the parametric family po, since in general the distributions px, py live in infinite dimensional spaces. Rather than a limitation, enforcing lim- ited complexity is helpful to avoid degenerate, deterministic joint probabilities (e.g. pxy (x. v) = 5(y - g(x:) > (z), where g(-) is any mapping which guarantees exact coupling), which would induce infinite joint entropy.
Definition 3.1. Given random variables X X' and Y E V. the continuous MEC problem with soft marginal constraints corresponds the optimization problem
min E Uypp
[
+ x KIL
P&'s
Px
(3)
Crucially, we note that the parametric portion of the joint distribution, namely y (ar (xlly. y). can be interpreted as a con- ditional generative model of the variable X given Y. As consequence, the conditional entropy from Definition 3.1 can be interpreted as an expected log-likelihood, leading to
Interestingly, the MEC problem has an intuitive interpre- tation connected to the problem of Mutual Information maximization. Indeed, ) - ) + (px) H((p))) and in the exact matching scenario Hx(x)
[- log (pxrr=u) + Ax [p& px]
min EE g.-pxk
(4)
(Px)), H H.py). In other words, whenever the marginal constraints are satisfied with reasonable quality, the MEC problem is a good approximation of the informa- tion maximization problem.
A maximum likelihood solution to the MEC problem in Equation ( is appealing, because it can be addressed by learning the parameters of an appropriate conditional gener- ative model, while approximating the marginal constraints on X through the unconditional version of the model. Nev- ertheless, this approach bears several challenges:
Early instances of the coupling problem express through the lenses of OT (Monge 1781 Kantorovich, 1942). In the simplest (albeit rich and interesting) scenario, the goal is to minimize the transportation cost between distribu- tions E, Rx,0rpx [||= y. with the implicit assumption of X x= R and under the requirement of exact match- ing (corresponding to Ax = xy = oo). Several inter- esting extensions, including additional constraints on the joint distribution such geometry or structural constraints, lead to made approaches (Villani 2009 Peyré & Cu- turi 2019). Other than the trivial relaxation of constraints from exact to approximate, a particularly useful extension concerns the entropy-regularized version of this problem, where the cost function is complemented by the entropic term H(p'y-). Although MEC is fundamentally different than OT, link between the two clearly exists. However, straightforward comparison is not possible, the entropic term enters the respective minimization problems with dif- ferent signs. Minimizing H)(pky) directly over other ge- ometric costs (like the euclidean norm considered in OT) has several advantages in terms of generality, as it does not require geometrically comparable spaces X' and y
Asymmetry: Equation (4) can be used to minimize the conditional entropy H The learned condi- tional generative model can be used to generate sam- ples from variable X given Y, but not vice-versa.
Marginal constraint: in principle, exact matching re- quires xx - oc, but this choice leads to degenerate solutions to the MEC problem. The marginal constraint from Definition 3.1 despite being soft, should strive to keep p& anchored to px(x), which is not known.
To address the first challenge, we introduce second fam- & ily of parametric models py x (x)x))))xx(( this time corre- sponding to conditional generative model of the variable Y given observations of X. Then, we can write a specular version of the MEC problem we defined as
min E
z.upp [-
[log log 1 (D11x==)] + AYKL (pf ll Pr
(5)
3

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Learning to Match Unpaired Data with Minimum Entropy Coupling
Recall that x Px and yxy = PyxxPxx it is then reasonable to strive, among all the possible solutions, for pxx px This joint constraint can be approxi- mated with a penalty term proportional to the KL divergence between the two distributions. Interestingly, this coupling allows to implement a practical method that exploits coop- @ eration: we use 241x to improve x and vice-versa.
v Bie with e (0, I). Interestingly, an arbitrary portion of this forward chain can be efficiently simulated through the equality in distribution 2; = v voLio (vI âz)e, with 20 Px and 0e = 1 - 3t. ā. = II.= 0tg.
The corresponding discrete-time reverse process, that has a Markov structure as well, used for generative purposes. The model generates data through the iterative sam-
To address the second challenge, and pave the way for our practical implementation, we break the optimization problem by first focusing on respecting the marginal con- straints. To do so, we pretrain unconditional models such that p (x) R (x( and p (y) R (v)) Then, we use 9, and Φx to initialize the conditional models, and anchor their parameters throughout the optimization such that they do not deviate too much from the pretrained models.
pling process p'x (Fo...) = I 1 ( (xT), (, and typically pl (IZ-1 is Gaussian transition kernel with mean and covariance 3.T. Intu- itively, starting from simple distribution 2T I)),,), samples are generated by a denoising network c², that removes noise over T denoising steps. A simple way to learn the denoising network e" is to consider a re-weighted variational lower bound of the expected marginal likelihood, where the problem arg ming KL [px p'x becomes
where pf(TT) =
Vai st -/- A_
D. c
Overall, the method we propose writes
min E. y pxxY
e log (PrIx=0)
)] + +xxKI
PX PX]
0
min a
log a (PxII=u) ) + Ay
Py ||
T (o
arg min
[[le - - (xx,, 011
(7)
(6)
where we additionally enforce the approximate joint constraint. Notice the difference with Equation f and Equation (): given the structure of the paramet- ric distributions we use, it is possible to show that Vobe.w-rky lo y x Vobx,grep, lo | whenever pxx.y = px,y (see Appendix A.1 for more de- tails). Striet adherence to the joint constraint, in princi- ple, allows "swapping" the roles of the conditional models without affecting the optimization dynamics, leading to a cooperative method. In practice, we found trough empir- ical exploration that such a cooperative formulation (See Appendix [ albeit approximate, proves to be much more stable than the original problem, and consequently decided to adopt it in our implementation, as described next.
This simple formulation has been extended to conditional generation (Ho & Salimans, 202T), whereby a condition- ing signal y injects "external information" in the iterative denoising process. This requires simple extension the denoising network such that it can accept the conditioning signal: c( y. t). During training, a randomized approach allows to learn both the conditional and unconditional vari- ants of the denoising network, for example by assigning a null value to the conditioning signal, e.g. y = 0.
In Equation (6). the likelikel emerges as critical quantity to address the MEC problem. In the ideal condi- tions of a perfect denoising network, the difference between predicted and actual noise can be used, in the limit of infinite number of denoising steps, to compute exactly such quan- tity (Kong et al. 2023). We use these results to compute the logelikelssoods through Monte Carlo estimation techniques
3.1. Practical implementation
In our implementation, we consider the parametric class of probability distributions associated denoising diffusion probabilistic models (DDPM) (Sohl-Dickstein et al., 2015 Ho et al. 2020). These models enjoy excellent performance in fitting complex multimodal data, and allow accurate esti- mation of information metrics (Franzese et al., 2024 Kong ct al. 2024: Bounoua et al. 2024b; Dewan et al. 2024).
T E. [[|e (2,,, [)]] (8)
- loo leog pe (zo) 20 const +
2
where the unspecified constant does not depend neither on To nor on 0, and is consequently irrelevant for optimization purposes. This approach can be trivially generalized to the case of conditional denoising network e e((ree y, t)-
DDPM. These generative models are characterized by forward process, that is fixed to a Markov chain that gradu- ally adds Gaussian noise to the data according to a carefully selected variance schedule Gr i.e. Zt = v1 B,5t-1 +
We being by pretraining uncon- ditional models such that px (x) R (a) and py (v) x
Our method: DDMEC
4

