Learning a Cross-Modal Schrödinger Bridge for Visual Domain Generalization
Authors: Hao Zheng
Jingjun Yi
Qi Bi
Huimin Huang
Haolan Zhan
Yawen Huang
Yuexiang Li
Xian Wu
Yefeng Zheng
Venue: NeurIPS 2025 poster ()
Keywords: Domain Generalization
Schrödinger Bridge
Image-Text Alignment
Optimal Transport
Primary area: applications
TL;DR: This paper proposes a Cross-Modal Schrödinger Bridge to align the domain-specific imges to the domain-invariant text, so as to enhance generalization to unseen domains.

Abstract: Domain generalization aims to train models that perform robustly on unseen target domains without access to target data. 
The realm of vision-language foundation model has opened a new venue owing to its inherent out-of-distribution generalization capability.
However, the static alignment to class-level textual anchors remains insufficient to handle the dramatic distribution discrepancy from diverse domain-specific visual features.
In this work, we propose a novel cross-domain Schrödinger Bridge (SB) method, namely SBGen, to handle this challenge, which explicitly formulates the stochastic semantic evolution, to gain better generalization to unseen domains.
Technically, the proposed \texttt{SBGen} consists of three key components: (1) \emph{text-guided domain-aware feature selection} to isolate semantically aligned image tokens; (2) \emph{stochastic cross-domain evolution} to simulate the SB dynamics via a learnable time-conditioned drift; and (3) \emph{stochastic domain-agnostic interpolation} to construct semantically grounded feature trajectories. 
Empirically, \texttt{SBGen} achieves state-of-the-art performance on domain generalization in both classification and segmentation. This work highlights the importance of modeling domain shifts as structured stochastic processes grounded in semantic alignment.

Learning a Cross-Modal Schrödinger Bridge for Visual Domain Generalization
Hao Zheng' i, Jingjun , , Qi Bi4 Huimin Huang', Haolan Zhan Yawen Huang',
Yuexiang Li Xian Wu', Yefeng Zheng
f: equal contribution
Tencent Jarvis Lab, China, "Westlake University, China
University of Alberta, Canada, *University of Amsterdam, the Netherland 'Monash University, Australia, University of Macau, Macau q @ieee org, yuexiang. li Li@ieee.on org zhengyefengèwestlake edu. cn
Abstract
Domain generalization aims to train models that perform robustly on unseen target domains without access target data. The realm of vision-language foundation model has opened a new venue owing to its inherent out-of-distribttion gener- alization capability. However, the static alignment to class-level textual anchors remains insufficient to handle the dramatic distribution discrepancy from diverse cemaii--specffc visual features. In this work, we propose novel cross-domain Schrödinger Bridge (SB) method, namely SBGen. to handle this challenge, which explicitly formulates the stochastic semantic evolution, to gain better generaliza- tion to unseen domains. Technically. the proposed SBGen consists of three key components: (1) text- text-guided domain-aware feature selection to isolate semantically aligned image tokens; (2) stochastic cross roos-domain evolution to simulate the SB dynamics via a learnable cine-conditioned drift; and (3) stochastic domain-agnostic interpolation to construct semantically grounded feature trajectories. Empirically, SBGen achieves state-of-the-art performance on domain generalization both classification and segmentation. This work highlights the importance of modeling domain shifts as structured stochastic processes grounded in semantic alignment.
Introduction
Distribution shift is a fundamental challenge in both machine learning and computer vision. Domain
source domains Generalization (DG) addresses this challenge by training models generalizable representation that can generalize well to unseen target domains [1 (5)
especially critical for trust-worthy artificial intelligence and plays a pivot role in safety-crucial
applications, such as autonomous driving [32 74 8 77 21 and medical imaging [1] 68 7 76
where target environments not available during training.
In visual domain generalization, the images from various domains are usually diverse in terms of the contrast, texture, illumination, and resolution [88] 59 48). The emergence of vision-language models (VLM) [59] has opened up a new venue to approach the DG problem. Its general idea is that, the category-wise text description is capable to anchor high-level semantics despite the distribution shift of the images from various unseen domains [31]. Specifically, existing VLM based DG methods usually treat the VLM static feature extractor and apply fixed alignment strategies such prompt learning [86). cosine matching [ and adversarial regularization [1 to enforce similarity between image features and class-level text queries.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Vision
Domain Domain 2
Ted
Car, road, sidewalk person, building
D
Domain 3
Static A Aignmeent
Stochastic A Aignmenn
Figure 1: Leveraging domain- -agnossic category- wise text embedding to align the domain-specinf visual features from different domains is a common paradigm for domain generalization in vision- language models (VLM). Left: Existing methods usually rely on a static alignment strategy, which can be insufficient to handle the dramatic discrepancy and spurious correlations across domains. Right: In contrast, this paper presents SBGen, stochastic alignment strategy to enhance the domain generalization ability of VLM.
Crucially, such fixed alignment strategies are developed under the assumption that visual embeddings can be directly and reliably projected onto the textual anchors, which may not necessarily hold under the domain shift. In fact, visual features extracted from unseen domains by an image encoder may exhibit dramatic distribution discrepancy (shown in Fig. D Such degraded semantics and spurious correlations may not be corrected by static alignment. Moreover, such a deterministic projection may not offer clear path to model the semantic drift caused by the domain shift, as it can be difficult to map onto the textual semantics in a single step. Conversely, we turn to the multi-step based mapping, how transform the partially-alignee intermediate states the semantic invariance the key challenge.
