Universal Multi-Domain Translation via Diffusion Routers
Authors: Duc Kieu
Kien Do
Tuan Hoang
Thao Minh Le
Tung Kieu
Dang Nguyen
Thin Nguyen
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Models
Multi-Domain Translation
Primary area: generative models

Abstract: Multi-domain translation (MDT) aims to learn translations between multiple domains, yet existing approaches either require fully aligned tuples or can only handle domain pairs seen in training, limiting their practicality and excluding many cross-domain mappings. We introduce universal MDT (UMDT), a generalization of MDT that seeks to translate between any pair of $K$ domains using only $K-1$ paired datasets with a central domain. To tackle this problem, we propose Diffusion Router (DR), a unified diffusion-based framework that models all central$\leftrightarrow$non-central translations with a single noise predictor conditioned on the source and target domain labels. DR enables indirect non-central translations by routing through the central domain. We further introduce a novel scalable learning strategy with a variational-bound objective and an efficient Tweedie refinement procedure to support direct non-central mappings. Through evaluation on three large-scale UMDT benchmarks, DR achieves state-of-the-art results for both indirect and direct translations, while lowering sampling cost and unlocking novel tasks such as sketch$\leftrightarrow$segmentation. These results establish DR as a scalable and versatile framework for universal translation across multiple domains.

Published as a conference paper at ICLR 2026
UNIVERSAL MULTI-DOMAIN TRANSLATION VIA DIFFUSION ROUTERS
Duc Kieu'* 1*
Kien Do
Tuan Hoang'
Thao Minh Le
Tung Kieu
Dang Nguyen!
Thin Nguyen!
Appliedd Artificial Intelligence Intiative, Deakin University
PPennsylvania State University AAalborg University
1 [v. kieu, do tuan.h, d.nguyen, thin nguyen dedeakin.edu.ar. 2mx1 622 24 @2p4s. edu "tungkvt@cs.aau. aau.dk
"Equal contribution.
ABSTRACT
Multi-domain translation (MDT) aims to learn translations between multiple do- mains, yet existing approaches either require fully aligned tuples or can only han- dle domain pairs seen in training. limiting their practicality and excluding many cross-domain mappings. We introduce universal MDT (UMDT), generaliza- tion of MDT that seeks translate between any pair of K domains using only K - 1 paired datasets with central domain. To tackle this problem, we pro- pose Diffusion Router (DR), unified diffusion-based framework that models all central--oo-ccentee translations with single noise predictor conditioned the source and target domain labels. DR enables indirect non-central translations by routing through the central domain. We further introduce novel scalable learning strategy with a variational- brruudd objective and an efficient Tweedie refinement procedure support direct non-central mappings. Through evaluation three large-scale UMDT henchmarks, DR achieves state-of-the-ar results for both in- direct and direct translations, while lowering sampling cost and unlocking novel tasks such as sketch---semeentaionn These results establish DR as scalable and versatile framework for universal translation across multiple domains. The code is available at httpss//ggtabbboom///mmeettttttt
INTRODUCTION
Paired domain translation, which aims to learn mapping between domains given aligned sam- ples, underpins wide range of applications. including image-to-image tramslation (Isola et al., 2017 Park et 2019) image captioning (Xu et al. 2015 Fang et al. 2015). and text-to-speech synthesis (van den Oord et al. 2016; Shen et al. 2018). Despite remarkable progress in two-domain settings, many real-world problems inherently involve multiple domains, motivating the study of Domai- Translation (MDT).
Existing MDT approaches usually fall into two paradigms: (i) training on fully aligned tuples across
domains (Wu& & Goodman) 2018 Shi et al.) 2019: Bao et al. 2023 Le et al. (202)). (ii) training on multiple paired datasets with shared central domain (Huang et al.) [022] /Zhang et al. 2023 Huang et al. 2023 Koley et al. 2024). The former quickly becomes impractical as the number of domains grows due to the difficulty collecting large-scale aligned tuples. The latter scales better but mainly supports translations between the central domain and each non-central domain, leaving cross non-central translations unaddressed.
In this paper, introduce Universal Multi- Dottiin Translation (UMDT), which combines the am- bition of enabling translations between any pairs of K domains with the practicality of only requiring K paired datasets involving central domain. UMDT captures many real-world scenarios, such

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as -mage--textt-auude translation multilingual machine translation, where fully aligned tuples are scarce but pairwise datasets with a pivot domain (e.g., text or English) are abundant.
To address UMDT, propose Diffusion Router (DR), novel diffusion-based framework that sup- ports arbitrary cross-domain translations with only single noise prediction network Ep. Inspired by network routers that determine paths using source and destination IP addresses, DR conditions <g on both source and target domain labels, guiding the denoising process along the correct trans- lation path. This design avoids training a separate model for each mapping, enabling scalability to large numbers of domains. DR can perform indirect translation between non-central domains via the central domain. To further enable direct non-central translations, we introduce scalable learning strategy that minimizes a variational upper bound on the KL divergence between indirect and parameterized direct mappings. This reduces to aligning two conditional noise predictions one conditioned on the source non-central-domaal sample and the other on its paired central-domain cumple-resembling the training objectives of diffusion models. improve efficiency, we develop Tweedie refinement, lightweight sampling procedure that approximates conditional samples in only a few steps, greatly reducing computational cost and facilitating scalable training.
Extensive experiments and ablation studies on three newly constructed large-scale UMDT datasets demonstrate that DR consistently outperforms state-of-the- GAN-, flow-, and diffusion based baselines for multi-domain translations, either indirect direct. Morcover, DR naturally generalize to more complex UMDT topologies such as spanning trees with multiple central domains.
In summary, main contributions are:
We formalize Universal Multi-Domain Translation (UMDT), general setting that aims to learn translations between any pairs of K domains using only 1 paired datasets.
We propose the Diffusion Router (DR), a unified diffusion-based framework that models all central--non-central mappings with a single noise predictor.
We develop scalable learning strategy with a variational-boun objective and Tweedie refinement to enable direct non-central translations.
We construct three new UMDT benchmarks and show that DR achieves state-of state-of-the-art results for both indirect and direct translations.
2 PRELIMINARIES
2.1 DIFFUSION MODELS
Diffusion models (Sohl- Dickstict et al., 2015 Song & Ermon 2019 Ho et al. 2020 Song et al.)
2021a) are a generative framework that generates data from noise by reversing a forward diffusion process. The forward process is typically Markov process with the transition kernel ( -r+-)) chosen such that the marginal distribution p (2)(ar) (with xo = x) is Gaussian distribution of the form N (f... This enables direct sampling of x4 from as:
Pt 0g + ott
Here, E ~ N N0, I); dt and or are predefined time-dependent coefficients satisfying ag > ar 0 and 0 =z of
< of x 1. At the final time step, p (xT 2)) p (IT) = (0.1).
has been shown that (Song et al. 202 the reverse transition kernel pe (#t-1 can be param- eterized as Gaussian distribution N (Per1--1 (zt) ( where the mean is defined as:
|0t..--1 (Ix) 4:-1
(Vo2 - 0204-1
(vt, ():
at
and the variance is wi_1lc = 1200 (1 of-1 df of
with 17 E [0, 1] (Song et al. 2021a).
af
The noise prediction network eg is trained to predict the noise e used to construct v, in the forward process (see Eq. [] using the following loss (Ho et al. 2020):
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Tranalation ?
Domain 3
Domain 3
Domain
Domain
traiing palls
traiiing pains
Trensbaton ?
Trensiation ?
Traniiaion
Transiation
Domain
Domain
Transsatioo
Transinton
Teembetto
Domain 2
Domain
(a) Conventional multi-domain translation
(b) Universal multi-domain translation
Figure 1: Illustration of conventional and universal multi-domain translation
(0) - Ez,c, ( (xz, 1)
=112]
(3)
where a ~ p (z)) 2 and /V (0, This objective can be interpreted variational
upper bound on - log (x) Sohl-Dickstein et al. 2015 Ho et al. 2020 Song et al. 20216).
While diffusion models were initially proposed model unconditional data distributions p (xx). they can be extended to conditional distributions p (zly) (Dhariwal & Nichol 2021 Ho & Sali-
mans 202T Rombach et al. 2022). by incorporating the condition y into the noise predictor, 1.c.,
60 (Te, 1, y) This conditioning can be implemented either by concatenating y with (Saharia et al.
2022). or through cross attention between 24 and y (Rombach et al., (002)).