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Learning to Match Unpaired Data with Minimum Entropy Coupling
Py (v). Then, we use 0. and d, initialize conditional models y and PÝzx, which use denoising networks that accept additional conditioning signals, following Zhang et al. (023). Next, interpret the optimization expressed in Equation (6) model fine-tuning objective, which is reminiscent of the work by Fan et al. (2023).
Algorithm DDMEC Training Loop
0., tb Initialize 0 +- 0.. o o, repeat
Input:
Call Algorithm
2 with py, Pr,0000,0 0, Call Algorithm
with x ~ px O 2
until Converged
min E
u. (g. + πx KL
pX PX
Algorithm 2 DDMEC Training Step
min E xxpxx r (x, + XL
py &
|| 1801
(9)
Input: y, 0, 0,
x ~ p PXIY= ~ TT, No I) Update e using Equations (TO) and
where r = - PoggP and r = - log x are re- ward signals striving to minimize the conditional entropies, and Ay are scaling factors used for fine-tuning, that no longer require to be extremely large. Furthermore, enforce the joint constraints via extra penalty terms IKL P ll 1
Update e using Vo [lle Es 3 1)
the parameters 6 of the specular model py x which we use as reward term. Then we adapt the parameters e to ensure Px.y = y this is achieved by noting that we can adapt Equation (7) to this purpose, whereby the parameters 0 are now fixed. In the second phase (which can described as the specular version of Algorithm ( the parameters of the model Py while fixing the parameters 0 of the model P using the corresponding reward term. Finally, in a specular manner to the first phase, we ensure coherency of the two models by adapting & such that p px.y, thus satisfying the joint constraint.
IKL Px,y )
Fine-tuning DDPMs introduces significant computational overhead. To address this, various studies have explored supervised methods (Lee et al. 2023 Wu et al. 2023) or re- inforcement learning. In (Clark et al. 2023
we optimize for
Xu et al. 2024). fine-tuning is achieved through direct propppappagttion through the reverse process, which can be costly. Alterna- tive methods use proximal policy optimization (PPO) (Fan iet al. 2023 Black et al., [2244] UChaaa et al. 2024). leading to improved stability. Note that (Fan al. 2023) incor- porates KL-regularization to maximize the reward signal, while ensuring fidelity to the pretrained model, which is analogous our soft marginal constraints.
4. Experiments
DDMEC is general method that can be applied across a variety of data domains, as it relies on an information- theoretic measure to match unpaired entities. Next, we demonstrate DDMEC versatility using two realistic pairing tasks that use various data modalities, including multi-omics and image data. We compare DDMEC art methods for each task, and measure performance using domain-specific metrics. More details about DDMEC im- plementation, and our experimental protocol are given in Appendix A
In our implementation, we compute gradients of the reward V EpE r (y. .)) as follows (Fan et al. 2023)
state-of-the-
E (u, x)
V log (12.1)
(10)
while the gradient of the marginal constraints Vo
px gradient of an upper bound (Fan et al. 2023)
p& are obtained as the approximate
4.1. Multi-omics single-cell alignment
Single-cell measurements techniques. such as mRNA se- quencing for trose-ttanseriptomee analysis at the single-cell level (Tang et al. 2009), have been adapted and commercial- ized by companies which developed platforms to facilitate scalable and efficient single-cell transcriptomics and multi- omies data collection. This data provides detailed snapshot of the heterogeneous landscape of cells in a sample, and can be used to study the cell developmental trajectories across time, for example. The availability of multi-omia measure- ments - capturing various properties of a cell, such as gene expression, mRNA transcriptomes, chromatin accessibility,
T VoE., [lle (I,, - e"
(11)
Similar expressions apply to the specular model.
Given pretrained models x Py dh. the pseudo-code of our DDMEC method in Algorithm| Dis extremely simple, as it materializes as alternating optimization steps, described (for the top Equation ( in Algorithm 2 First, we optimize for the parameters 0 of the model Px while fixing
5

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Learning to Match Unpaired Data with Minimum Entropy Coupling
histone modifications, name -clls for data integra- tion methods combine variety of modalities (Xi al. 2024). Unfortunately, current measurement techniques are destructive: it is hard to obtain multiple types of measure- ments from the same cell. Furthermore, is well-known that different cell properties, such as transcriptional and chromatin profiles, cannot be matched using the geometric properties of features in the two domains. Then, pairing single-cell data modalities requires methods that do not rely on either common cells common features across the data types (Welch et al. 2017: Amodio & Krishnaswamy 2018 Welch et al.
Method
Celltpe Acc
Subcelltype Acc
PBMC
UnionCom*
34.8 : 10,9
22.9 # 7.2
MMD-MA*
28.3 6.4
10.2 # 4,8
SCOT +
12.9 1.1
2,4
scTopoGAN
8.6
41.3 6.5
DDMEC
66.3 = 2.6
46.0 = 0.5
BM
UnionCom
51.8 3.7
20.9 = 2.6 8,4
MMD-MAR
38.8 + 17.9
SCOT* scTopoGAN
90.5 0.0
31.6
50.9 14.7
2,,5
2019 Stuart et al. 2019)..
DDMEC
77.340.1
44.220.1
Baselines. We compare our proposed method DDMEC to several baselines from both the machine learning and bioinformatics literature, including SCOT (Demetci et al. 2022), MMD-MA (Liu et al., 2019), UNIONCOM (Cao et al. 2020), and ScToPOGAN (Singh et al. 2023). SCOT pro- poses a variant of an OT formulation based on the Gromov- Wasserstein distance, which preserves local neighborhood geometry when transporting data points. MMD-MA is global manifold alignment algorithm based on the max- imum mean discrepancy (MMD) measure. UNIONCOM performs unsupervised topological alignment for single- cell multi-omics data, emphasizing both local and global alignment. SCTOPOGAN uses topological autoencoders to obtain latent representations of each modality separately; topology-guides Generative Adversarial Network then aligns these latent representations into a common space. We compare our method to INFOOT (Chuang et al. 2023) in Appendix I All alternative methods we consider require a choice of distance or similarity measures, which is a pain point that our method DDMEC completely eliminates.
Table I: Single-Cell alignment experiments
quality of the coupling, we compute cross-modks neighbor- hood consistency; for each cell in one modality, we identify its h = 5 nearest neighbors in the aligned space from the other modality using Euclidean distance. We then evaluate the proportion of cases where the cell's class or subclass label matches the majority label among its neighbors. The resulting metrics are reported as the Celltype Ace and Sub- celltype Acc, respectively. In this experiment, DDMEC is trained once, and inference is conducted five times with different seeds.
Results. Section 4.1 palserts the quantitative results for the single-cell alignment task. Results marked with reported directly from (Singh et al., 2023). We observe that DDMEC consistently outperforms existing baselines on the PBMC dataset, achieving superior performance in aligning both coarse-grained cell types and fine-grat one-grained cell subclasses. On the BM dataset, DDMEC obtains the best performance for subclass-level alignment and ranks second for cell-type alignment.
Datasets. We evaluate our method on single-cell multi- omics datasets: the peripheral blood mononuclear cells (PBMC) dataset and the bone marrow (BM) dataset. The PBMC dataset comprises healthy human peripheral blood mononuclear cells, profiled using the 10x Genomies multi- ome protocol, which enables simultaneous measurement of gene expression (RNA) and chromatin accessibility (ATAC) from the same cells. This dataset includes a total of 1 11,910 cells, encompassing 7 major immune cell types that are fur- ther subdivided into 20 finer-grained cell subclasses. The BM dataset consists of human bone marrow cells profiled using the CITE-seq protocol (Stoeckius et al. (207]). which jointly captures gene expression (RNA) and protein abun- dance via antibody-derived tags (ADT). A set of 10,235 cells are randomly selected from each modality based on the major cell type labels.
Notably, DDMEC is the only method that demonstrates robust performance across both datasets, whereas alterna- tive approaches exhibit inconsistent behavior-e., SCOT performs well on BM but fails to generalize to PBMC. In contrast to existing methods that learn deterministic one- to-one mappings between modalities, DDMEC funda- mentally generative. It learns sample from a coupling distribution rather than enforcing fixed correspondence. To compute alignment metrics, we draw sample in the tar- get modality conditioned on source cell, then identify the closest observed cell in the dataset using Euclidean distance. Figure T illlstrates conditional generation with DDMEC using UMAP projections.
4.2. Unpaired image translation
For both datasets, we adopt the data preprocessing and eval- uation pipeline described in (Singh et al., 2023), resulting in 50-dimensional embeddings per modality. To assess the
This is a well-known problem in computer vision, where, in the absence of paired data (the joint distribution), the
6