This paper pushes this frontier by approaching VLM based DG from fundamentally different perspective. Rather than enforcing the static similarity between image and text features, we ask, what model their alignment stochastic semantic evolution? Is it possible to gain better generalization to unseen domains?
We propose SBGen, a novel cross cross-modal Schrödinger bridge for visual domain generalization, to realize the above objectives. Its general idea to formulate the alignment from domain-specific visual representations to domain- agnnosicc textual semantics, as controlled stochastic process that interpolates between two distributions while remaining close to prior. It comprises three main stages. First, lext-guided domain- aawre feature selection component is proposed to extract local visual tokens from source images that align with class-level textual queries, which focuses the model on semantically relevant content while avoiding domain sppecfc interruption. Next, stochastic cross-doman evolution component is proposed to model the Schrödinger Bridge as a time-indexed stochastic differential equation (SDE) with lcarnable, modyycoonditioned drift. This process is then discretized and simulated to generate a trajectory of evolving features. Finally, a stochastic domain- agnostic interpolation component is proposed, using these features to bridge source representations with semantic anchors.
Notably, each stage in the proposed SBGen is differentiable and can be jointly trained with loss function that balances task supervision and stochastic consistency. It enables us to simulate sequence of latent feature states that progressively reduce domain-specite bias and converge toward semantic consistency with textual queries. In contrast to prior methods, our approach supports structured semantic interpolation, and models interpretable and probabilistic trajectory from biased source features toward text-grouut -ext-grounded, domain-agnosta representations. The proposed SBGen is evaluated on standard domain generalization benchmarks for both classification and segmentation. consistently outperforms the state-of-the-ee methods.
Concretely, our contributions can be summarized as follows.
We propose cross-modal Schrödinger Bridge for visual domain Generalization (SBGen), novel framework that aligns domain-specific image features with domain-agnostie textual semantics via Schrödinger Bridge- Bridge-guided guided stochastic evolution.

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We introduce principled three-stage pipeline that performs text-guided feature selection, stochastic cross-domain evolution, and semantically anchored interpolation.
We provide a theoretical justification of the proposed SBGen through provably tighter generalization bound, and demonstrate its effectiveness multiple DG benchmarks in both classification and segmentation settings.
2
Related work
Vision-Language Models (VLMs) have emerged effective tools for capturing deep semantic relationships across modalities. Some typical VLMs include CLIP [59]. ALIGN (34] and EVA02 [26
25
The expressive representations have proven effective for more complex downstream vision-
language tasks 40
87
Domain Generalization (DG) has been extensively studied in the machine learning community 7
12
73 46 [0 More recently. the emergence of vision-language model (VLM) [59
86 has
paved new path for DG. One research line focuses leveraging its inherent out-of-distribution
generalization ability [T] 23 42 Another closer research line usually leverages the text embedding
to augment the domain diversity to statically align the domain-specinn visual features [22 B 50
13
13, [9]. However, the majority of these approaches rely on the assumption that Nlomain-speciffe visual features can be directly and statically matched to domain- -omainnagnostic textual embeddings, which may overlook the dynamic and multi-faceted characteristics of semantic shifts caused by domain changes.
65 16 36
39
Domain Generalized Semantic Segmentation (DGSS) aims to learn a generalizable segmenta- tion model trained only on source domain. Earlier works usually use normalization [53 54),
whitening 57 or mask attention (21 Other works use style hallucination or randomization
80). More recently, vision foundation models 32 have been used for DGSS. Despite these advancements, these approaches usually implement direct and static alignment between the image and text embeddings, or enrich the domain diversity guided by the text description. They may still be insufficient to handle the semantic drift caused by the dramatic domain shift.
techniques for domain augmentation 38
84
35
(VFM) 774 77 8 and VLMs [22
Schrödinger Bridge and Stochastic Feature Transport These frameworks define stochastic processes that interpolate between distributions via entropy-
20
70 have drawn increasing attention.
regularized optimal transport. Recent work has explored SBs for generative modeling [72 641
score-based dynamics 19 66 69), depth estimation (28), and modality translation [68 4 33
However, how to explicitly model the stochastic alignment from domain-specifte features to domain- agnostic semanties using SB-driven dynamics for DG remains underexplored.
Preliminaries
Problem Definition. Let X' and y denote the space of input images and the space of structured labels
from a certain task (e.g., classification). Given set of labeled source domains DS = {(= y Na with x X, v. e e, and a set of unseen target domains D = {(>li ym N J the objective is to train model source domains that generalizes these unseen domains.
Definition Optimal Transport (OT). Let p8 and pú be two probability distributions over R respectively. The classical OT problem seeks deterministic transport map M. RRC + R minimizing a transport cost:
min M-M P8 Ess-ps [||= - (12))
(I)
where My Ps denotes the pushforward measure of pS through M.