UNIVERSAL MULTI-DOMAIN TRANSLATION
We study more general and challenging extension of conventional multi-domain translation (MDT) problem (Fig. [h), which we term universal multi-domate translation (UMDT) (Fig. [pp). In this setting, we consider K distinct domains, X X² ... X with training data consisting of K -1 paired datasets between each domain X and shared central domain X (where S S K):
DE.- {(**, n=1' for all k + The goal is to learn a model that can translate between any pair of domains (X* X²) for all + j. Since paired data between non-central domains are unavailable, we assume that samples from the central domain X share overlapping information across the paired datasets. This a mild assumption and typically satisfied in practice.
UMDT is highly practical. For example. in multi-modal translation across images, text, and audio, is often difficult to obtain large-scale datasets with fully aligned triplets (image, text, audio). How- ever, paired datasets such image-text (e.g., image captions) and text-audio (e.g., audiobooks) are more common. In this case, text naturally serve as the central domain, with image and audio as non- without
central domains. Image++audio translation can then be achieved indirectly via text, direct image-audio training pairs. Importantly, text samples across datasets need not match exactly; loose overlaps-sun such shared vocabulary semantics sufficient for iaudio translation.
More generally, UMDT can be extended to cases where the tree over the K domains. Since spanning trees take arbitrary structures, in this work restrict our attention to the star-shaped configuration. Nonetheless, the method proposed in the following section naturally generalizes to the broader spanning-tree setting.
paired datasets form a spanning
'1 (1, denotes uniform distribution of time steps hetween and T.
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4 DIFFUSION ROUTERS
4.1 INDIRECT TRANSLATION BETWEEN NON-CENTRAL DOMAINS VIA THE CENTRAL ONE
From a probabilistic perspective, bidirectional translation between two non-central domains X' and X can be viewed as sampling from the conditional distributions p (x*1x>) and p (x* (z'). These can be expressed via the central domain X as:
f (1d (1))2=)) (1 (1)) de,, p (1d((1))= f D (x"1x=) p (x dxf
(4) Here, we assume X 1 X1IX for all i.j + which leads to p (a+l)1)) = (at (00,,,)) x²) and p (x'12²) p (2* (xx, x²). In other words, once the central domain X is known, the non-central domains become conditionally independent of each other.
p (222
This formulation extends the transitivity property of equivalence relations into probabilistic frame- work. It implies that if we can learn bidirectional mappings between the central domain X and each non-central domain X (k: A e) that capture the couplings p (a* (x²) and p (a: (=**) from the training data (i.e.. solving the conventional MDT problem), then we can perform indirect translation between any pair (Xx,, ) through X even without direct supervision between X and X².
Notably, p ( and p (x91=>) can be effectively modeled using conditional diffusion models as discussed in Section or diffusion bridges (Liu et al. 2023a Li et al., [023 Zhou et al.) 2024] Kieu et al., 2025) (with details in Appdx. B. However, naively constructing a separate model for each distribution would require up 2(K - models to cover all translations between X and the non-central domains (X* + c). which becomes impractical as the number of domains grows.
To address this scalability challenge, we propose a unified framework called Diffusion Router (DR) that learns all bidirectional mappings between the central and non-central domains using a single network, thereby avoiding redundancy and enabling efficient multi-domain translation.
Inspired by network routers, which rely on source and destination IP addresses to determine the routing path, our framework incorporates the labels (or indices) of the source and target domains into the network eg. This design allows eq to infer the correct translation path for given noisy input Ie- Specifically, ep takes the form (z)" t, tgt, ste) where sre and tgt denote the source and target domain labels. We train DR with the following objective function:
Cpaired (0) Ee (0) Daccttik ( 1lea (xf 1, aaahhe k. =112 + (1 (1-C) llea (16 ,,,,,,,,))))) <
(5)
where 1 ~ (1, ~ N' (0,T), and ( B (0.5 Next, xt u + are for the standard (or diffusion-based) DR variant and x ag30 + Bex. + are for the bridge-based DR variant.
Once trained, DR can translate between two non-central domains X and X indirectly via X using a two-stage process: 1) generate central-domain samples zf conditioned on source sample x² (or 2d), then 2) generate target samples x (or x²) conditioned on the intermediate samples x
4.2 DIRECT TRANSLATION BETWEEN NON-CENTRAL DOMAINS
Indirect translation requires generating intermediate samples r: of the central domain, which is com- putationally expensive and sensitive to sample quality. To overcome these drawbacks, we propose a novel approach that enables direct translation between non-central domains X and X by explicitly modeling pe (x (x=|==)) and pe [ar). ). Our method can either finetune DR pretrained with Eq. 5 or train a new DR from scratch for direct cross-domain translation.
To learn pe (xd [xr) (or similarly pe (x*)<))), we minimize the following KL divergence:
Ep(ay)) [DEL [p
(x') lipe (<r [e(l=))]]
Es .ar (rD) E (p((z))e) [xt) [log (Ep(z)quz) [p (z'(x"")) - log pe [<)]
(6)
2B (0.5) denotes Bernoulli distribution with the probability of getting equal 0.5.
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where x/ denotes samples of X that are distinct from x², The detailed derivation of Eq.
provided in Appdx. A.1 The main bottleneck here is the term log (E (Ep(zr)zz)) [p (a> a First,
sampling from (2 0 typically requires hundreds to thousands of denoising steps if using a
pretrained DR. Second, even if we obtain samples x' from (x'((c") evaluating p (ar (r') remains intractable due to the lack of a flosed-form expression.
To overcome this, we approximate (p(z)(zz)) [p (x"|=""] by p (> [z7) where x" comes from the pair (x*, v²) Di,e. This leads the following tractable training objective:
(-) 16 pp(a) (ut) [log (Ep())))))) [p (x²)x"))
log (<'[x')]
(( Di. pp (as) [log p (a² (z*) Po (x*(x")) [x']]
((rt. ~De. [Dxr. (p (</|=²") lipo [x)]
(8)
Although this approximation introduces bias due to the logarithm operating on (single-sample) Monte Carlo estimate, empirically observe that its impact on learning is manageable, particularly when the conditional distribution (xx" (x'|x²) is sharply peaked at its mode.
The objective in Eq. 8 suggests that we can learn a direct mapping from X to X by ensuring that a and x are semantically aligned (i.e., appear as pair in D.,.), then the conditional distribution over the target domain X given x² should closely match that given x². Note that p (a (r) can be viewed as the path distribution of the stochastic process X - X modeled using pretrained DR introduced in Section 4.1 Likewise, pe (>²|x²) represents the path distribution for the process X X2, which we also model using the same DR. This is implemented by providing i, j as inputs to the noise prediction network eg of the DR. As a result, learning po (ad (x²) corresponds to finetuning this pretrained DR to support direct mapping from X to X3.
Let peer (x+) denote the parameterization of p (x=)/x:)) given by the pretrained DR. Instead of minimizing the KL. divergence between Pret (ar (x²) and po (x*(>>" directly, we minimize the sum of KL divergences between their respective transition kernels. This sum acts as a variational upper bound on the original KL objective, as shown in Appdx. A..2
E E(rt,
DrL (Pter (x=") I|pe (x)))
T
)) const (9)
< )
E (o((lel Σ
[DKL (Pres (zl ( 20) llpo (2/_1ar)) x0
Ideally, the sampling distribution in Eq. should be Peet (xflx²) so that after training, pe (al-al x²) can accurately model the transition dynamics from X' to X3, However, the actual sampling distribution is Prer (x:1=>) l For baddee-based DR, the stochastic processes X + X and X X starts from different initial states x' and xf, respectively, making their path distribu- tions, Pref (u2(lze) and prer (ar la*)) inherently different. Therefore, bridge-based DR are ill-suited for finetuning with the objective in Eq.
9 By contrast, standard DR provide more viable solution. Since both X + X and X + X2 originate from the same Gaussian prior, they share a com- mon stochastic path, allowing Pref (x/(2²) to serve as proxy for Pee (x/]="). For this reason, restrict our focus to finetuning standard DR.
By applying the standard reparameterization trick for diffusion models (Ho et al. 2020), we refor- mulate Eq. as noise prediction loss:
Cunpaired (0) E (rr an paa (e) (Flit, Gotl t2/, J.
(10)
where Get is the frozen noise prediction network of the pretrained DR.
To prevent catastrophic forgetting of previously learned mappings between X and X* we combine Eq. 10 with Lpaired (see Eq. 5) resulting in the final loss:
Linaal (8) Xx Lumpurred (0) + A puiired (0)
(11)
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(or) :) Dje (,,2)) ~ Dio
Pret(ag a*))
Refined noisy semple (2,(n)
Figure 2: Tweedie refinement with {0, 1, 3,5,,,,, 5, 7 Faces-UMDT-Latemn Left: A conditional sample x" and random target-domain sample x². Middle: A ground-truth noisy target-domain
sample z aligned with x" (not available during training). Right: Tweedie refinement progressively transforms x². ~ [ into x² ~ (of (x)(x²) as n increases.