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Learning to Match Unpaired Data with Minimum Entropy Coupling
which are related to our method, have also been explored for unpaired image translation. UNIT-DIPM (Sasaki et al. 2021 learns two conditional models along with two ad- ditional domain translation models, incorporating GAN- like cycle-consistency loss. ILVR (Choi et al.) ( and SDEDIT (Meng et al., 2 utilize a diffusion model in the target domain while conditioning on a source im- age refine the sampling procedure for image translation. EGSDE (Zhao al., 2022) employs an energy function pretrained on both source and target domains to guide the inference process. Similarly, SDDM (Sun et al., 2023) intro- duces manifold constraints, forcing distributions at adjacent time steps to be decomposable into denoising and refine- ment components. Compared to these methods, DDMEC leverages two conditional models, one per domain, which can be initialized using pretrained unconditional diffusion models. By design, our method does not require comparable domains and does not rely on specific image similarity measure. We report results for two values of the guidance coefficient, a parameter influencing conditional generation.
LINAU2
ATAC
UMAP1
cells
CD4 cells
UMAPP
CDB cells
Dendritic cells
HSPC
Monocytos
Datasets. We adopt the same experimental validation pro- tocol as described by /Zhao et al. (2022). where all im- ages are resized to resolution of 256 x 256. We use the AFHQ (Choi et al., 2020) dataset, consisting of high- resolution animal face images across three domains: CAT, DOG, and WILD. This dataset exhibits relatively large vari- ations within and between domains, with 500 test images per domain. We compute the performance of our method DDMEC and compare it to the baselines on CAT->DoG and WILD-DOG tasks. Furthermore, we employ the CELEBA- (Karras 2017) dataset, which comprises high- resolution human facial images categorized into two distinct domains: MAL.F and FEMALF. To evaluate the efficacy of our proposed approach relative to existing baselines, we conduct experiments on the domain translation task from MALE to FEMALLF
UMNN1
Figure 1: Conditional generation with DDMEC on PBMC dataset. Top: UMAP visualizations of source and generated data. Bottom: points are colored by cell type, illustrating how well DDMEC preserves cell type separation.
objective is to discover the correct mapping between two image domains. In this work, we show that unpaired image translation can be framed as a MEC problem, where the goal is learn the correct joint distribution between two unpaired image domains, X' and V respectively. Given the growing popularity of diffusion models in image-related tasks, pretrained weights for various image domains are available: we leverage them in our method DDMEC as done e.g. by Zhang et al. (2023).
In addition, due to the bidirectional architecture of DDMEC which leverages two conditional generative models our framework also inherently supports translation in the reverse direction. The corresponding results for the FEMALE + MALE task are presented in Appendix B
Baselines. As the literature on image translation is vast, here we primarily focus on the unpaired case, and compare our method to vast range of alternatives. Generative Ad- versarial Network (GAN) have been widely applied to this domain (Pang et al. 202T). These methods can be broadly categorized into those focusing on cycle-consistency, which enforces bidirectional mappings between image domains, such as CYCLEGAN (Zhu et al., 2017), DUALGAN (Yi et al. 2017). SCAN (Van Gansbeke et al. 2020). and U- GAT-IT Kim 2019); the second category uses distance- based methods, such as DISTANCEGAN (Benaim & Wolf 2017), GCGAN (Fu et al.
Results. In Table 2 we present the quantitative results of DDMEC results for alternative methods, marked with x, are reported as obtained in (Sun et al., 2023] Zhao et al.) 2022 DDMEC results are reported using seeds. The evaluation is based on generation quality, measured by the Fréchet Inception Distance (FID) score (Heusel et al. (017) (lower is better), and the fidelity to the source domain, as- sessed using structural Similarity Index Measure (SSIM) score (Wang et al., 2004) (higher is better). Note that quality and fidelity can be thought of as divergent objectives: high
2019). CUT (Park et al.) 2020).
and L.SESIM (Zheng et al.
2021 Diffusion-basee models,
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Learning to Match Unpaired Data with Minimum Entropy Coupling
quality does not imply high fidelity and vice-versa.
Table 2: Quantitative image translation results.
On the AFHQ dataset, GAN-based methods generally suffer from low image quality, except for STARGAN, which achieves low FID but performs poorly on SSIM. In contrast, iittusiionbbsede methods demonstrate superior performance compared to GAN-based approaches. DDMEC achieves the best FID score the CAT->DOG task and the highest SSIM in the WILD->DO task while maintaining comparable results on the remaining metrics. Overall, DDMEC strikes the best balance between high-quality image generation and accurate alignment with the target domain.
Model
FID
SSIM
CAT-DDO
CycleGANx
85.9
MUNIT*
104.4
DRIT*
123.4
Distancex
155.3
SelfDistancce
144.4
GCGAN+
96.6
LSeSim+
72.8
ITTR (CUT)*
68.6
StarGAN v2*
54.88 1: 1.01 76.21
0.27 = 0.003
Our results on the CELEBA- HQ dataset, demonstrate that DDMEC outperforms competitors on both FID and SSIM even with only 50 sampling steps. Specifically, 50 steps the improves by approximately point and the SSIM by 0.02 points, with FTD reduction) when using 100 sampling steps: it is well- known in the generative modeling literature that these im- provements are significant. With the TELEBA-HQ dataset, DDMEC benefits from a larger training set than in AFHQ animal dataset, and achieves tatee-o-thheaar performance on image translation.
CUT*
0.601
SDEdit
74.17 + 1.01
0,423 = 0.001
ILVR*
74.37 + 1.55
0.363 = 0.001
EGSDE*
65.82 + 0.77
0.415 + 0.001
greater improvement (a point
SDDM*
62.29 + 0.63
0.422 0.001
50 Sampling steps DDMEC (guidance=9)
60.70 + 1.07
0.410 + 0.001
DDMEC (guidance=7)
58.50 + 0.96
0.404 = 0.001
100 Sampling steps
DDMEC (guidance=9)
60.51 1.01
0,403 1 0.001
DDMEC (guidance=7)
57.89 + 0.37
0.397 = 0.001
WILD DDG
SDEdite
68.51 + 0.65
0.343 + 0.001
ILVR*
75.33 1.22
0.287 + 0.001
EGSDE*
59.75 + 0.62
0,343 + 0.001
SDDM*
57.38 + 0.53
0.328 = 0.001
50 Sampling steps DDMEC (guidance=9)
62.03 + 1.18
0.360 + 0.002
DDMEC (guidance=7)
60.67 + 1.01
0.353 = 0.004
100 Sampling steps
DDMEC C (guidance=9)
62.09 1 0.59
0.356 0.001 0.346 = 0.001
DDMEC (guidance=7)
59.22 + 0.35
MALE FEMALE
SDEdit=
49.43 + 0.47
0.572 = 0.000
ILVR*
46.12 + 0.33
0,510 + 0.001
EGSDE*
41.93 + 0.11
0.574 + 0.000
SDDM*
44.37- 0.23
0.526 0.001
50 Sampling steps DDMEC (guidance=2.5)
40.73 +0.61
0.593 0.003
DDMEC (guidance=2)
36.99-t 0.83
0,556 + 0.002
100 Sampling steps
DDMEC (guidance=2.55
38.93 1 0.37
0.588 + 0.002
DDMEC (guidance=2)
34,86 1 0.70
0.549 0.002
This outcome aligns with expectations, as DDMEC designed to reduce uncertainty and enforce adherence marginal constraints. In Figures 2 and 3 qualitative results further confirm the performance of DDMEC in this task. Additional results are available in Figures Gand
Source
Output
Source
Output
Qualitative results of DDMEC -100 (guid- ance=2.5) on CELEBA-HQ. Left: -MAAE-FEMALE and Right: MALE Source domain image is used to generate the target female image.
Figure 2:
We investigate the effect of the guidance scale which acts as a temperature like parameter that controls conditioning strength- in our diffusion- based unpaired image translation
8