Definition 2. II(P8 pU) introduced improve the robustness of OT, minimizing:
OT. A stochastic coupling tI T22 2²) with marginal constraints
min xEI[ pU)) f 1| J 2 (2 20) + (LT(=|R))
3

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Domain agoostic Text Distribution
prompt
bur
Transfommer
Decoder
bicycle"
"rood
Topek Selection (Rig 8)
Task-spocific
Greaadd
frruan
Huad
bockoaad
P
M2
Image
Domoinn-ppci Distritution
Input
Domain-Iware Festure Selection
Stockevtic Cross-Domain Evolation with Interpolation
Prediction
Figure 2: Overall pipeline of SBGen. Step 1: We generate initial textual object queries qf from the K class text embeddings {t=) k= Step 2: To improve the segmentation capabilities of these queries, we incorporate text-to-pixel attention within the pixel decoder. This process enhances semantic clarity of pixel features, while reconstructing high-resolution per-pixel embeddings Z. Step 3: The transformer decoder refines these queries for the final prediction. Each prediction output is then assigned to its corresponding ground truth through fixed matching, ensuring that each query consistently represents the semantic information of one class.
where R is reference measure and & > 0 controls the regularization strength, enabling stochastic transport but lacks notion of dynamics over time.
Definition Schrödinger Bridge (SB). OT is extended to the dynamic setting by introducing continuous-tiin stochastic process {Pi [e that evolves from ps to PV, while being minimally
deviated from prior diffusion process P (e.g., Brownian motion). The SB formulation is:
min (L[ subject to P Q
where Q denotes the law of the interpolating process over latent features. This yields a family of time-indexed distributions P; modeling the optimal evolution of visual features across domains.
Methodology
We propose SBGen, a cross-modal Schrödinger Bridge framework for visual domain Generalization. Its general idea is to learn time-indexed stochastic process over feature distributions, evolving from the source domain toward target-aligned representations, guided by the domain-agnosti class-level textual queries. Specifically, consists three components, namely, Domain-aware Visual Feature Selection, Stochastic Cross-Domain Evolution, and Stochastic Donain-Agnostic Interpolation. The rigorous generalization error analysis on its upper bound is provided the supplementary material.
4.1 Domain-aware Visual Feature Selection
Assume we have an image encoder & (c.g., CLIP-ViT), which extracts the visual features from an encoder T and set of class names Q. (c = 1, ... No). where N₂ is the number of semantic classes. Following the textual query generation protocol [52. each class name is converted into a class-specific textual query and embedded via T. resulting in q. T(Q.) E R IxC The visual features F include information such as background, textures, lighting, or object co-occurrence, which can be usually domain-specific. In contrast, the class-specific textual embedding q. tends to capture high-level and domain-invann semantics. To leverage the class-wise text embeddings q. as domain-agnostic anchors for semantic alignment, the domain-awar visual feature selection component proposed.
image x E X, given by F E(x) E I H We also assume access

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Rather than using all visual tokens uniformly. we identify consicondd regions in the image feature space that exhibit high alignment with textual descriptions. This targeted selection filters out the irrelevant content and yields semantically grounded set of features for downstream modeling. Specifically, the visual feature F E H Wxx can regarded as set of feature vectors (Fh.,u I.. IR and each Fh.w corresponds to the representation of a certain spatial position (h, w) in the image feature. Then, the cosine similarity between each spatial feature Fu,, and each class embedding q, computed as
Sn,uc (1
Fho |FFmwoll'
9c l|
vh, w.
Then, the top-k spatial locations for each class that exhibit the highest alignment scores are selected, which distinguishes the semantically relevant and domain-robuse feature locations, avoiding noisy or background-dominet
Let {=u J==1 I denote the corresponding set of (8) (
inputs 60 89
visual tokens, which serve as the class- and domain-aware initial feature set. These empirical samples from the source distribution Po are used as input to our Schrödinger Bridge evolution process.
4.2 Stochastic Cross-Domain Evolution
Most prior VLM based DG methods adopt deterministic feature mappings prompt tuning strategies, which are limited in capturing uncertainty or adapting to structured variation between domains. We explicitly model the evolution from the Nommin--pecciic visual features to the domain-agnoss text embeddings, so as to enhance the generalization unseen target domains. To realize this objective, this evolution is modeled as a stochastic process governed by Schrödinger Bridge.
The selected features 2o Specifically, we define a time-indexed stochastic process {zz (Et00100.1 C IRC governed by the following stochastic differential equation (SDE), given by
the initial samples form the empirical source distribution 20 ps...
dz, = fa t) dt + dW,
(5)
where fe RC x [0.1] + RC is learnable drift function, 5 > a fixed diffusion coefficient, and W, denotes standard Brownian motion. The process begins at zo Is and is regularized to terminate near distribution pe implicitly defined by the textual query embeddings {qe}-
Following best practices in recent Schrödinger Bridge literature 20 70 we parameterize the drift function fo using a Multi- Perceptron (MLP) conditioned on both the time index and class semantics. This drift parameterization is computed as
(6) where (t) E I a sinusoidal time embedding and 9, is the target text embedding for class c. This design encourages smooth and semantically aligned evolution under time-aware control.
fa t) MLPp (LayerNorm(z: + (t)) q.)).
Concretely, the proposed stochastic cross-domain evolution allows the alignment between the domain- specific visual features and the semantic anchors defined by domain-agnoste text embeddings, while maintaining ficxibility to domain shifts.