Here, the coefficients À₁ A 2 0 balance the trade-off between learning new translations X X and preserving existing mappings X ++ X
Notably, Limal (0) is highly flexible. It can train DR from scratch by treating <ref an online network with frozen parameters rather than pretrained model (Appdx. D. Moreover, it supports in the case where paired domains form a spanning tree with multiple central domains (see Section 5).
4.2.1 SAMPLING FROM CONDITIONAL DISTRIBUTIONS WITH TWEEDIE REFINEMENT
The main challenge Eq. 10 is sampling zf from Peot [x] (ella)). A straightforward approach is to perform backward denoising from time T to using the pretrained DR, but this computationally expensive and does not scale well. To address this, we propose a novel sampling method:
(((++1 = (m,(0)) o (E- (Z) x² 2.,c))
(12)
where (0, and (n))n) denotes the refined sample after n steps, initialized with ((,(0)
Pret x
A sample x.. ~ Prel x can be obtained by first drawing (x), from Dj.e and then
applying the forward diffusion process x.. a, ogc.
We refer to this procedure as Tweedie refinement due to its connection with Tweedie's 's formula find that Tweedie refinement
approximate samples from l with only few refinement steps (see Fig, 2). Compared with existing refinement tech- niques (Song et al. 2021b Yu et al. 2023), our approach (1) introduces a distinct formulation, (2) converts unconditional samples into conditional ones rather than projecting off-distribution samples back onto a marginal distribution, and (3) is applied during training rather than inference.
Efron
201T). Empirically,
EXPERIMENT
EXPERIMENTAL SETUP
5.1.1 DATASETS
Since the proposed UMDT problem is novel, datasets currently exist for it. To address this, create three benchmark datasets for evaluating our method, namely Shoes-UMDT, Faces-UMDTT, and COCO-UMDT. Detailed descriptions of these datasets are provided below.
UMDT This dataset is adapted from the Edges2Shoes dataset (Isola et al. 2017) used for paired image-to-image translation. From the original 50K (shoe, edge) pairs, we randomly sample two disjoint subsets of 20K pairs each. One subset remains unchanged, while in the other, the "edge" image in each pair is replaced with a "grayscale" version the corresponding "shoe" image, rotated by 20 degrees and scaled by 20% smaller. This yields two disjoint sets: 20K (shoe, edge) pairs and 20K (shoe, grayscale) pairs. this setup, "shoe" the central domain, while "edge" and "grayscale" non-central domains. The remaining 10K (shoe, edge) pairs used to generate

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FID
Method
Shoes-UMDT
Edget+>Shoe
Shoee
Edge+>Grayy
Ske.t>
Seg. Pace
Ske.t>Seg.
StarGAN
9.92/20.18
19.73/42.61
18.64/27.41
Rectified Flow 2.88/30.92
3.75/43.38
20.14/18.83
20.22/97.76
10.85/81.44
50.82/17.31
UniDiffuser
2.98/11.94
2.72/4.40
4.81/12.26
13.13/55.46
11.02/46.04
36.13/12.52
iDR
1.66/5.15
0.53/1.60
1.85/5,48
9.07/23.88
6.12/19.12
15.37/6.15
dDR
2.01/5.76
0.57/1.69
2.74/6.51
9.62/27.09
3,43/21.26
19.42/5.52
Table 1: FID scores Shoes- and Faces-UMDT-Latent. Translations without paired data are marked in brown. The best results are shown in bold, and the second-best are underlined.
begee
(a) Shoes-UMDT
(b) Mioes-UMDTT-Ltemn
Figure 3: Qualitative results Shoes-UMDT and Faces-UMDDT-Latenl)
grayscale images following the same procedure as in the training data, producing a test set of 10K
(shoe, edge, grayscale) triplets. All images are resized to 64x64x3.
Faces-UMDT We build Faces-UMDT by combining CelebA-Mask- (Lee et 2020 (30K
(face, segment) pairs) and FFHQ (Karras et al. 2019) (70K face images). From FFHQ, we randomly select 30K face images and use Sobel filter followed by sketch simplifier (Simo-Serra et 2018) to generate corresponding sketches, producing 30K (face, sketch) pairs. These are merged with 25K randomly selected (face, segment) pairs from CelebA-Mask-HHt to form the training set in which "face" is the central domain and "segment", "sketch" are non-central domains. For testing, generate a sketch for each face image in the remaining 5K CelebA-Mask-HHC pairs, resulting in 5K (face, segment, sketch) triplets.
This setup reflects real-world scenarios where the two disjoint subsets of face images associated with CelebA-Mask. HQ and FFHQ follow distinct distributions, making sketchs+segment translation via the face domain more challenging. For this dataset, we consider two settings: 1) resizing input images 128 128x3 and translating in the pixel space, and 2) encoding 256 x 256x256x3 images using a VAE encoder (Rombach et al.) 2022) and translating in the latent space of shape 32x32xx x4. These settings result in two versions which are Paces-UMDT-Piigg and Face-UMDD-LLtentl respectively.
COCO-UMDT-Star and COCO-UMDT-Chaiin COCO-Stuff (Caesar et al. 2018 a large and diverse image segmentation dataset with 118K (color, segment) pairs for training and 5K pairs for evaluation, covering 80 "thing" classes, 91 "stuff" classes, and one "unlabeled" class. Following (Mou et al. 2024), we generate additional domains by applying the Pixel Difference Network (Su jet al.) (021) to extract sketches and MiDaS (Ranftl et al.) (020)) to produce depth maps from all color images. For COCO-UMIDTSStar, construct three training subsets of 70K color images each,

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FID
Method
Skol++Color
Seg. Seolor
Depth++Coloo
Skc.++Seg.
Ske.++Depth
Segt++Depth
Rectified Flow
23.18/80.80
54,00/142.15
17.32/112.64
64.47/75.58
78.41/28.69
79.20/35.53
UniDiffuser
15.39/40.93
35.81/89.58
12.64/59.72
39.62/38.44
28.12/15.72
38.39/23.41
iDR
10.72/21.73
21.64/29:28
7.25/24.19
22.77/22.96
17.88/8.63
23.19/12.00
dDR
10.12/20.94
21.23/28.32
7.00/23.20
26.73/23.64
20.75/9.42
24.91/14.87
Table 2: FID scores COCO-UMDT-Star Translations without paired data are marked in brown. The best results are shown in bold, and the second-best are underlined.
FIDL
Method
Skelo+Color
Seg. Seg.e+Color
Depthe-SColor
Skc.4+Seg.
Ske.++Depth
Seg.++Depth
Rectified Flow
22.33/85.03
57.53/148.57
27.49/146.27
65.18/79.91
68.27/21.02
94.55/42.40
UniDiffuser
15.43/45.51
36.89/92.38
14.40/68.64
39.97/38.70
26.49/12.40
39.85/24.51
iDR
10.47/26.73
23.14/38.60
8,55/39.66
26.52/25.02
14.73/7.54
26.08/14.15
dDR
11.11/28.39
24.68/38.47
10.69/44.23
28.13/26.94
14.92/7.82
30.98/18.61
Table FID scores on COCO-UMDT-Chain. Translations without paired data are marked brown. The best results are shown in bold, and the second-bes underlined.
paired with segmentation maps, sketches, and depth maps, respectively, yielding (color, segment), (color, sketch), and (color, depth) training pairs. For DOCO-UMDT-ChainD we form three paired subsets: (segment, color), (color, sketch), and (sketch, depth), each with 70K pairs. Both COCO- UMDT-Star and COCO-UMDT-CCMEE
share a common test set of 5K (color, segment, sketch, depth) quadruplets derived from the original evaluation split. All images are resized to 256x256x3 and mapped into latent space of size 32x32x4 using VAE encoder from (Rombach et al. 2022
5.1.2 BASELINES AND METRICS
We consider two versions of DR. The first is trained with the loss Lparred (0) in Eq. 5 which can only perform indirect translations between non-central domains. The second is finetuned from the first using the loss Lnal (9) in Eq. 11] allowing direct cross-domain translation. We refer these two versions as iDR and dDR, respectively. We also train version from scratch using Crinal (0) and compare with the finetuned version in Appdx. D.3.3
As with datasets, methods for addressing the UMDT problem remain largely unexplored. To estab- lish baselines, we adapt several approaches originally designed for the conventional MDT setting. These include StarGAN (Choi et al., 2018), UniDiffuser (Bao et al. (023). and Rectified Flow (Liu) al. 2022) as representatives for GAN-based, diffusion-based, and flow-based methods. Further implementation details for both our method and the baselines are provided in Appdx. D.1
Following previous works on MDT, we use FID (Heusel et al., (017) and LPIPS (Zhang al. 2018) to measure distributional fidelity and perceptual similarity, respectively. In all tables, each entry for translations A reported as X/Y, where X is the metric for A<BB and Y is the metric for A-B.