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Learning to Match Unpaired Data with Minimum Entropy Coupling
framework. A test-time ablation study (Figure reveals trade-off: higher guidance improves SSIM (structural similarity) but degrades FID (image realism), while lower guidance yields the opposite. This illustrates how guidance balances modality fidelity and mutual information maxi- mization under the MEC framework.
cific domains, such images or language. However, exist- ing methods typically rely on geometric spaces compute costs, mappings, and similarities between data points.
We proposed a novel method that shifts the focus toward information and uncertainty quantification, thereby cir- cumventing the limiting assumptions of prior approaches. Specifically, we studied the coupling problem through the lens of minimum entropy coupling, Since prior work on MEC has largely been confined to discrete distributions, extended this framework to continuous distributions. Our key idea lies in introducing a parametric class of joint dis- tributions reinterpreted as conditional generative models, augmented with terms to enforce adherence to marginal con- straints. Our approach uses two models, which alternately optimize their objectives while approximately satisfying marginal constraints.
The resulting method enables sampling and generation in either direction between modalities, without requiring spe- cialized embeddings or strict geometric assumptions. Fur- thermore, it adaptable to complex settings beyond one-to- one matching between modalities. We validated the perfor- mance of our approach in two domains. First, we applied it to multi-omics sequencing data, and compared method against several sate-offthee-r alternatives that rely predefined measures for data comparison and coupling cost definition. Our approach, being more general and free from stringent assumptions, achieves performance on par with superior to these alternatives. Second, we evaluated our method in the image translation domain, comparing range of approaches from the literature. Our method demonstrated superior performance across widely recog- nized metrics for image quality and coherence, by striking good balance between these often conflicting measures.
Source
Output
Source
Output
Figure 3: DDMEC (guidance=7) CAT->DoG (Left) and WILD-DOG image (right) translation examples. Source domain image used generate the target dog image.
5. Conclusion
The machine learning community has recently directed sub- stantial effort toward designing multimodal models, as they reflect the inherently multi-faceted nature of the real world. These models often achieve superior performance on down- stream tasks compared unimodal counterparts. However, the intrinsic complexity of multimodal data introduces sig- nificant challenges. In this work, we addressed the critical problem of coupling data represented by diverse modali- ties. The coupling problem has been widely studied in the literature, often framed as an optimal transport problem or approached with specialized architectures tailored to spe-

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Learning to Match Unpaired Data with Minimum Entropy Coupling
Acknowledgment
In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 11618-11622 11628. IEEE, 2020. doi: (AA00///CVPR220022220..000
Pietro Michiardi was partially funded by project MUSE- COM2 Al-enabled MUltimodal SEmantic COMmunica- tions and COMputing, in the Machine Learning- based Com- munication Systems, towards Wireless AI (WAI), Call 2022, ChistERA.
Cao, K., Bai, X., Hong, Y., and Wan, L. Unsupervised topo- logical alignment for single-cell multi-omics integration. Bioinformatics, 36(Supplement. 1)::48ttt)))1-ii: i56, 2020.
Chen, M.. Xing. L., Wang. Y., and Zhang. Y. Enhanced multimodal representation learning with cross-modal kd. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition (CVPR), pp. 11766- 11775, 2023.
Impact Statement
This paper aims to advance Machine Learning by improv- ing unpaired data translation, with potential benefits for applications such as multi-omics alignment in biology and image translation. However, these advancements may also increase risks associated with generative models, including the generation of deepfakes. Responsible and ethical use is advised.
Chen, S., Lake, B. B., and Zhang, K. High-throughput sequencing of the transcriptome and chromatin accessi- bility in the same cell. Nature Biotechnology, 37(12): 1452-1457, 2019.
Chen, W. Hao, X., Wu, Y., and Liang, Y. Terra: A multimodal spatio-temporal dataset spanning the earth. In Advances in Neural Information Processing Systems, 2024. URL
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15

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Learning to Match Unpaired Data with Minimum Entropy Coupling
A. Additional Details
DDPM
P
X
Paired data
X
(1) ming log P AX (p&
Unpaired data
(x, y)
Pxx = P XY Cooperative
(2)
x))x
min, log (xy)) 1y ApLl(p8 (p\lls
y
DDPM
p
y
Figure 4: Overview of the DDMEC methodology. The phase (1) corresponds to the procedure described in Algorithm 2 Lines 2 and Z and involves generating samples (depicted in red and blue) conditioned on inputs æ and y drawn from their respective marginals. These samples are then used evaluate the loss defined in Equation (9). which is practically optimized using PPO (Fan et al.) (202)). The phase (2) corresponds to Line of Algorithm? 2 wherein the joint consistency constraint is enforced by updating the model with the previously generated sample pairs. Both phases require coordination between the two models, alternating their roles as outlined in Algorithm
A.1. Details on Swapping the Roles of the Conditional Models In this part we provide more details about the justification of
V VeE Fe [- loo [ogpX(r] Ro Vol. VoLe. |- log prr
(12)
whenever xx Px
Equation reads:
Vo f pr (v) P (xly)) log fogr' (xly) da dy.
(13)
Moving the gradient V inside the integral and applying the chain rule, we obtain:
I pr(()) Vo (pxix (xlv)) log p& (alv) dr dy
+ f pr(u)) Vo ( (xlv)) dx dy.
(14)
16

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Learning to Match Unpaired Data with Minimum Entropy Coupling
The second term simplifies to zero:
f py |rr()) (x)r) vo (10g)()v(x)))) dæ dy = py (v) px(()) Vo (1x)))))))) dr dy
V. / pr pr(() p (xl1) dx dy
Vol = 0.
(15)
Assuming pxx,y = & PxxY, px = Px, and p = py, the first term rewrites as:
/ pr((v) Vo (+l)) log px (xlx))xxxx) (z)
dx dy
pr(u)
= = pr(v)) Vo (xl)) (logrrix (u|x) + logpx(x) logpr(s)) de dy
= Vo / pr( px (zl log P Prxx (y/z) dr dy,
(16)
which corresponds the right-hant hand side (r.h.s) of Equation Indeed, the additional terms vanish:
J (Dr(w) Vo (pxrr (zlw)) log px (x) de dy = / Ve (p}(2)) logpxx>)))
/ Vo (ox
(17)
and similarly,
p Vo (xl))) logpy py (v) dz dy = O.
pr())
(18)
A.2. Diffusion Models Training with Reinforvement Learning
Our methodology begins by training unconditional diffusion models for both data modalities. We then use a reinforcement learning technique to train two conditional models (initialized from the first step) in a cooperative manner, allowing them learn from each other to optimize the joint coupling under MEC constraints and objectives. We formulate this second phase as training diffusion models with reinforcement learning and IKLL-reuularization We follow the training scheme presented by Fan et al. (2023), where samples are generated conditionally using classifier guidance (Ho & Salimans] 2021] with DDIM sampler (Song et al. 2020). The generated trajectories are then used to update the diffusion model, which framed as a Markov Decision Process (MDP), using a policy gradient RL algorithm.
Reward Estimation In DDMEC the reward signals are log-likelln ligeliieli values mutually generated by the two conditional models. Accurately estimating this signal is crucial for steering training towards the optimal MEC solution. To achieve this, multiple Monte Carlo steps to estimate Equation 8
Policy Gradient Training We follow the training procedure of Fan et al. (2023) where, at each step, a batch of samples is generated using DDIM (Song et al., [2020). These generated trajectories are then used to perform multiple gradient updates. Additionally, we apply importance sampling and ratio clipping (Schulman et al. 2017) to improve training stability.
Classifier- Guidance We employ classifier-fre guidance (Ho Salimans 2021 in all experiments. This technique enables conditional sampling in step [ The denoising loss in 2 is optimized to account for the guidance mechanism by randomly dropping 10% of the conditional signal. thereby stabilizing the unconditional model.
A.3. Technical Details and Hyperparameters
The source is publicly available.
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Learning to Match Unpaired Data with Minimum Entropy Coupling
Single-Cell Alignment For the PBMC and BM datasets, utilize the preprocessed versions provided in the official code repository of Singh et al. 2023 along with the accompanying evaluation protocols. Prior training, we normalize the data by subtracting the mean and scaling to unit variance, while applying outlier mitigation. The model is first trained unconditionally using DDPM for 100,000 steps with T = 1000 diffusion steps. To stabilize training, we additionally incorporate a nearest neighbor retrieval step after each generation, where generated samples are projected back to the closest real data points using Euclidean distance. Subsequently, we train the two conditional models, following the procedure outlined in Algorithm[] [ For each training step, we use batch size of 256 and perform four gradient updates corresponding line followed by four updates for 2ine We find it beneficial accumulate generated samples during training and reuse them in optimizing _ We use simple MLP network with skip connections and use the Adam optimizer (Kingma 2014) with learning rate of x 10-4, The KL divergence regularization weight is set to À 0.01 for PBMC and X 0.02 for BM.
Unpaired Image Translation DDG and WILD->DOG Tasks: We utilize the pre-trained model from the official implementation Choi et al.
1ttpl/11///vvaallee to initialize the dog modality conditional model. For the other domains (CAT, WILD): We train diffusion model from scratch using the same architecture and sypper-prraetent as done the target domain. MALE MMALE-FEMALE FEMALE Task: use the publicly available pre-trained model by Zhao et al. (2022) (attps: / git.ub...uu....oggg.o com/ ML- GSAI EGSDE) for the Female modality. For the Male modality, train diffusion model from scratch using the same architecture and papee-pprameters done in the target domain.
To introduce additional conditioning into the pre-trained diffusion model, we follow the work in (Zhang et al. 2023). where the encoder part of the U-NET duplicated and used as a conditional encoder. The various hyperparameters are summarized Table 3
We follow the evaluation protocol described in (Zhao et al. 2022).
General Settings
Dataset
AFHQ
CelebA-HQ
Batch Size
16
Learning Rate:
2e
Optimizer
ADAM
ADAM
Training Steps
2000
Weight Decay
0.0
Diffusion Model
Noise Scheduler
Linear
Linear
Number of Diffusion Timesteps (T)
1000
1000
Sampler
DDIM
DDIM
Guidance Scale (training)
7.0
7.0
Sampling steps
50
50
Exponential moving average
Yes
Yes
Reinforcement Learning
Reward (Monte Carlo steps)
3
3
Policy Update Steps
4
4
Importance Sampling Clipping
le - 4
le
\ X2
le 3
4e
Gradient Accumulation
12
Gradient Clipping
1.0
1.0
Table 3: Hyperparameters used for training.
htttaar//ataaatcco////aaaaaaaaae 18