4.3 Stochastic Donain-Agnostic Interpolation
Nevertheless, the proposed stochastic cross-domain evolution only specifies the initial state from the domain-specific visual features from the source domain and the end state from the domain- agnostic text embeddings, which does not take the transitional dynamics between domain-specifie representations and Nomain-invarian semantics. To address this issue, in the proposed SBGen, a stochastic interpolation mechanism is introduced. Its general idea is to model continuous-time evolution of feature states that gradually transforms source-domain features into semantically aligned, agnosic representations.
By simulating the Schrödinger Bridge dynamics from = 0 to /. we generate trajectory (2:) that smoothly interpolates between domain- biased initial state from the source domain and a domain- agnostic text embeddings. Unlike deterministic mappings, this stochastic evolution accounts for uncertainty and allows for explicit control over the extent of semantic alignment. Each intermediate
5

---

feature 2, can be interpreted as partial semantic abstraction, providing generalization to unseen domains and offering flexibility in selecting the optimal representational point. Specifically, we numerically simulate the SDE in Eq. (5) using Fuler-Maruyana discretization. Given step size Al A1=1/ and in
n. At, the discretized evolution can be computed as
i tt,, + [Z (n) At v 2e En:
En ~ ~O,,1,
with initial condition zt. 20 ~ Po- The sequence { captures the full evolution. We implement this via Monte Carlo estimation over minibatches and simulate each path using Eq. 7
We adopt this discretization to efficiently simulate the evolution from domain-specifie features zo to domain-invariant features 2T through a sequence of stochastic updates. The terminal feature 2tz is given by the unrolled summation:
T-1 fo tx)
Stg
too +
At
Σ
22 Sn.
(8)
n=0
n=0
which represents a stochastic interpolation path toward the domain-agnoste class-wise text semantics.
4.4 Prediction, Optimization & Implementation Details
The evolved features ZT E Kxx obttained through the SB trajectory, are not directly used for task prediction. Instead, they serve as refined class-aware feature anchors that are written back to the original visual feature map F E RCx H WW For each class c, the corresponding evolved embeddings =)6 are broadcasted to their original locations. This update process yields an enhanced image feature map F², in which the selected regions are aligned toward the class-conditional textual semantics.
The updated visual feature map F' and the class queries }qo} are then fed into the decoder 'D for the final task prediction (e.g.. classification and segmentation) on unseen target domains, where classification uses global pooling followed by linear projection, and segmentation employs per-pixel decoding via cosine similarity with class embeddings.
Afterwards, the learning objective is to minimize the expected total loss over the data distribution and simulated paths, given by
min ExnDs Ezo-Po sCsu (D(F', (qct).v) + A KL [[] 0
where Suap denotes the task-specific loss, Qo denotes the forward path distribution induced by the learned drift fe to transport initial visual features 2g ~ Po toward class-conditional textual anchors
qr IP denotes the reference Brownian motion starting at 2o, and À E R>o is a regularization weight.
The KL loss (L || IP) is approximated by ΣT- (z At// + || qell2 for each pair of zo and the corresponding 9.
For the classification task, the image encoder & and the text encoder T use the pre-trained CLIP in align with the prior DG methods. The task-specific decoder 'D is a linear layer followed by a Softmax layer. For the segmentation task, following prior domain generalized semantic segmentation methods [52] (7), the image encoder & and the text encoder T use the pre-trained EVA-02 (25). The task-specific decoder D integrates the pixel decoder of the Mask2Former model [14]. The hyperparameters and configurations of both tasks are detailed in the supplementary material.
Experiments
5.1 Results on Domain Generalization in Classification
Datasets & Evaluation Metrics. PACS 4ЛЛ, VLCS 24), OfficeHome (71]. TerraIncognita [5]. and DomainNet (58) comprise of 1,991, 10,729, 15,588, 24,330 and 0.6 million images from four, four, four and six domains, respectively. In line with prior work 22 12], the evaluation protocol is adopted, where domain is held out the unseen target domain, while the remaining domains are used for training the model. Performance is reported using classification accuracy (percentage, %) as the evaluation metric.
Compared Methods. Existing VLM based domain generalization methods are involved for com- parison, namely, SWAD 122. CLIP (59 SMA (3). DUPRG (50), CoOp (86), MIRO B3] SEDGE

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Table 1: Comparison with the state-of- oaee--ff-hhe--eee the-arr art methods on PACS, VLCS, OfficeHome, Domain and Terralnc. By default the results are cited from [1]
65
16
36 79
Evaluation metric is classification
accuracy (in %).. Top three results highlighted
best second and third respectively.