5.2 RESULTS
We report the quantitative results for Shoes-UMDT and Faces-UMDTT-Ltentt in Table for COCO- UMDT-Star in Table [ and for UMDT-Pixel in Table of Appdx [.2.2] iDR consistently outperforms all baselines by significant margin across all benchmarks, highlighting the effective- ness of conditioning 6o on both source and target domain labels to guide translation. The weak performance of StarGAN in central<-non--centee main factors: (i) its relatively outdated generator architecture, and (ii) its original design for unpaired rather than paired translation. Despite being trained on aligned (z x*) pairs, Rectified Flow (RF) and UniDiffuser struggle to learn robust X
translations (Xe <> X²) can be attributed to two
X mappings. For RF, performance degrades most
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noticeably when the target domain is diverse and high- variance (e.g.. Edge->Sher Seg. Face), as its deterministic formulation cannot capture the stochasticity of p (x*1/=)) and (x"1=") For UniDiffuser, the repeated substitution of missing domains with Gaussian noise during training and inference likely undermines its ability to accurately modeling the joint distribution over all domains.
dDR shows slight decreases in performance for translation tasks without paired data compared to iDR, yet still outperforms all baselines significantly. The performance drop can be attributed to imperfections in our refinement procedure and to bias introduced by the Monte Carlo approximation inside the logarithm in Eq. 7 Importantl, dDR reduces the number of sampling steps for non- central translations by half relative iDR a substantial efficiency gain, given that iDR typically requires hundreds to thousands of steps for cross-domain translation. As shown in Fig. 3 iDR and dDR produce higher-quality samples compared to the baselines.
in Table 3 compared to COCO- UMDT-Star: dDR improves efficiency by 2-3 times for translations without paired data while ineur- ring only marginal performance degradation compared to iDR. This empirically validate the gener- alizability of our learning strategy for UMDT beyond star-shaped structures.
We further observe consistent behavior on
5.3 ABLATION STUDIES
Owing space limitations, conduct comprehensive ablations to quantify the effect of key hy- perparameter choices and report at Appdx. [.3] Specifically, we (i) evaluate Tweedie refinement by varying the number of refinement steps n {0, 1, 3, 5) (Appdx. D.3.I): : (ii) study the effect of the rehearsal coefficient X₂ E {0, 0.3. 1, in the loss Limal during finetuning (Appdx. D.3.2); compare dDR trained from scratch versus finetuned from a pretrained iDR (Appdx. :: (iv) compare the
performances of iDR and dDR across different numbers of sampling steps (Appdx. D.3.4); and (v) benchmark standard DR against several bridge-based DR variants on UMDT (Appdx. D.3.5).
RELATED WORK
Due to space constraints, we primarily review diffusion-based approaches for multi-domain transla- tion (MDT), with other methods discussed in Appdx. [ Building the success of Stable Diffusion (SD) (Rombach et al. 2022) in text-to-image generation, many works adapt SD to MDT by finetun- ing it to condition on additional modalities such as edges, segmentation maps. depth, or poses (Zhang
cet al. 2023 Huang et al. 2023 Mou et al. (2024). Versatile Diffusion (Xu et al.) (2023) extends this idea with a modular cross- modal architecture, but its model size scales linearly with the number of domains and experiments are limited to text and image. CoDi (Tang et al. (023) adopts two-stage framework: first, moodlltty-specitn
encoders are trained to align samples into shared latent space; second, separate latent diffusion models are trained for each modality conditioned on encoder out- puts and noisy paired samples. While this supports any-to-any generation, it requires contrastive pretraining and multiple diffusion models that scale linearly with domain count. UniDiffuser (Bao) ct al. 2023) instead models the joint distribution of all domains using a single transformer- based noise prediction network that treats domain samples tokens, with independent noise schedules per domain. This design supports arbitrary modalities but requires fully aligned tuples across do- mains and long training times. Moreover, UniDiffuser must process all domains jointly, making generation costly when only subset of domains is desired. One Diffusion (Le et al. (025) follows a similar design but replaces the noise prediction network with a velocity network trained via flow matching (Liu et al.) 2022) and introduces different transformer architecture (Zhuo et al., 2024). Like UniDiffuser, depends on fully aligned tuples where samples of missing modalities are syn- thesized. OmniFlow (Li et al., (025) is another flow-based model for multi-modal generation like One-Diffusion. However, instead of training from scratch, it extends from SD3 (Esser et al.) 2024).
In summary, most diffusion-based MDT methods either rely on contrastive learning or synthetic fully aligned data, suffer from linear growth in model size, or require processing all domains si- multaneously. By contrast, our method trains directly from domain pairs, avoids model-size scaling with domain count, and flexibly adjusts sampling cost to the desired number of output domains. Although not the primary focus of this work, Diffusion Routers also enable generation of a single target domain from multiple source domains by combining the scores of models conditioned on these sources. This capability makes method adaptable for any-to-any generation.

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7 CONCLUSION
We introduced the universal multi-domain translation (UMDT) problem, which seeks to learn map- pings between any pair of K domains using only K --1 paired datasets with central domain. The main challenge lies in learning non-sentral+-non-ceme are unavailable. To tackle this, we proposed Diffusion Router (DR), which supports both indirect NNTs through the central domain and direct NNTs. The direct variant can be obtained either by fine- tuning from the indirect version or by training from scratch. We introduced novel vuriational-boum objective and conditional sampling method for learning the direct variant. Empirical evaluations on three newly constructed UMDT datasets demonstrated that our method consistently outperforms existing baselines. In future work, aim to extend DR to large-scale multimodal generation across image, text, and audio, particularly addressing scenarios where paired datasets (e.g., image++audio) scarce in practice.
translations (NNTs), where training pairs
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LLM USAGE
Large Language Models were not involved in the design of our approach. We used them solely to improve the manuscript's readability (grammar and style); none of these uses influenced the method, training, or reported results.
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A THEORETICAL RESULTS
A. DERIVATION OF EQ. 6
The detailed derivation of Eq. i given below:
Ex(ay) [DE (p (x)) Ippo (xr [x))]]
(at)) [s (cr))o)) (logp (o0)<z") log pa (o (x')) dod dx²
(13)
-s p (a²) (f (202l ( (27)2²) dor) (log (x")x²) log pe (xd) xx)) dxl dæ
(14)
J / (xx) (x (x2)x²) (log (x")xx") log (x² (ellx²)) dx da"
(15)
J p (at, 29) (at,
/ (z0) (logp (x')ax") - log po (1²1x²)) dx da.
(16)
Eppai
and aas ( (ppaailr))) [log (x')x")) po [x*)] Eptat an Bpla) (nn) [log (p(zelatt (p (x')=²")) log pa (<'1x")
(17)
(18)
Note that in derivation, p (ar (xt) is substituted with its expression in described in Eq.
A.2 DERIVATION OF THE VARIATIONAL. UPPER BOUND EQ.
The detailed derivation of the variational upper bound in Eq. 9 is as follows:
[D& (Pres (x²1Z²) (x-) llpo (x']="))]
< (at wa Dice
DEL Pree (xdrlle))
llpo (#bzrl)))) )]
(19)
Pref ( (o
Prof (Bdrllee) log
(e
(20)
Po ( 100
I Prer ( lio
= (at a
Pret (2durla")
log
dro.r
(21)
p x loo
IT Po (4 z0))
T
Prer
-aa ae)
E Pir (nr are)
Σ
log
const
(22)
po
Pref 14-1 x
= (a
Σ
E
narl (2)
TE Bona (
log
const
(23)
x lar xrd po
-E(at, Diie
Σ
E Pal (2/(2)) Dret. (prer (wl
Ilpo
const
(24)
Here, the inequality in Eq. 19 is a variant the data processing inequality.
B ADDITIONAL PRELIMINARIES
B.1 DIFFUSION BRIDGES
Diffusion bridges (Liu et al. 2023a| Albergo et al. [023] Liu al. 2023b Li al. 2023 Zhou et al.)
2024 Kieu et al. 2025 Nguyen et al. (025) offer an alternative approach to modeling conditional
distributions (z ()) where the generation process starts from the observation y rather than from
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standard Gaussian noise, as in traditional diffusion models. Specifically, the compption process is characterized by the marginal distribution p (z) xr y). representing a stochastic trajectory between endpoints x and y- This distribution is assumed to Gaussian: p (2g [z. y) = N (orr Bry, of). which allows za be directly sampled from a training pair (xr, 3) using:
Ft 0F + BLy + ove,
(25)
where c ~ I0, at. 3t. di are time-dependent coefficients satisfying boundary conditions: ag 3T and oT Bo 0o or = 0.