---

Learning to Match Unpaired Data with Minimum Entropy Coupling
B. Additional Results
B.1. SNARE-seq additional experiments
In this experiment we use the SNARESEQ (Chen et al. 2019) dataset, which links chromatin accessibility with gene expression data on mixture of four cells types. We use the same preprocessing procedures detailed in (Demetci et al.) 2022), which deal with filtering spurious data affected by technical errors, and normalization. Data samples have 1-1 correspondence information, which constitute the groud-truth information used for our performance evaluation. We the average "fraction of samples closer than the true match (FOSCTTM)" metric introduced by Liu et al. (2019): given a sample in one domain, this amounts to compute the fraction of samples that are positioned more closely to than its true match after pairing. Results report the average FOSCTTM across all samples, where lower values indicate better performance. We also report the label transfer accuracy as done by Cao et al. (2020), which measures how well sample labels are transferred between domains based on neighborhood alignment. k-nearest neighbor classifier is trained on one domain and used to predict labels in the other. this experiment, DDMEC trained once, and inference conducted five times with different seeds. the FOSCTTM metric, and is on-par with the best method in terms of accuracy. In this experiment, DDMEC is trained once, and inference is conducted five times with different seeds. Unlike other methods, DDMEC is conceptually different as it generates samples rather than learning a deterministic, 1-1 mapping. To compute the different metrics, given sample from one modality, we use DDMEC to generate a coupling to the other modality, then select the nearest sample from the dataset based on Euclidean distance.
SNAREseq
FOS
Acc f
UnionComx
0.265
42.3
MMD-MA*
0.150
94.2
SCOT*
0.150
98.2
InfoOT*
0.156
98.8
DDMEC
0.147
98.6
Table Performance results on SNAREseq dataset
1.5
GT
GT Generated
Generated
K362
K562
0.6
0.0
-0.5
-0.4
H1
HL
-066
-1.0
-0.5
0.0
1.0
-1.
-1.0
-0.5
0.0
0.5
Figure 5: Conditional generation using DDMEC on the SNAREseq dataset. The cell types are indicated by colors. Top: generation of chromatin accessibility data using gene expression, Bottom: generation of gene expression using chromatin accessibility data.
19

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Learning to Match Unpaired Data with Minimum Entropy Coupling
Source
Output
Source
Output
Figure 6: DDMEC (guidance=7) CAT (Leff) and WILD-+DOG image (right) translation examples. Source domain image is used to generate the target dog image.
20

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Learning to Match Unpaired Data with Minimum Entropy Coupling
Source
Output
Source
Output
Figure 7: DDMEC (guidance=7) DOG-CAN (Left) and DOG-WILD image (right) translation examples. Source domain image is used to generate the target Cat/Wild image.
21

---

Learning to Match Unpaired Data with Minimum Entropy Coupling
steps=50
55
5
steps=100
50
45
FID
2.5
40
2.0
35
1.5
30
0.50
0.55
0.60
0.65
0.70
SSIM
(a) the guidance scale ablation study, we report FID and SSIM function of the guidance scale on the CelebA-HO HQ dataset. We notice that an increase in the guidance scale results in more information transfer between the two modalities, leading to worse FID, and vice versa.
22

---

Learning to Match Unpaired Data with Minimum Entropy Coupling
Source
Output
Source
Output
Figure 9: DDMEC (guidance=2.5) MALE FEMMAE translation examples. Source domain image is used to generate the target female image.
23