Method
Venue
PACS
VLCS
OfficeHome DomainNet TerraInc
Avg
ResNet- Pre-trained by ImageNet:
DANN 27
2016
83.6
78.6
65.9
38.3
46.4
65.6
Fish 663
ICML'2022
85.5
77.8
68.6
42.7
45.1
63.9
DAC-SC 337
CVPR'23
87.5
78.7
70.3
44.9
46.5
65.6
SAGM 73
CVPR 2003
86.6
80.0
70.1
45.0
48.8
66.1
VIT-B/16 Pre-trained by CLIP:
SWAD [2
NIPS'202
91.3
79.4
76.9
51.7
45,4
68.9
CLIP 59
ICML'2021
96.2
81.7
82.0
57.5
33.4
70.2
SMA
NIPS'2022
92.1
79.7
78.1
55.9
48.3
70.8
DUPRG
ICLR'2023
97.1
83.9
83.6
59.6
42.0
CoOp 86
UCC222022
96.2
77.6
83.9
59.8
48.8
73.3
MIRO 13
ECCV
95.6
82.2
82.5
54.0
54.3
73.7
SEDGE +3
ArXiv'2022
96.1
82.2
80.7
54.7
56.8
74.1
DPL. 82
TAT'2023
97.3
84.3
84.2
56.7
52.6
75.0
CLIPOOD
ICML''0023
97.3
85.0
87.0
63.5
60.4
78.6
Promptstyler 16
ICCV
97.2
82.9
83.6
59.4
KAdaptaion 36
WACV'
97.5
83.0
90.3
62.7
51.9
77.1
GESTUR 39
ICCV 2023
96.0
82.8
84.2
58.9
55.7
75.5
DPR 15
CVPR 2024
97.5
86.4
86.1
62.1
57.1
77.8
CLIPCEIL+
79
NeurlPS' 2024
97.2
85.2
87.7
63.6
62.0
79.1
Ours
2025
97.4
86.7
89.9
64.4
63.5
80.4
Promptstyler [6]. KAdaptaion [36.. GESTUR 33), DPR 15 and CLIPCEIL+ ++ 79 Several prior ImageNet pre-trained domain generalization methods, namely, DAC-SC
43
DPL
CLIPOOD 65
DANN 27 Fish 63
37 and SAGM 773, also compared for boarder reference.
Results. Table I reports the outcomes on the five datasets. The proposed method shows the state-of- the-art performance over the existing VLM based DG methods, yielding classification accuracy of 86.7%, 64.4%, and 63.5% on VI.CS, DomainNet and TerraIne, respectively. Notably, DomainNet and TerraInc are particularly large-scale, indicating the scalability of the proposed method. Its performance is also very close to the statc-of-the -f-th--rr--- on PACS and VLCS, where both benchmarks have been highly saturated. Overall, the proposed method shows the best performance the average accuracy of five datasets, outperforming the second-bes by 1.3%.
5.2 Results on Domain Generalized Semantic Segmentation
Datasets & Evaluation Metries. Four driving-scene semantic segmentation datasets that share 19 common scene categories are used for validation. Specifically, CityScapes (C) (18) consists of 2.975 and 500 images for training and validation. respectively, The images were captured under the clear conditions in tens of Germany cities. BDD-100K (B) 78 has 7. 7,000 and 1,000 images for training
and validation, respectively. The images were captured under diverse conditions from variety of global cities. Mapillary (M) [7 is large-scale semantic segmentation dataset, which consists of 25,000 images from diverse conditions. GTA5 (G) [1 is another synthetic dataset, which has 24,966 simulated images from the American street landscape. Following the evaluation protocol of existing foundation model based DGSS methods 724, [2), two commonly-used evaluation settings are: I) + C, B. M; and 2) C - B, M. respectively. The evaluation metric is mean Intersection of Union (mloU, in percentage %). All the experiments report the average outcomes from three independent repetitions.
Compared Methods. We compare with existing DGSS methods from three major categories: 1) ResNet based methods, namely, ISW [I7] GTR [56). SHADE 33 SAW [57], WildNet [88] AdvStyle 185) SPC [30] and BlindNet 2 2) Mask2Former based methods, namely, HGFormer
21 and CMFormer 9 3) VFM and VLM based methods, namely, DIDEX (4), REIN 74, SET
77 FADA 8 tqdm 52 and MGRNet [67]. By default, the performance is directly cited from
52 67), and we report two decimal results. *** denotes that the original paper only reported one decimal results.
prior works 99
7

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Table 2: Performance comparison between the proposed method and existing DGSS methods. C: CityScapes [8]; B: BDD-100K [78]; M: Mapillary [47]; S: SYNTHIA 62]; : G: GTA5 [1]. : results were not reported and official source code is not available; ** only reported one decimal
official results; "t': re-implementation with official source code under default settings. Evaluation
metric is mloU in %. Top three results are highlighted as best second and third respectively.
Method
Encoder
G
G Avg.
C-
Avg.
ImageNet Pretrained:
ISW [17]
2021
ResNet-101
36.58
35.20
40.33
50.73
$8.64
GTR
TIP'2021
ResNet-101
34.52
50.75
SHADE 83
ECC2
ResNet-101
44.65
39.28
43.34
50.95
60.67
SAW 57
CV2222002
ResNet-101
39.75
37.34
41.86
$2.95
59.81
WildNet
CVPR-2022
ResNet-101
44.62
38.42
46,09
50.94
58.79
AdvStyle
35
Neu2
ResNet-101
39.62
35,54
37,00
SPC
CV22
ResNet-101
44.10
40.46
45.51
BlindNet
CVPR-2024
ResNet-101
45.72
41.32
47.08
51.84
60.18
HGFormer
21
2V23
Swin-T
53.4
66.9
CMFormer
AAAI 2024
Swin-B
55.31
49,91
60.09
59.27
71.10
VLM Prefrained:
DIDEX 49
20CV
Stable Diffissinn
62.0
54.3
63.0
59.7
VLTSeg
ACCV
CLIP-L
55.6
52.7
59.6
56.0
REIN" 74
CVPR-2024
EVA02-1.