The transition distribution pa (+t-1 |xt, v) for generating æ from y (with DT = 3) is modeled as N [0 (It, 62 of
where the mean is given by (Kieu et al.
2025):
0r-1 Et + B/-11 Brat-1) 04
Gt-1
07
6 6-
0,00-1
10,,1-1 (2):
y +
eo (1t, 1,3)
an
Ot
at
(26)
Here, 01-1/c
Oz) controls the sampling variance and is typically defined as 8z-1|c
V o of
a² with 7j E [0, 1] in the case of Brownian bridges (Li et al. 2023).
af
The noise prediction network in Eq. 26 trained by minimizing the noise matching loss:
L (0) = (( [Be.
[lea (Iz, (, - -<12]]
(27)
where (I, ~ (x, (), 1 u (1,
I0,
Bidirectional diffusion bridges (Kieu et al. 2025 extend this framework to jointly model both p (xly) and (y|z) using a single shared noise prediction network.
C ADDITIONAL RELATED WORK
Multi-domain translation (MDT) methods can be classified according to the underlying generative models. This section focuses reviewing traditional VAE-based and LAN-based approaches.
C.I VAE-BASED METHODS
VAE-based methods generally aim to learn latent representations that facilitate translation across domains or modalities. JMVAE (Suzuki et al., (2066) captures shared representations with a joint encoder go (z x x²,x²" and handles missing modalities at test time by aligning unimodal encoders go | and go (z x.2) with the joint encoder through KI. divergence minimization. TELBO (Vedantam et al. 2018) adopts a similar encoder design but differs in its training strategy. Instead of jointly optimizing all encoders, it first trains the joint encoder and then fits the unimodal encoders while keeping the joint encoder's parameters fixed. MFM (Tsai et al.) 2019) factorizes the mul- timodal latent space into shared discriminative and modality-peecifi generative factors, enabling inference of missing modalities at test time from observed modalities. While these methods are sufficient for two modalities, they do not generalize to the truly multi-modal case.
More recently. MVAE (Wu & Goodman 2018) innovatively factorize the joint encoder into go (= x go (z x²)) p notable advance that scales to multiple domains by training with randomly masked modalities and seamless in- ference when any modality missing at test time. Alternately, MMVAE (Shi et al.) 2019) con- structs the joint encoder mixture of experts (MoE) of unimodal encoders, alleviating the preci- sion-miscen miscalibration issues inherent to PoE. Despite strong results in conventional MDT, both MVAE and MMVAE assume fully aligned tuples across domains, requirement rarely satisfied in practice. In contrast, our method trains directly on domain pairs, removing the need for fully aligned triplets and improving practicality in real-world settings.
product of experts (PoE), i.e., do | x xx."
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LPIPS
Method
Shoes-UMDT
Faces-UMDD-LLa UMDTT-Ltent
Shge
Gray. Shoee
Edge++Graay
Ske.+>Face
Seg. Pace
Skg.
StarGAN
0.128/0.222
0.095/0.214
0.191/0. 144
Rectified Flow
0.063/0.17 175
0.012/0. 146
0,138/0.083
0.278/0.560
0.165/0.548
0,419/0.274
Unidiffuser
0.066/0.170
0.019/0.091
0.090/0.091
0.263/0.483
0.165/0.511
0.393/0.223
iDR dDR
0.050/0.129
0.003/0.069
0.069/0.058
0.221/0.427
0.129/0.471
0.377/0.177
0.050/0.128
0.004/0.063
0.077/0.090
0.222/0.428
0. 0.126/0.466
0.392/0.192
Table 4: LPIPS scores of our method and baselines on Shoes-UMITT and Faces-UMDDT-Lanee
Translations without paired data are marked in brown. The best results are shown in bold, and the second-best are underlined.
C.2 GAN- GAN-BASED METHODS
Another line of research uses GAN (Goodfellow al.) 2014) for MDT. Pix2pix (Isola et al. 2017 uses paired datasets to train a conditional GAN with a reconstruction loss, aligning outputs with ground-truth targets while encouraging realism. CoGAN (Liu & Tuzel 2016) instead learns a joint distribution across domains by sharing weights in high-level layers, enabling related samples across domains without aligned pairs. However, weight sharing constrains model design and limits scal- ability to high-resolution images. A popular approach for MDT is to leverage sycle-consistency constraint when training cross-domain translators (Zhu et al. Kim al. 2017 Yi et al.] 2017] Liu et al. 2017), enforcing meaningful relation between the input and the translated image. Never- theless, these GAN-based methods do not scale gracefully with more domains, leading to higher computational cost.
Notably, StarGAN (Choi et al. 2018) introduces unified GAN framework for MDT that uses a single generator and discriminator with auxiliary domain classifier during training. Its successor, StarGAN-2 (Choi et al., 2020), injects continuous style codes via AdaIN (Huang & Belongie (017) to deliver diverse outputs. Like StarGAN, UFDN (Liu et al., 2018) also ofters a unified MDT model and and extends the framework to disentangle dovain-invariant features. More recently, MultimodalGAN (Zhu al. 2024) proposes MDT framework trained on fully aligned tuples across domains.
Our work related StarGAN in that both introduce frameworks for MDT, but differs in pervision and translation. We exploit paired datasets for training and perform indirect translations between domains lacking training pairs. In contrast, StarGAN treats such translations as an un- paired translations. We empirically show that leveraging paired supervision leads to more faithful input-output mappings and superior performance.
D ADDITIONAL EXPERIMENT
D. ADDITIONAL IMPLEMENTATION DETAILS
D.1.1 DIFFUSION ROUTER
We adopt the standard Diffusion Router variant with DDPM (Ho et al. 2020) as the underlying diffusion process. comparison with the bridge-based variant is provided Appdx. D.3.5| The
noise prediction network has a U-Net architecture following (Dhariwal & Nichol 2021). For
Shoes-UMDT (64 x4) and Faces-UMITT-Pice ( (128x) 128), 60 operates directly on raw images with 128 and 256 base channels, respectively. For UMDT-Chain, the 256 x 256 images are first encoded into 32x32x4 latents using a pretrained VAE Rombach et al. 2022), which are then processed by ea with 128 base channels. To ensure efficient training within single H100-80GB GPU, we use a batch size of 128 for the 128-channel U-Net variant and 32 for the 256-channel variant. iDR is trained using AdamW (Loshchilov & Hutter, 2019) with a learning rate of 1c-4, 3 = 3₂ = 0.9 and 3000 warm-up steps across all datasets. The number of training steps is 250k for Shoes-UMDD' and Faces-UMDDi and 500k for COCO-UMDT.
COCO-UMDT-SEE and COCO-
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Ske. - Face
Seg. - Face
Seg. - Sko.
Ske. Seg.
UNION
Figure 4: Qualitative results of our method and baselines on Brres-UUMT-PPiEEE
Training takes roughly three to four days. dDR is finetuned from pretrained iDR with reduced learning rate of 5e-5, loss coefficients Å = 1 and 2 = 1, and 5 refinement steps. The number of finetuning steps is 100k for Choes-UMDDY and Faces-UMDT, and 150k for COCO-UMDT. We run 2021a) with 1000 steps to translate from one domain to another.
DDIM (Song et al.
D.1.2 BASELINES
We implement StarGAN using its official repository While more recent GAN-based approaches for MDT exist, such as MultimodalGAN (Zhu et al. (022). we exclude them from our baselines due to the lack of publicly available code and instead discuss them Related Work.
UniDiffuser (Bao et al., 2023) is designed to model noisy samples across all domains at random noise levels. Since our training data provide only domain pairs rather than full tuples, we substi- tute missing domains with Gaussian noise and set the corresponding domain-specific time step to 999 during training. Rectified Flow (Liu et al. 2022), which natively supports only two-domain translation with each domain boundary distribution, is adapted to the multi-domain setting by conditioning the velocity network on both source and target domain labels, similar to our approach. For faireess, implement UniDiffuser and Recified Flow using the same architecture and settings as our method.
D.2 ADDITIONAL RESULTS
D.2.1 SHOES-UMDT AND FACES-UMDT-LATENS
ssows the LPIPS scores for our method the baselines. Overall, iDR and dDR consistenly outperform the baselines across all translation tasks. On tasks with paired supervision, both methods achieve comparable results. For translations without paired data, dDR exhibits a slight performance drop relative to iDR, but supports direct non-central translations. These observations align with the FID results presented in the main text.