Reviews:
[{"review_id": "jwBhSCrGDW", "reviewer": "Reviewer_zrWS", "summary": "The manuscript presents a novel method for matching unpaired data through Minimum Entropy Coupling (MEC). By extending MEC to continuous distributions and leveraging denoising diffusion probabilistic models (DDPMs), the authors propose a cooperative framework that alternates between two conditional generative models. The method is evaluated on single-cell multi-omics alignment and unpaired image translation. Experimental results show competitive performance compared to existing state-of-the-art approaches.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "We thank the reviewer for recognizing the contribution of our method for matching unpaired data and the insightful feedback which we address in the following. As suggested by the other reviewers as well, we performed additional experimental campaigns to challenge our DDMEC method.\n\n> 1- Could you provide a more detailed discussion and analysis on the sensitivity of your method to various hyperparameter choices?\nThe hyperparameters on which our method depends are:\n\n* **Classifier Guidance Weight**: This parameter acts as a temperature scale that determines the influence of conditioning in the diffusion model. Since our reinforcement learning framework for diffusion models requires sampling from the model during training, we use classifier guidance and set it to cfg = 7, following (Fan,2023). In the plot available here [ https://anonymous.4open.science/r/icml2025_ddmec-7798/celeba/ablations.png ], we show the results of a new ablation study on the guidance scale in the unpaired image translation case, which is performed at test time. The plot displays the trade-off between FID and SSIM scores, as a function of the guidance scale for two different number of sampling steps (50 and 100).\nIntuitively, we observe that the guidance scale controls the tradeoff between information transfer between modalities and fidelity to the marginal constraint. A higher guidance scale improves SSIM but worsens FID, while a lower guidance scale favors FID but negatively impacts SSIM. Under the MEC framework, guidance allows us to balance the maximization of mutual information (by lowering the joint entropy) and maintaining fidelity to the marginals. \n\n- **Lambda Weight**: We find that a low KL divergence weight is essential for ensuring reward convergence, thus set \n $\\hat{\\lambda_X} = \\hat{\\lambda_Y} = 0.001$, in equation (9) for all experiments. Indeed, as noted in (Fan, 2023), the  $\\hat{\\lambda_{(\\cdot)}}$ parameter helps the diffusion model to remain close to the unconditional model and prevents reward overfitting, thus avoiding the degenerate solution where the marginal constraints are ignored. Recall that our method initializes the model with the unconditional model, so we start from a state where the marginal constraints are fully respected and the KL divergence is near zero. Thus, the role of lambda is weight to the KL to keep the conditional model close to the unconditional one while optimizing the joint entropy coupling.\n\nFan, Y., Watkins, O., Du, Y., Liu, H., Ryu, M., Boutilier, C., ... & Lee, K. Dpok: Reinforcement learning for fine-tuning text-to-image diffusion models. Neurips2023.\n\n> 2 - Could you elaborate on the computational cost and runtime of your approach compared to baseline methods?\n\nWe discuss computational costs of our and baseline methods according to an application-specific division.\n\n- **Image Translation**: Given that diffusion models (DMs) outperform GANs for this task, we compare DDMEC against other diffusion-based methods. The inference process of our method behaves similarly to an unconditional model, with additional conditional layers to incorporate guidance. The generation requires an iterative process with a guidance mechanism. We find that 50 steps are sufficient to achieve superior performance. This is significantly more efficient than SDEdit and EGSDE, which require 1000 steps, and SDDM and ILVR, which require 100 steps.\n\n\n-  **Single-Cell RNA (ScRNA) Analysis**: For the SNARE dataset, optimal transport (OT) methods are computationally lightweight and achieve results comparable to DDMEC. However, this dataset is limited, with dimensionalities of 19 and 10 for the two modalities and only 1,047 samples. To further assess scalability, we conducted additional experiments on larger and high-dimensional datasets such as PBMC (see Question 1 of Reviewer 2cNi). The results indicate that OT methods fail entirely on this dataset, whereas alternative methods like DDMEC and ScTopoGAN perform significantly better. In general, approaches to solve the OT problem in high dimensions face challenges related to the cost matrix computation and fundamental difficulties of distance estimation in high dimensions. \n\nComparison with ScTopoGAN: In terms of computational time, DDMEC is comparable to ScTopoGAN. However, ScTopoGAN involves a first topological autoenconding and a GAN model trained via multiple distinct phases and multiple runs to select the best model over multiple seeds, making a direct computational comparison challenging.\n\n>3 - Have you conducted repeated experiments to measure the stability of your method?\n\nInference-time results are run over several seeds to improve statistical significance. Our new results show confidence intervals in terms of standard deviation.", "claims_and_evidence": "The claim that a cooperative diffusion-based approach can be used to solve the continuous MEC problem is supported by both theoretical derivations and experiments. The author claim that DDMEC is applicable across diverse domains. Although the experiments on two tasks are promising, these evaluations may not be sufficient to claim full generality. More extensive testing on a broader range of datasets or modalities could better support this claim. Claims that the proposed method outperforms state-of-the-art methods are supported by quantitative results. However, the margins of improvement is relatively small, and no repeated experiments are conducted to measure the stability of the method.", "methods_and_evaluation_criteria": "The methods and evaluation criteria make sense for the problem at hand. The authors propose method is well aligned with the challenge of matching unpaired data in multimodal settings.\nThe choice of benchmark datasets and metrics also appears appropriate.", "theoretical_claims": "The proofs and the theoretical claims appear correct.", "experimental_designs_or_analyses": "The experimental designs and analyses are fundamentally sound. The authors did not provide sensitivity analyses, and computational cost discussion. No repeated experiments are conducted to measure the stability of the method.", "relation_to_broader_scientific_literature": "This paper extends the Minimum Entropy Coupling (MEC) framework to continuous distributions. By reinterpreting the joint distribution as two conditional generative models, the work is related to generative diffusion models.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "The paper makes contributions by extending the MEC framework from discrete to continuous settings. \nThe authors provide derivation of the optimization objectives, including soft marginal constraints and the joint cooperative scheme. \nExperimental results on two distinct tasks demonstrates the method’s performance\n\nThe manuscript would benefit from more detailed discussion and analysis on how sensitive the method is to these choices could strengthen the empirical section.\nA discussion of computational cost and runtime compared to baseline methods would help assess the practicality of the approach.\nRepeated experiments could be helpful in measuring the stability of the method.", "other_comments_or_suggestions": "No.", "questions_for_authors": "1. Could you provide a more detailed discussion and analysis on the sensitivity of your method to various hyperparameter choices?\n\n2. Could you elaborate on the computational cost and runtime of your approach compared to baseline methods?\n\n3. Have you conducted repeated experiments to measure the stability of your method?", "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": "", "extra_scores": {}}, {"review_id": "Z9kjXhLONu", "reviewer": "Reviewer_Acbm", "summary": "This paper proposes minimum entropy coupling (MEC) to align unpaired multimodal data. MEC seeks a joint distribution with the desired marginals that is optimal in the sense of minimum entropy, in comparison to optimal transport approaches which minimize an integrated cost. This sidesteps the difficulty of specifying a cross-modal cost function which in previous work was either achieved using auxiliary labels or with Gromov-Wasserstein OT. To solve MEC over continuous spaces, the joint is factorized into known marginals and unknown conditionals, for which minimum entropy becomes maximum conditional likelihood. This is done for both possible factorizations and the conditional models are trained as denoising diffusion. Experiments are performed on single-cell and image translation datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "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": "Thank you for recognizing the relevance of the MEC framework in coupling unpaired data and for the insightful feedback, which we address next. \n\n> Another single-cell dataset.