65.3
60.5
64.9
63.6
64.1
69.5
66.8
SET 27
2024 2024
EVA02-L
66.4
61.8
65.6
64.6
FADA
NeurlPS 2024
EVA02-L
66.7
66.1
64.9
iqdm 52
BCCV"2024
EVA02-L
68.88
59.18
70.10
66.05
64.72
76.15
70.44
MGRNet 67
2021'2025
EVA02-L
69.53
61.14
69.97
66.88
64.70
76.43
70.56
Ours
EVA02-L
71.24
62.26
71.91
68.74
66.03
77.90
71.97
11.71
11.12
1199
11.59
11.33
11.47
11.41
Table 3: Generalization test various vision- language mod-
Table 4: Comparison between stochas- tic evolution and static alignment
els. ***; only reported one decimal official results.
DINOv2S11
CLIPE9
methods.
Method
G-st
G c
6
Avg.
G-C
G- B
Avg.
REIN 74
66.4
60.4
66.1
64.3
57.4
Method
G-CC
G-+B
G-G
Avg.
SET
68.06
65.79
58.3
Baseline
68.88
59.18
70.10
66.03
58.9"
DCM
69.78
60.92
70.84
67.18
Ourx
70.10
70.01
w.o. TID
70.01
61.16
71.13
67.43
1-102
70.68
10.38
10.0
10.86
11.33
10:47
11.02
Ours
71.24
62.26
71.91
68.74
Table 5: Ablation studies on each component of the proposed
Table 6: Impact of time step T.
method. Evaluation metric is mIoU in %.
7
G-+C
Avg.
Compenent
G-43
G-;
Avg.
C- M
Avg.
2
68.97
59.83
70.35
66.38
Baseline
68.88
59.18
70.10
66.05
64.72
76.15
70.44
3
70.15
60.24
70.85
67.08
3)
DPS, SCE
69.74
61.02
71.09
67.28
64.68
76.93
70.81
4
70.82
60.77
71.06
67.55
4)
DES, SDI
70.68
61.55
71.36
67.86
65.38
77.16
71.27
5
71.24
62.26
71.91
68.74
SCE,
71.03
71.38
68.10
Results, Table reports the outcomes. The proposed method outperforms all the compared methods. Specifically, with the same EVA02-L VLM backbone, outperforms the second-best MGRNet [67] by 1.71%, 1.12% and 1.94% in mloU on the C, B, and unseen domains, respectively, when using G as the source domain. It outperforms the second-best MGRNet (67] by 1.33%, and 1.47% in mloU on the B, and M unseen domains, respectively, when using C as the source domain.
Generalization on Various Foundation Models. We further test the generalization ability of the proposed method when using other foundation models, namely, DINOv2 [51] and CLIP [59]. Since the proposed method requires the class-wise text input, we use CLIP text encoder under all the experiments. The experiments are conducted when using GTA as the source domain. Table reports the outcomes. The proposed method shows better generalization ability over these foundation models than the prior arts.
Effectiveness over Static Alignment. To validate the effectiveness of the stochastic evolution in the proposed method, we compare with two static alignment methods, namely, direct cosine matching (DCM) and without time-indexed dynamics (w.o. TID). The experiments are conducted when using GTA as the source domain. The results in outperforms both static alignment methods, indicating its contribution the overall performance.
that the stochastic evolution clearly

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Unseen Image
Groud Truth
WildNet
CMForme
FA DADA
taddm
Ours
Figure 3: Exemplar segmentation results of existing DGSS methods (WildNet [38, CMFormer (
FADA [8, tqdm (52]), and the proposed SBGen on unseen target domains.
Table 7: Impact of the optimal transport solving methods generalization performance.
Table 8: Computational cost analysis.
The GPU hour refers to one single
Method
C
G
A100 GPU hardware.
Baseline
68.88
59.18
70.10
66.05
64.72
76.15
70.44
CFM
70.73
61.05
70.64
67.47
64.81
76.57
70.69
Morell
Sinkhorn
69.62
60.18
70.57
66.79
65 65.29
76.48
70.89
Baseline
79.0
788.59M
5.60GB
Ours
71.24
62.26
71.91
68.74
66.03
77.90
71.97
5.3 Ablation Studies
On Each Component. On top of VFM and task specifcc head, the proposed method consists of three key components, namely, Domain-aware Visual Feature Selection (DFS). Stochastic Cross- Domain Evolution (SCE), and Stochastic Aggnosi- Interpolation (SDI). Table 5 leverages four experiment settings to inspect how each component impacts the overall performance. Overall, all the components contribute positively to the generalization performance. Specifically, DFS leads to up to 0.47% mloU improvement on GTAS->C/BLL (Avg.) setting. SCE further improves the performance by 0.76% mloU on the same setting. SDI brings an additional 0.58% mIoU improvement, reaching the final performance of 68.74%.