D.2.2 FACES-UMDT-PIXXE
For completeness, we compare iDR and dDR against baselines on Faces-UMDDT-Pixcd (see Table 5 The results from Table mirror the trends in our main experiments at Section with iDR consis- tently surpassing all baselines and showing the largest gains on high uncertainty translations (e.g., Ske. and Seg. FFace, For Rectified Flow (RF), we observe poor generalization the test set, particularly on arises because training uses FFHQ for Ske.-Facc while testing use CelebA-HQ for Ske.->aace, causing a mismatch between the training and test distributions and encourging model's generaliza- tion, where RF fails.
yielding underperforming results. This underperformance likely
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FID4
LPIPS|
Method
Ske.t+Pace
S-acee
Ske.t+Seg.
Ske.e+ Face
Seg.
Ske.t>Seg.
StarGAN
38.16/91.61
32.98/75.41
50.17/76.40
0.280/0.44 479
0.282/0.471
0.408/0.364
Rectified Flow
9.88/261.96
19.54/122.87
45.01/109.37
0.160/0.63
0.137/0.464
0.362/0.398
Unidiftuser
18.49/36.99
14.03/25.36
36.89/15.79
0.181/0.548
0. 0.125/0.442
0.367/0.191
iDR dDR
9.25/13.32
4.12/10.54
13.51/3.88
0.159/0.329
0.101/0.412
0.361/0.146
9.02/12.81
3,57/12.88
28.46/3.91
0.139/0.347
0.106/0.421
0.422/0.167
Table 5: Results on Lices-UMDT-Pixel of our method and baselines. Translations without paired data are marked in brown. The best results are shown in bold, and the second-best are underlined.
In comparison, dDR performs comparably to iDR on most translations while maintaining clear margin over the baselines. We attribute dDR's lower results on Seg. to FID's sensitivity for sketches: background differences and minor detail changes can depress the score even when outputs are visually similar to the targets (see Fig. 4). These results empirically indicate that iDR and dDR can generalize to high-resolution data.
D.3 ABLATION STUDIES
Unless otherwise specified, ablation studies are conducted on Laces-UMDDT-Lteent to evaluate the impact of different design choices. To reduce both training and sampling costs, we substitute large U-Net used in the main experiments (304.8M parameters) with smaller variant (32.3M pa- rameters).
FID
FID
A
Skc.e-Face
Seg.t+Face
Skc.e+Sog.
Skc.esFace
Seg.s>Face
Ske.e>Seg.
1DR
14.21/39.06
10.18/24.95
20.82/10.85
IDR
14.21/39.06
10.18/24.95
20.82/10.85
16.11/39.82 82
13.75/25.77
55.30/13.35
405. 12/30666
355.78/307.15
403.19/528.75
14.14/40.61
11.77/23.65
37.45/13.13
0.3
19.44/45.94
17.01/30.90
52.31/13.18
dDR
13.73/41.14
11.58/25.26
26.77/10.96
dDR
16. 16.39/99.82
13:75/25.77
55.30/13.35
13.52/37.26
11.52/22.73
26.27/11.37
3
14.47/39.38
10.19/25.50
82.77/13.23
Table 6: FID scores of dDR finetuned
Table 7: FID scores of dDR finetuned on Faces-
Faces-UMDT-Lattes w.r.t. different number of refinement steps n.
UMDT-Latent w.r.t different values of Az. Tweedie refinement is not applied in this case (i.e, 0).
D.3.1 IMPACT OF THE NUMBER OF REFINEMENT STEPS
We analyze the effect of the number of refinement steps in the proposed Tweedie refinement (Eq. 12 by varying n E (,,,, with results summarized in Table 6 When (i.c.,
0045
Corpoved
0,25
30
0835
40
60
100
20
60
10
100
Figure 5: Learning curves of finetuned dDR on Face-UMDT-Latent w.r.t. different number of Tweedie refinement steps E 5, The task is Segment--Settch translation.
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Saurcce
Iaggt
m=0
Figure 6: Comparison of translated images from Segment to Sketch w.r.t different value of m2
3072
35
3977
0.002
3.70
0.07
0003
0062
Enstuning tanatore (x1000)
Pieetnin Io uetoom (=1000)
Aisturing heratoon (x1000)
FID
pairedd on E [0, 250) (Ske.-Face)
Lumpuired (Ske.+Segg
Figure 7: Learning curves of dDR finetuned with different values of Xg.
no refinement), dDR performs poorly on direct translations between non-central domains (e.g., Sketch<-Segmennt, though it remains comparable to iDR on translations involving central domains (e.g., Sketch<-Face, Segment<>Face). The failure is most pronounced for the Segment-rSketch translation task, where the target domain has limited detail and the FID scores are highly sensitive to minor variations. This indicates that without refinement, dDR struggles to capture the correct mapping from Segment to Sketch. Increasing n consistently improves translation quality across all settings, particularly in unpaired translations, reflected in the performance curves in Fig. 5 and the qualitative results in Fig. [ The improvement arises because the refined noisy sample
(,(()) provides closer approximation to samples from Prel
[x:] (r/]1=)) thereby yielding more aceu-
rate predictions from Gee (zt, x. approximates the variational bound in Eq. 9
(Eq. 10p. Consequently, the unpaired loss Compaired better
D.3.2 IMPACT OF THE COBFFICIENT A2 IN THE LOSS LANAL
We investigate the effect of the coefficient A2 by experimenting with different values in (0, 0.3, 1, As shown in Table 7 setting À 0 causes the FID scores for all translation tasks to diverge. This happens because the unforgetting term Gurred is discarded from Limat. causing the finetuned dDR to forget previously learned centrals--onn-eentaa mappings (e.g.. Sketch<+Face and Segment--Face). Consequently, non-ceetrall-moon-cmme
translations such as Sketch are also learned in- correctly, since training them depends on noisy samples from the central--non-centrr mappings. When A2 > 0, the fine-tuned model must balance preserving old translations with learning new ones. Increasing X2 improves the FID scores of the finetuned dDR on non-centrall-non-cooe translations, enabling them to match those of the pretrained iDR, but at the cost of degrading per- formance on the centrale--onnceenral translations. This trade-off is more clearly reflected in the learning curves in Fig. 7 Empirically, we found that X2 provides the best balance across translation tasks.
D.3.3 IMPACT OF TRAINING FROM SCRATCH WITH FEINAL
We study the effect of training dDR from scratch versus fine-tuning it from a pretrained iDR, both strategies using roughly the same loss Linal in Eq. [] In this experiment, we set n = 3. Training from scratch is computationally demanding-requiring over three days to run 200K iterations of a small U-Net on a single H100 GPU-so we limit the from-scratch run to 200K iterations and compare it with 100K fine-tuning iterations under identical settings. Fig. presents the loss and FID curves for both strategies on central--non-centie and non wam--cetral--oon-ooo Overall, dDR trained from scratch performs poorly in the early stages (as indicated by high FID
translation tasks.
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0.70
0.25
0.00
Staruse
0.10
033
0.00
C:44
0.0
160
Finturing hamatone (-3000)
Freturing haratons (X2000)
Biaturing farstorn (+1000)
123
325
140
142
122
25
30
175
200
106
125
150
330
122
150
175
Aeoduring 100000
Nodounin Ravasiess 1+10000
Hectaring
(a) Sketch->Facc
(h) Segment-FFace
(c) Segment-<Sketel
Figure 8: Loss and FID curves of trained- from-scratch and finetuned dDRs.
Faces-UMDDES UMDT-LLaent
COCO-UMDT-Star
NFF
Skc.t+Seg.
Skg.
Ske.ta SketthDepth
Sepphh
iDR
dDR
iDR
dDR
iDR
dDR
iDR
dDR
74.33/7.95
39.29/6.66 65
98.14/79.17
31.29/34.19 19
78.65/13.96
27.10/12.09
79.95/24.97
35.31/16.48
20
42.25/6.98
23.79/5.82
50,09/32.07
28.26/25.67
42.95/11.30
22.69/11.26
32,93/15.35
26.03/15.66 63
30
31.77/6.67
22.31/5.72
36.01/25.99
27.93/24.81
31.37/9.83
22.09/10.35
26.12/14.71
25.57/15.02
50
23.37/6.39
21.85/5.61
27. 27.84/33445
27.19/24.06
19.39/9.03
21.65/9.59
24.87/13.95
25.21/14.74
100
18.05/6.25
20.82/5.58
24.70/23.16
26.87/23.78
18.81/8.70 79
21.18/9.29
24.08/12.49
25.18/14.62
1000
19.
19.42/5.52
23.06/23.01
26.73/23.64
18.15/8.86
20.75/9.42
23.37/12.11
24.91/14.87
Table 8: FID across NFE for non-central--noon-cenaa translations with iDR and dDR on Faces- UMDT-Latent and COCO-UMDT-Star
scores), but gradually improves and eventually approaches the performance of the fine-tuned variant as training converges. Interestingly, the Lunquired curve for from-scratch training behaves differently from Lpaired increases from near zero rather than decreasing from a large value. This occurs because, the start of training, eg is untrained, making the expectations of Eg (at x j.i) and of
Eo (2l,t,z', ..c) t.,)) approximately equal.