\n\nWe perform new experiments using the peripheral blood mononuclear cells (PBMC) dataset, as the CITE-seq dataset is used in a semi-supervised setting: while we can ignore label information with DDMEC, we lack the time to run competitors. The PBMC dataset is large and high-dimensional, consisting of simultaneous profiling of gene expression (RNA) and chromatin accessibility (ATAC) of healthy patients (refer to question 1 of Reviewer 2cNi). Our new results indicate that OT fails likely due to issues related to data dimensionality, while DDMEC achieves the best performance in all cases.\n\n> Image translation is not really multimodal. Use image caption pairs.\n\nBoth modalities are images though the unpaired setting is a significant challenge. CycleGAN lags behind DMs. DDmec outperforms DM alternatives without the need to define a similarity function like in EGSDE and SDDM ( additional CelebA experiments in RV 2cNi question 1). While interesting, due to time limits, we differ the suggested txt/img case to a camera ready version.\n\n> The claim on l190 has no justification.\n\nWe apologize, $- \\log p_{Y|Y}$ is a typo and should be $- \\log p_{Y|X}$.\n\nThe l.h.s term of l190 reads $\\nabla_\\theta \\int p_Y(y) p^\\theta_{X|Y}(x|y)  \\log p^\\theta_{X|Y}(x|y)dx dy$. Moving the $\\nabla_\\theta$ inside we obtain\n\n$$ \\int p_Y(y) \\nabla_\\theta\\left(p^\\theta_{X|Y}(x|y)\\right)  \\log p^\\theta_{X|Y}(x|y)dx dy +  \\int p_Y(y) p^\\theta_{X|Y}(x|y)  \\nabla_\\theta\\left(\\log p^\\theta_{X|Y}(x|y)\\right) dx dy\n$$\n\nThe second term simplifies to zero: \n$$ \\int p_Y(y) p^\\theta_{X|Y}(x|y)  \\nabla_\\theta\\left(\\log p^\\theta_{X|Y}(x|y)\\right) dx dy=\n\\int p_Y(y) \\nabla_\\theta\\left(p^\\theta_{X|Y}(x|y)\\right) dx dy=\\nabla_\\theta\\int p_Y(y)\\left(p^\\theta_{X|Y}(x|y)\\right) dx dy=\\nabla_\\theta1=0  $$\n\nAssuming $p^\\theta_{X,Y}=p^\\phi_{X,Y}$ and $p^\\theta_X=p_X,p^\\phi_Y=p_Y$, the first term rewrites as\n$$\n\\int p_Y(y) \\nabla_\\theta\\left(p^\\theta_{X|Y}(x|y)\\right)  \\log \\frac{p^\\phi_{Y|X}(y|x) p_X(x)}{p_Y(y)} dx dy =  \\int p_Y(y) \\nabla_\\theta\\left(p^\\theta_{X|Y}(x|y)\\right)  (\\log p^\\phi_{Y|X}(y|x) +\\log p_X(x)-\\log p_Y(y)) dx dy=\n$$\n$$\n= \\nabla_\\theta\\int p_Y(y)\\left(p^\\theta_{X|Y}(x|y)\\right)  \\log p^\\phi_{Y|X}(y|x)\n$$\nwhich is the rhs of l190. Indeed $\\int p_Y(y) \\nabla_\\theta\\left(p^\\theta_{X|Y}(x|y)\\right) \\log p_X(x) dx dy=\\int \\nabla_\\theta\\left(p^\\theta_{X}(x)\\right) \\log p_X(x) dx=\\int \\nabla_\\theta\\left(p_{X}(x)\\right) \\log p_X(x) dx=0$, and similarly, $-\\int p_Y(y) \\nabla_\\theta\\left(p^\\theta_{X|Y}(x|y)\\right) \\log p_Y(y) dx dy=0$.\n\n> Clarify Eqn 3 and Eqn 4.\n\nTo clarify, we recognize that \n$$\nH(p^\\theta_{X,Y})=-\\int p^\\theta_{X,Y}(x,y) \\log p^\\theta_{X,Y}(x,y) dx dy=-\\int p_Y(y)p^\\theta_{X|Y}(x|y)  \\left( \\log p^\\theta_{X|Y}(x|y)+ \\log p_Y(y) \\right) dx dy=\n$$\n$$\n-\\int p_Y(y)p^\\theta_{X|Y}(x|y)  \\left( \\log p^\\theta_{X|Y}(x|y) \\right) dx dy+H(p_Y).\n$$\n\nThe term $H(p_Y)$ does not influence optimization being independent of $\\theta$, while the first term, can be either rewritten as $E_{y\\sim p_Y}[H(p_{X| Y=y}) ]$ (as in eqn 3) or as $- E_{x,y\\sim p_{X,Y}}[\\log p_{X|Y}(x|y)]$, which is the expression in eqn 4.\n\n> How does Eqn 6 finds a MEC? \n\nTo see that eqn 6 is a proxy for a MEC loss, we start from equations 4 and 5, which correspond exactly to two (uncoupled) MEC problems. We combine them in a system where we use the assumption from line 190, which allows swapping the loglikelihood terms: from the perspective of a gradient based optimizer, two losses which induce the same gradients are equivalent. Attention: we need to enforce the joint constraint throughout training, as discussed in L 241.\n\n> Is the approach in Eqn. 11 a reasonable approximation to the marginal? \n\nWe apologize, the term $\\epsilon^\\theta(x_t, y = \\emptyset, t))$ is a typo and should be $\\epsilon^\\theta(x_t, y, t)$. To enforce the marginal constraint, we keep the conditional model close to the frozen unconditional model by following Fan et al. (2023), which upper-bounds the KL divergence (see their Equation 6). In their work, fine-tuning preserves proximity to the original model, whereas we learn a conditional model while maintaining closeness to the frozen unconditional model. Our Equation 11 reformulates Equation 6 from (Fan et al,2023) in terms of denoisers.\n\n> Why MEC is a good objective that aligns unpaired data? \n\nThe MEC problem is well-established and grounded in the domain of information theory. By minimizing entropy while satisfying marginal constraints, MEC maximizes mutual information, capturing shared structure between distributions. Unlike optimal transport, which relies on a predefined ground metric and may overlook structural relationships, MEC aligns unpaired data without requiring an explicit similarity measure.", "claims_and_evidence": "Yes, I think the claims made in the paper are supported by evidence.", "methods_and_evaluation_criteria": "The MEC method makes a lot of sense as an alternative to OT for the problem of multimodal alignment, which solves the problem of the marginals being in different spaces trivially. I can't really comment on whether the diffusion approach makes sense as I am unfamiliar with MEC, but it seems like a reasonable approach.\n\nThe evaluation leaves much to be desired, unfortunately. The single-cell dataset from the SCOT paper appears to be a poor choice for demonstrating the strengths of the MEC method, as SCOT already performs so well in this case. Another single-cell dataset with ground-truth pairings available is the CITE-seq dataset from the NeurIPS benchmark from 2022 that many other works in this area also evaluate on. \n\nThe image translation dataset makes little sense to include as it is not really multimodal, and the authors do little to support why MEC would be preferable to existing methods such as conditional diffusion or CycleGAN. Note the authors write (l373)\n> By design, our method does not require comparable domains and does not rely on a specific image similarity measure.\nWhich I agree is a strength but is not demonstrated through the experiment itself. As a suggestion, it would be much more interesting to look at image--caption pairs (which are abundant), artificially split them, and then evaluate on this instead.", "theoretical_claims": "The paper makes few theoretical claims. There is a claim on l190 under Equation 6 about the approximate equivalence of gradients that has no justification, and it is not clear to me that it would be true (these are conditional distributions over completely different spaces). \n\nAlso, is there a typo here? $-\\log p_{Y \\mid Y}$ should maybe be $-\\log P_{Y \\mid X}$, and is the gradient on the RHS supposed to be w.r.t. $\\theta$ or $\\phi$?)", "experimental_designs_or_analyses": "The actual design and analyses seem fine but I find them insufficient, see \"Methods And Evaluation Criteria\".", "relation_to_broader_scientific_literature": "This is an important problem that has been especially significant for single-cell biology, where measurement processes can usually only be taken once per cell, but different measurements can nonetheless capture additional information. The authors do a good job of reviewing the literature here.", "essential_references_not_discussed": "The authors do a good job at reviewing the relevant literature. There are a few methods that bring both modalities into a shared latent space and then do OT alignment on that space, e.g., MatchCLOT (Gossi et al., 2023) or propensity score alignment (Xi et al., 2024), but this usually requires some sort of supervision (ground truth pairs, or shared labels) to first learn the latent space so it may not be as relevant here. One simple baseline (though I don't expect this to do well unsupervised) might be to just do separate dimension reduction on both modalities and align them with OT. \n\nGossi et al., Matching single cells across modalities with contrastive learning and optimal transport, Briefing in Bioinformatics, 2023.\n\nXi et al., Propensity score alignment of unpaired multimodal data, NeurIPS 2024.", "other_strengths_and_weaknesses": "### Strengths\n\n- The paper does great job reviewing the literature ranging from the problem, applications and solutions, as well as effectively introducing the MEC problem. The writing was clear and approachable.\n\n### Weaknesses\n\n- I actually think the paper undersells the relevance of MEC for this problem, since it completely avoids the problem of different modalities. In Section 4.1 the authors briefly mention\n\n>All alternative methods we consider require geometric distances or similarity measures, which is a pain point that our method DDMEC lifts completely.\n\nIn my opinion this should absolutely be a central selling point of the method.