On Time Step T. Table further studies how the time step impacts the generalization performance. By default, 7 is set to 5 under all of our experiments. We further test the situation when it is 2, 3, 4, 5, and 6. The results show that the generalization performance achieves the optimal when it set to be 5. A too-small time step may lead to the under-training problem, while too-large time step may already saturate the performance but lead to more computation overhead.
Impact of Optimal Transport Solving, We compare the proposed method with Conditional Flow Matching (CFM) [69] and the commonly-used Sinkhorn transport (Sinkhorn). Table 7 shows that these methods achieve a very similar result. The proposed method shows a slight improvement, which may be explained that is more tailored for the alignment between image and text embeddings.
Computational Cost Analysis. We'r 'ee compared the proposed method with the baseline in terms of the training time, parameter number and model size under the DGSS experimental setting. Ta- shows that although the proposed method achieves an acceptable trade-off between computa- tional cost and performance improvement over the baseline. Specifically, the increase of GPU hour is 0.2 hours, the parameter number increase is 1.58 million, and the model size increase is O.O The GPU hour refers to the A100 GPU hardware.
X. The and the cross-modal Schrödinger Bridge loss. To test its impact, we conduct the experiments when is set 0.01, 0.1, 1, 10 and 100, respectively. The results in Table 9 show that when is set to 1, the generalization performance achieves the optimal. A too small (e.g.. 0.01) may not impose sufficient alignment between the Nomain-agnostie class text and the domain- speciffc visual features. too large (e.g., 100) may overwhelm the task loss, leading
On prper--aramedde
À in Eq.9 balances the impact of the task-specific loss
performance drop.
On Feature Selection Ratio K. By default, K set to 0.3 under all of our experiments. To inspect its impact, conduct the experiments when it is set to be 0, 0.1, 0.2, 0.4 and 0.5, respectively.

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Table 9: Impact of hhper--paamenn X. Evalua-
Table 10: Impact of haper- parameter
tion metric is mloU in %.
K. Evaluation metric is mloU in %.
G G-+C C
G -
G-> M
Avg.
K
G-C
G-+8
G-SG
Avg.
0.01
69,83
60.67
70.68
67.06
70.39
60.57
70.54
67.17
0.1
70.57
60.81
70.90
67.43
0.1
70.65
61.72
71.34
67.90
0.2
71.04
61.58
71.59
68.07
71.24
62.26
71.91
68.74
0.3
71.24
62.26
71.91
68.74
10
71.01
61.17
71.27
67.82
71.15
62.04
71.37
68.19
100
70.37
61.26
71.28
67.64
70.92
61.75
71.16
67.94
The results in Table show that when K is set to 0.3, the segmentation performance on unseen target domains achieves the optimal performance. A too small K (e.g., 0 and 0.1) may not select sufficient visual features to align with the domain-agnostic class features, which may under-fit the representation. A too large K (e.g. 0.4 and 0.5) may introduce more visual features that are not domain-specific, which results slight performance drop.
5.4
Qualitative Results
On Visual Prediction Maps. Fig. displays some visual prediction maps on unseen CityScapes, BDD and MAP target domains, when using GTAS as the source domain. The proposed method shows more precise per-pixt ver-pixel prediction than existing state-of-the-art DGSS methods, namely, WildNet
38
CMFormer [9), FADA [8) and tqdm [52]
On Feature Space. We further inspect if the proposed method can alleviate the domain gap between the source domain and unseen target domains over the baseline. For each sample in each domain. we extract the features before the task- specific decoder, flatten them into feature embedding, and then project the embed- ding into a latent space by t-SNE visual- ization. All the experiments are conducted under the G C, B, M setting. As shown in Fig. 5 the samples from three unseen tar- get domains are more uniformly distributed and aligned closer to the source domain by the proposed method, indicating its ef- fectiveness to mitigate the domain gap by aligning the domain domain-specific visual features to the domain-agnosse agnostic text embedding.
Baseline
Ours
BDD1OK
Mapillary
GTA5
CiyyScapss
Figure 4: t-SNE visualization. Feature embedding is extracted from the last VFM layer. Left: baseline; Right: ours.
Conclusion
This paper introduced SBGen, principled stochastic domain generalization framework, which bridges domain-specific image features and domain-agnostic textual semantics through Schrödinger Bridge dynamics. leverages textual queries to guide visual feature selection and employs time- conditioned stochastic evolution to model continuous trajectory from source domain representations to semantic targets, enabling robust generalization to unseen target domain samples. Extensive experiments show its superiority over domain generalization in both classification and segmentation.
Future Work, Limitation & Societal Impact. Future work may explore extensions multimodal generalization across more complex modalities (e.g., audio and video), and efficient approximations of high-dimensional Schrödinger Bridge dynamics. However, the proposed SBGen requires simulation of multiple-step stochastic differential equation (SDE) for each batch, which additionally adds multiple forward passes and increases the complexity over the baseline. Still, it exhibits a good trade-off between the complexity and the clear performance improvement on unseen target domains. This work can benefit domain generalization in various real-world applications, contributing to more reliable artificial intelligence systems. We do not envision its negative societal impact.
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References
[1] Sravanti Addepalli, Ashish Ramayce Asokan, Lakshay Sharma, and R Venkatesh Babu. Leveraging vision-language models for improving domain generalization in image classification. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 139222239922 2024.