D.3.4 COMPARISON OF IDR AND DDR ACROSS SAMPLING STEPS
We evaluate iDR and dDR on Faces-UMDTTLatenl and COCO-UMDT-SIIE while varying the total number of sampling steps (NFE) from 10 to 1000. We focus on www.erral--aoneeneeeeee transla- tions because the central=-mon-ceentaal results for iDR and dDR are already comparable matched NFE in the main experiments. Both methods use 304.8M 8-paramtter U-Net models and are evaluated with the same total NFE for each non-centralk--oon-eenna translation.
From Table 8 we observe that iDR's performance deteriorates sharply NFE decreases, whereas dDR remains much more stable, with only a moderate drop in FID. At NFE < 30, dDR achieves 50- 100% lower FID scores than iDR on several translation tasks, such as Seg. SKee. on Faces-UMDTT This discrepancy
Latent, and Skg. Ske.+ -Dpphh and Deeph on COCO-UMDT-SHIE arises because iDR relies on an intermediate central-domain sample whose quality degrades signifi- cantly when NFE is small, thereby impairing the final non-central -oottanl--ominn output. By contrast, dDR directly generates the non-central domain and thus avoids this issue. These results strongly indicate
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FID4
LPIPS
Model type
Model
Ske. ++Face
Seg.t+Face
Ske.
Skeee+Face
Seace+Face
Ske.
BBDM
25.69/41.79
19.02/27.76
27.92/20.76
0.244/0.457
0.144/0.492
0.399/0.196
I-BB
16,56/31.81
10.58/24.56
21.37/13.16
0.241/0.456
0.153/0.492
0.396/0.208
Bridge
DDBM
15,51/42.55
10.32/27.77
19.51/11.41
0.246/0.477
0.156/0.506
0.408/0.213
BDBM
13.67/33.98
6.12/26.73
24.88/6.63
0.24480.479
0.151/0.500
0.382/0.204
Diffusion
DDPM
9.07/23.88
6.12/19.12
15.37/6.15
0.221/0.427
0.129/0.471
0.377/0.177
Table Results on Faces-UMDT- UMDT-Latent comparing diffusion- bbsed and bridge-based DRs.
Ske. Face
Seg. Face
Seg. Skee
Ske. Seg.
P50
Figure 9: Qualitative comparison of diffusion- and bridge-based DRs on Faces-UMDT-Latent
that dDR not sensitive to the central domain sample quality and is the clearly preferable choice when sampling with a limited number of steps.
D.3.5 COMPARISON OF STANDARD AND BRIDGE- BASED DRs
We compare diffusion- based against bridge-based DRs on Pices-UMDT-LLtent. For diffusion-based DR, adopt DDPM schedules with the noise parameterization. For bridge-based DR. we follow schedules and parameterizations of state-of-iee satte-of-the-an the-art bridges, including BBDM (Li et al. 2023), S (Liu et al.) (203a),, DDBM (Zhou al.] (024)) and BDBM (Kieu al. 2025). In all cases, a single 304.8M -aaram--rr U-Net models all conditionals between the central and each non-central domain; translations between two non-central domains are executed indirectly via the central domain.
Table 9 shows that the ceffusion-basee DR outperforms bridge variants on most translation direc- tions, with especially large margins on high-variance mappings (e.g., Skec-Face, Sege-Face). We attribute this advantage to diffusion's 's Gaussian prior and iterative denoising, which recover missing detail under sparse conditioning (e.g., sketches), whereas bridge-based sampling tightly anchors to the input and suppresses plausible completions. Consistent with this, Fig. 9 shows that bridge- based DRs on Ske.-Facce often over-condition on noisy sketch details, producing unrealistic ar- tifacts. Moreover, diffusion models partially share generative processes across domains (from common Gaussian), while bridge models learn separate source-to-targe couplings for each paired dataset. single U-Net thus faces harder multi-coupling approximation under bridge designs, particularly when paired datasets differ substantially. Overall, these results indicate that diffusion- based DRs are better suited MDT and can be fine-tuned support direct non-central translations without paired supervision.
D.3.6 IMPACT OF THE LEARNING RATE FOR FINETUNING
Table I0 examines the learning- rate trade-off when finetuning iDR into dDR for direct translation. Higher rates accelerate learning of the new Ske. Seg. mappings but significantly degrade the pre-
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FIDJ
Skcc+FFace
Scg.++Face
Skc.e>Sog.
IDR
14.21/39.06
10.18/24.95
20.82/10.85
le-4
17.05/43.18
15.16/28.90
52.41/12.87
16.11/39.88 82
13.75/25.77
55.30/13.35
dDR
15.55/39.56
13.60/25.13
71.09/15.09
14.58/40.01
13.51/24.58
76.96/16.50
1c-6
14.90/9.944
13.01/25.12
82.04/18.84
Table 10: FID on Locs-UMMDT-Latett for dDR across finetuning learning rates. All dDR models were fine-tuned without Tweedie refinement (n = 0).
trained mappings, e.g., S and Seg. Face, relative to iDR. At le-4, dDR achieves the best FID Ske.t>Seg. yet yields the worst FIDs Face and Seg. Face compared with iDR and dDR finetumed at lower rates. Reducing the rate reduces catastrophic forgetting and restores the pretrained directions, reaching 14.90/39.99 94 with Ske. and 13.01/25. 12 on Seg.e+aace le-6. This improvement comes at the cost of the direct translation mappings, with Ske.-Seg. FID increasing from 52.41 82.04 and Seg.+tke. FID increasing from 12.87 to 18.84. Overall, the results show a clear trade-off between retaining pretrained tasks and learning the new one. Among the tested settings, 5e-5 offers the best balance.
D.4 QUALITATIVE RESULTS COCO-UMDT-STAR AND COCO-UMIT-CHAN
We visualize dDR's s results on COCO-UMDT-Sta and COCO-UMDT- to illustrate its ability handle UMDT. Specifically, Figs. 10
and show cross-domain samples conditioned on color images, sketches, depth maps, and segmentation maps, respectively. Because segmentation maps are not ideally suited to a continuous diffusion process in latent space and predicted class labels are hard to decode perfectly, apply DBSCAN (Ester et al. 1996) as post-processing step that merges noisy pixels into the nearest region with the smallest label difference.
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Input Color
Color Ske.
Color - Seg.
Color - Depth
J.P.Morgan
IPMODAA OAA
J.P.Moraen
POLL
(a) Results on COCO-UMDT-Star The classes for each segmentation map (from top to bottom) are: (snow,
person), (wall, person, floor), (cow, grass).
Input Color
Ske.
Color Seg.
Color Depth
J.P.Morgan
J.P.Morann
POLO
(b) Results on COCO-UMPE UMDT-Chain. The classes for each segmentation map (from top to bottom) are: (snow, person, sky), (wall, person, floor), (cow, grass).
Figure 10: Results of finetuned dDR on COCO-UMDT-Star (a) and COCO-UMDT-Chain (b) for translation tasks from Color domain to Sketch, Segment, and Depth domains.
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Input Ske.
Ske. Color
Ske. Seg.
Ske. - Depth
(a) Results on COCO-UMDT-SHAE The classes for each segmentation map (from top to bottom) are: (build-
ing, bus, road, bush), (tree, train), (elephant, sky, gravel).
Input Color
Ske. Seg.
Ske. Depth
(b) Results on COCO-UMDT-Chai The classes for each segmentation map (from top to bottom) are: (bus, road, wall), (tree, train, sky), (elephant, sky, tree).
Figure 11: Results of finetuned dDR on COCO-UMDT-Star (a) and COCO-UMDT-Chain (b) for translation tasks from Sketch domain Color, Segment, and Depth domains.
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Input Depth
Depth Color
Depth - Seg.
Depth Ske.
(a) Results on COCO-UMDT-Star The classes for each segmentation map (from top to bottom) are: (sky,
airplane, bridge). (airplane), (banana, table).
Input Depth
Depth Color
Depth - Seg.
Depth Ske.
(b) Results on COCO-UMDT-Chait (bridge, airplane, sky, undefined), (airplane, sky, road), (banana, floor).
The classes for each segmentation map (from top to bottom) are:
Figure 12:
Results of finetuned dDR on COCO-UMDT-Star (a) and COCO-UMDT-Chain (b) for translation tasks from Depth domain Color, Segment, and Sketch domains.
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Input Seg.
Seg. Color
Seg. Ske.
Seg. Depth
(a) Results on COCO-UMDT-SHH. The classes for each segmentation map (from top to bottom) are: (bed,
floor, window, light, wall), (pizza, undefined, table, fork), (grass, giraffe, bush, sky).