\n\n- The authors do not support the usage of MEC for alignment theoretically nor intuitively. Why should we expect minimum entropy to be a good objective that aligns unpaired data? What are your assumptions about how these data became unpaired in the first place? \n\n- There is no theoretical support for the DDMEC itself: the maximum likelihood part seems fine to me but I am not sure about the alternating optimization.", "other_comments_or_suggestions": "### Typos/unclear points\n\n- See also \"Theoretical Claims\". \n\n- Eqn 3 has $E_{y \\sim p_Y}[\\mathbb{H}(p_{X\\mid Y})]$. Is this a typo? I think it should probably be $\\mathbb{H}(P_{x,y}) = E_{x,y}[-\\log p_{X \\mid Y}^\\theta - \\log p_y]$, the latter term is a constant for Eqn 4. \n\n- I am trying to wrap my head around Eqn 6. How does this find a MEC? Doesn't $E_{x, y \\sim P_{X,Y}^\\theta}( -\\log p_{Y\\mid X}^\\phi(y \\mid x))$ control the __relative__ entropy (ie KL divergence) instead of the actual entropy? I can kind of understand how this enforces $P_{X,Y}^\\theta = P_{X,Y}^\\phi$, since\n$$\n\\mathbb{KL}(P_{X,Y}^\\theta \\mid P_{X,Y}^\\phi) = E_{x, y \\sim P_{X,Y}^\\theta}( -\\log p_{Y\\mid X}^\\phi(y \\mid x) - \\log p_X^\\phi(x)) = E_{x, y \\sim P_{X,Y}^\\theta}( -\\log p_{Y\\mid X}^\\phi(y \\mid x)) + \\mathbb{KL}(P_{X}^\\theta \\mid P_{X}^\\phi)\n$$\nbut I don't understand how it gives a MEC. \n\n- In Eqn. 11 you compute the marginal $p_X^\\theta$ by $p_{X \\mid Y}(x \\mid y = \\emptyset)$. However there is a true marginal corresponding to your model $p_X^\\theta = \\int_{y} P_{X \\mid Y=y}(x \\mid y) dy$. Is the approach in Eqn. 11 a reasonable approximation to the marginal?", "questions_for_authors": "I really like the idea of using MEC to solve this problem (see weakness 1 above). However the paper falls short on two counts. I would be happy to argue for acceptance (at least, weak accept) if satisfactory improvements/answers to both of these were made.\n\n### (1) Experiments\n\nSee \"Methods And Evaluation Criteria\"\n\n### (2) Technical Details\n\nSee \"Other Comments Or Suggestions\".", "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": "", "extra_scores": {}}, {"review_id": "j9qdYIP038", "reviewer": "Reviewer_2cNi", "summary": "This paper proposes a novel method to solve the continuous Minimum Entropy Coupling (MEC) problem. Specifically, it incorporates generative diffusion models to learn the joint distribution with the minimum joint entropy, while enforcing a relaxed version of the marginal constraints.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "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": "We appreciate the reviewer's insightful feedback and suggestions. Below, we provide responses and additional experiments to address their concern.\n\n > 1. Could you add one dataset to each application? \n\n- **Additional Image translation experiment**:\n\nWe consider the CelebA-HQ dataset (Karras et al., 2017), which features high-quality human face images divided into two domains: Male and Female. Each domain includes approximately 10,000 and 17000 training samples for the male and female modalities. We compare the performance of DDMEC against several baseline methods on the Male-to-Female translation task, following the same evaluation protocol as in (Zhao et al. 2022).\n\n| **Model**                     | **FID ↓**       | **SSIM ↑**       |\n|-------------------------------|-----------------|------------------|\n| **CelebA-HQ (Male → Female)** |                 |                  |\n| SDEdit*                       | 49.43 ± 0.47    | 0.572 ± 0.000    |\n| ILVR*                         | 46.12 ± 0.33    | 0.510 ± 0.001    |\n| EGSDE*                        | 41.93 ± 0.11    | 0.574 ± 0.000    |\n| SDDM*                         | 44.37 ± 0.23    | 0.526 ± 0.001    |\n| **50 sampling steps**         |                 |                  |\n| DDMEC (guidance=2.5)          | **40.73 ± 0.61**| **0.593 ± 0.003**|\n| **100 sampling steps**        |                 |                  |\n| DDMEC (guidance=2.5)          | **38.93 ± 0.37**| **0.588 ± 0.002**    |\n\n\nIn our experiment, we follow the same training procedure as for the AFHQ dataset available in the main paper. Our new results, demonstrate that DDMEC outperforms competitors on both FID and SSIM even with only 50 sampling steps. Specifically, at 50 steps the FID improves by approximately 1 point and the SSIM by 0.02 points, with an even greater improvement (a 3-point FID reduction) when using 100 sampling steps: it is well-known in the generative modeling literature that these improvements are significant. With the CelebA-HQ dataset, DDMEC benefits from a larger training set (in the AFHQ animal dataset, each modality has approximately 3000 images) and achieves state-of-the-art performance on image translation. Qualitative results supporting the quantitative evaluations can be viewed via ( https://anonymous.4open.science/api/repo/icml2025_ddmec-7798/file/celeba/qualitative.png?v=7d6cc987 ).\n\n- **Additional Single-Cell data experiment:**\n\nWe performed new experiments on a more complex single-cell data alignment task. Since the SNARE dataset we used in the submitted paper is relatively small and simple, our goal is to further substantiate the superiority of DDMEC using the peripheral blood mononuclear cells (PBMC) dataset: this dataset consists of healthy human PBMCs with simultaneous profiling of gene expression (RNA) and chromatin accessibility (ATAC). PBMC contains 11,910 cells, spanning 7 major cell classes that are further divided into 20 cell subclasses. We use the data processing and evaluation pipeline from (Singh, 2023), which results in a 50-dimensional representation for each modality.\n\n\n\nThe obtained results available here (https://anonymous.4open.science/r/icml2025_ddmec-7798/PBMC/table.png )  demonstrate that DDMEC performs extremely well also in a high-dimensionality scenario, whereas OT-based methods completely fail. Although OT-based methods are relatively lightweight, they do not scale, motivating the need for more advanced methodologies such as DDMEC. Our method outperforms the best competitor, scTopoGAN, scoring superior performance in terms of both cell type matching rate and subcell matching rate.\n\n> 2. The applications shown in the paper are existent, and the proposed method only marginally improve existing results (e.g., InfoOT, SDDM). As DDMEC is general, could you present another application of it?\n\nThe experiments (presented in Q1) include results on the extra CelebA-HQ dataset and an additional PBMC single-cell dataset, both of which are more elaborate and sophisticated. The results demonstrate that DDMEC outperforms and improves upon the state-of-the-art. The reviewer’s suggestion about considering additional applications is both valid and insightful. As suggested by reviewer Acbm, it is possible to consider image-text pairs or multilingual text alignment. However, due to rebuttal  time constraints, we leave this to the camera-ready version.\n\n> 3. data-scalability of DDMEC?\n\nWe discuss the scalability of our method against the baselines in our answer to the question 3  of Reviewer   zrWS.\n\n> typo in line 190 \n\nWe apologize, indeed it should be $Y|X$.\n\n-Singh, et. al, scTopoGAN: unsupervised manifold alignment of single-cell data. Bioinformatics Advances 2023.", "claims_and_evidence": "In general, the claims made in the paper are supported by clear and convincing evidence. However, I believe the experiments are not extensive enough, and require additional datasets/applications.", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "I’ve checked the soundness of the experimental designs, and I don’t see any issues.", "relation_to_broader_scientific_literature": "The key contribution of the paper is related to the broader scientific literature through interesting ideas (e.g., combining diffusion models with MEC).", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strengths:\n1.\tThe paper considers an important topic in multimodal learning – learning from naturally unpaired data.\n2.\tThe paper is generally clear and easy to follow\n\nWeaknesses:\nSee questions.", "other_comments_or_suggestions": "I believe there’s a typo in line 190 (should be Y|X)", "questions_for_authors": "Question For Authors\n1.\tThe evaluation of each of the two applications is done using only a single dataset. I find it not convincing enough. Could you add one dataset to each application?\n2.\tThe applications shown in the paper are existent, and the proposed method only marginally improve existing results (e.g., InfoOT, SDDM). As DDMEC is general, could you present another application of it? Please support this with empirical evaluation.\n3.\tI’m afraid about the data-scalability of DDMEC. Could you elaborate about it? How is it compared to other translation methods such as CycleGAN and SDDM? And compared to InfoOT?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes  method for alignment of continuous  multimodal data using Minimum Entropy Coupling. Reviewers have appreciated the contribution and its usefulness in real-world scenario. I recommend to accept the paper while requesting the authors to incorporate the reviews comments and experiments that came up during the rebuttal.
Decision: Accept (poster)