Input Seg.
S
Seg. Ske.
200
(b) Results on COCO-UMDT-CHan Chain. The classes for each segmentation map (from top to bottom) are: (bed, floor, window, light, wall), (pizza, undefined, table, fork), (grass, giraffe, bush, sky).
Figure 13: Results of finetuned dDR on COCO-UMDT-Star (a) and COCO-UMDT-Chain (b) for translation tasks from Segment domain to Color, Sketch, and Depth domains.
27


Reviews:
[{"review_id": "hKQStTfyOJ", "reviewer": "Reviewer_AFk3", "summary": "This paper introduces Universal Multi-Domain Translation (UMDT), a novel and practical problem setting for translating between any pair of K domains using only K-1 paired datasets structured around a central domain. To address this, the authors propose Diffusion Router (DR), a unified diffusion model that conditions a single noise predictor on both source and target domain labels to handle all translations involving the central domain. The authors create new large-scale benchmarks to evaluate their method, demonstrating that iDR outperforms existing baselines and that dDR can achieve direct translation, albeit with a trade-off in performance versus computational efficiency.", "questions": "See Weakness", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The formalization of UMDT is a significant contribution. It addresses a critical limitation of existing multi-domain translation methods, which either require impractical fully-aligned data or are limited to a hub-and-spoke translation model. This problem setting is highly practical and forward-looking, with clear real-world applications.\n- The core idea of the Diffusion Router is an elegant and parameter-efficient solution for the standard central-to-non-central translation task. The strong performance of iDR against other baselines validates this architectural choice and establishes it as a powerful model in its own right.\n- The theoretical contribution for enabling dDR is clever. The formulation of a variational upper bound (Eq. 9) to align the direct and indirect translation paths is a non-trivial approach to learning from \"pseudo-supervision\" generated by the model's own indirect capabilities.", "weaknesses": "- The central weakness of this paper is that the main technical contribution, dDR, consistently performs worse than the simpler, two-step iDR baseline on the very tasks it was designed to improve. This is evident across all three benchmarks, where iDR's FID scores are superior to dDR's.The paper motivates dDR by claiming it overcomes the computational expense and potential quality degradation of the two-step iDR process. However, the results show that dDR introduces quality degradation. This undermines its primary justification and calls into question whether the proposed learning strategy is genuinely effective.\n- While dDR is computationally cheaper at inference, the paper does not adequately quantify this benefit against the measured drop in quality. The faster inference speed is valuable, but if it comes at the cost of a significant increase in FID score, iDR remains the superior method in terms of output quality. The paper would be much stronger if it provided a clear analysis of this trade-off, perhaps identifying scenarios where the speed of dDR is worth the quality sacrifice.\n- The authors acknowledge that their tractable objective relies on a single-sample Monte Carlo estimate inside a logarithm, which introduces bias. This technical compromise is a likely source of dDR's underperformance. The work would be more convincing if it included analysis isolating the effect of this bias.", "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": "", "extra_scores": {}}, {"review_id": "ksaduMUB1r", "reviewer": "Reviewer_4zEu", "summary": "They propose a new approach for multi-domain translation using a diffusion model. Unlike existing settings, where we assume the presence of many fully aligned tuples or unlabeled paired domains, they assume access to one central domain that has the paired samples with other domains. Then, they propose an approach that can achieve domain translation between unpaired domains by using the central domain as the bridge between the unpaired ones. Specifically, they propose an architecture that can translate between different image domains and training objectives. They conduct experiments on image translation and show that their approach outperforms existing diffusion based approaches with a large margin.", "questions": "Please answer the concerns described in the weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Their idea of using the central domain as the bridge between two unpaired domains sounds reasonable and novel. Their derivation of the objectives also looks reasonable. \n\n2. The proposed architecture to achieve a single translation model, which specifies the origin and the target domain, also looks reasonable and might be novel. \n\n3. This paper is easy to follow. Presentation is overall clear.", "weaknesses": "1. They repeat the discussion on translating image, text, and audio. However, they lack experiments on such cases. I wonder why they do not include such results. Also, my concern is that in such a modality translation setting, the proposed framework might not work well. They propose to apply a single network to exchange the domains of inputs. Since their experiments are conducted only in the translation of images, the effectiveness of swapping modality is not clear. \n\n2. Lacking comparison to non-diffusion techniques in experiments. Readers should be curious about the performance difference between the proposed one and non-diffusion methods. \n\n3. The novelty in the proposed method is not very clear to me. I guess the field of domain translation using a diffusion model is a popular topic. Since I am not familiar with this field, I could not figure out how novel the proposed method is, compared to the existing ones. They need a more careful discussion in explaining their approach in the introduction and method sections. \n\n4. What if they have a designated model for one pair of domain translation and combine two models to bridge the domains? A single model still outperforms such an approach in terms of FID?", "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": "", "extra_scores": {}}, {"review_id": "xQMWzLNCBq", "reviewer": "Reviewer_ARK8", "summary": "This paper proposes a universal multi-domain translation model wherein a single model can be used to do pairwise translations across all the K considered domains with only requiring K-1 paired datasets.", "questions": "- How does the performance of the model change depending on what you consider as the central domain? Currently, the central domain is always fixed. \n- How well does the model perform on translating different modalities (audio-text-image) given that this is one of the motivating example in the introduction? \n- Can the model work if the different domains contain images of different classes? For example consider translating between sketch-image-grayscale. For domain pair is image to sketch you have shoes as examples and for the image-grayscale you have faces. The central domain image contains both face and shoes. This setup satisfies your K-1 paired examples with overlap in the central domain. How well can the model now translate between the other domains? \n- How about domains where structure is not preserved such as image to cartoon style? Current text-to-image models can generate a bunch of these paired examples, and different artist styles could also be considered as different domains.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The proposed diffusion routers is novel\n- The results are promising and outperforms baselines based on both qualitative and quantitative results\n- They construct 3 different datasets and conduct extensive experiments on them to validate their approach", "weaknesses": "- The domains considered for all the datasets they constructed are quite limited. They only consider the image, segmentation, sketches, and depth, which are quite similar in structure only different appearance / details. \n- The baselines compared are also quite old (2018, 2022, 2023). Granted that there are relatively few works on multi-domain translation models, I think it would still be good to also compare with more recent single domain translation models see how good its performance is compared to a dedicated model. This would also show whether or not training on more domains could improve performance due to having more auxiliary information.\n- For the domains involved, the recent text-to-image models can also already achieve this kind of translation. It would be good to also compare with them since in a way it is also able to perform multi-domain translation with a single model.\n- The authors motivated a lot of their work with image-text-audio domains. However, none of the experiments show any translations between these domains.", "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": "", "extra_scores": {}}, {"review_id": "IhCweWZlju", "reviewer": "Reviewer_es9u", "summary": "This paper introduces universal Multi-Domain Translation, a problem setting for learning translation between any pair of K domains using only K-1 paired datasets. Specifically, this paper proposes a unified diffusion-based framework that avoids the need for training separate models for each domain pair. The framework is presented in two variants: iDR that routes translations through the central domain, and a dDR that distills iDR’s indirect path distributions into direct non-central mappings via a variational-bound objective and Tweedie refinement. To validate this approach, the authors construct three new image-based benchmark datasets and demonstrate that DR achieves state-of-the-art performance.", "questions": "Minor clarifications for presentation. The X/Y format is not explicitly defined in tables. Please add a sentence to the table captions clarifying that the two numbers correspond to the forward and backward translation directions, respectively (e.g., \"A->B / B->A\").", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The problem formulation is well-motivated, and the background is clearly explained. \n2. Extensive experiments validate the effectiveness of the proposed method on the constructed star- and chain-structured image-domain conditional generation tasks.\n3. The writing is generally clear.", "weaknesses": "1. The universal claim is not supported by intra-modality experiments. The authors motivate the problem with compelling cross-modal examples like image/text/audio translation. However, all experiments are within the image-to-image translation domain, and the current results only prove the method's capability as a multi-style image translation system.\n2. The paper's contribution appears less like the introduction of a new paradigm and more like a new training setup for an existing problem. When viewed as a solution for multi-style image translation under a specific data-sparsity assumption, its conceptual novelty is reduced.  \n3. iDR relies on multi‑step inference, and its outputs can accumulate approximation and routing errors. The proposed training of dDR distills from iDR’s generations, thereby inheriting these accumulated errors, which bounds dDR’s quality.  In addition, although DR is a single network whose parameter is roughly independent of the number of domains K, the number of non-central directed pairs to cover grows roughly as O(k^2). Together with Tweedie refinement for conditional sampling, this implies that total training time requirements scale up significantly with K.", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)