Woo-Jin Ahn, Geun-Yeong Yang, Hyun-Duck Choi, and Myo-Taeg Lim. Style blind domain generalized semantic segmentation via covariance alignment and semantic consistence contrastive learning. Pro- ceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3616-3626, 2024.
[3] Devansh Arpit, Huan Wang, Yingbo Zhou, and Caiming Xiong. Ensemble of averages: Improving model selection and boosting performance in domain generalization. Advances in Neural Information Processing Systems, 35:8265- 8277, 2022.
[4] Eslam Mohamed BAKR, Liangbing Zhao, Vincent Tao Hu, Matthicu Cord, Patrick Perez, and Mohamed Elhoseiny. Toddle:Diffusion: Interactive structured image generation with cascaded Schrödinger bridge. In International Conference Learning Representations, 2024.
[5] Sara Beery, Grant Van Horn, and Pietro Perona. Recognition in terra incognita. In Proceedings of the European Conference on Computer Vision. pages 456-473, 2018.
[6] Shai Ben-David, John Blitzer. Koby Crammer, Alex Kulesza. Fernando Pereira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine learning, 19:151-175. 2010.
[7] Qi Bi, Jingjun Yi, Hao Zheng, Wei Ji, Haolan Zhan, Yawen Huang. Yuexiang Li, and Yefeng Zheng Samba: Severity aware recurrent modeling for doss-n medical image grading. Advances in Neural Information Processing Systems, 37:75829-75852, 2024.
[8] Qi Bi, Jingjun Yi, Hao Zheng, Haolan Zhan, Yawen Huang, Wei Ji, Yuexiang Li, and Yefeng Zheng. Learning frequency-adapter vision foundation model for domain generalized semantic segmentation. Advances in Neural Information Processing Systems, 37:94047-94072, 2024.
[9] Qi Bi, Shaodi You, and Theo Gevers. Learning content-enhanced mask transformer for domain generalized urban- scene segmentation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 819-827, 2024.
[10] Qi Bi, Jingjun Yi, Haolan Zhan, Wei and Gui- Xia. Learning fine- tie-ggrined domain generalization via hyperbolic state space hallucination. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 1853-1861, 2025.
[11] Qi Bi. Jingjun Yi. Hao Zheng, Wei Ji, Yawen Huang, Yuexiang Li, and Yefeng Zheng. Learning generalized medical image representation by decoupled feature queries. IEEE Transactions on Puttern Analyxis and Machine Intelligence, 2025.
[12] Junbum Cha, Sanghyuk uk Chun, Kyungjae Lee, Han-Cheol Cho, Seunghyun Park, Yunsung Lee, and Sungrae Park. Swad: Domain generalization by seeking flat minima. In Advances in Neural Information Processing Systems, volume 34, pages 22405-22418. 2021.
[13] Junhum Cha, Kyungjae Lee, Sungrae Park, and Sanghyuk Chun. Domain generalization by mutual- information regularization with pre-trained models. In European Conference on Computer Vision, pages 440-457, 2022.
[14] Bowen Cheng, Ishan Misra, Alexander G Schwing, Alexander Kirillov, and Rohit Girdhar. Masked- attention mask transformer for universal image segmentation. In Proceedings the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1290- 1100001112222220 2022.
[15] De Cheng, Zhipeng Xu, Xinyang Jiang, Nannan Wang, Dongsheng Li, and Xinbo Gao. Disentangled prompt representation for domain generalization. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 23595-23604, 2024.
[16] Junhyeong Cho, Gilhyun Nam, Sungyeon Kim, Hunmin Yang, and Suha Kwak. PromptStyler: Prompt- driven style generation for source-free domain generalization. In Proceedings of the IEFF/CVF Interna- tional Conference on Computer Vision, pages 15702-15712, 2023.
[17] Sungha Choi, Sanghun Jung, Huiwon Yun. Joanne T Kim. Seungryong Kim, and Jaegul Choo. RobustNet: Improving domain generalization in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 11580- 11590, 2021.
segmentation via instance selective whitening. In
11

---

[18] Marius Cordts, Mohamed Omran, Sebastian Ramos, Timo Rehfeld. Markus Enzweiler. Rodrigo Benenson. Uwe Franke, Stefan Roth, and Bernt Schiele. The Cityscapes dataset for semantic urban scene understand- ing. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3213-3223, 2016.
[19] Valentin De Bortoli, James Thornton, Jeremy Heng, and Amaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34: 177099 2021.
[20] Valentin De Bortoli, James Thomton, Jeremy Heng, Bernhard Schölkopf, Martin Arbel, and Arthur Gretton. Diffusion Schrödinger bridge with applications to score- based generative modeling. In Advances in Neural Information Processing Systems, volume 34, pages 17089-17105 2021.
[21] Jian Ding, Nan Xue, Gui-Song Xia, Bernt Schiele, and Dengxin Dai. HGFormer: Hierarchical grouping transformer for domain heneralized semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages
2023.
[22] Mohammad Fahes, Tuan-Hung Vu, Andrei Bursuc, Patrick Pérez, and Raoul de Charette. A simple recipe for language-guide guidded domain generalized segmentation. arXiv preprint arXiv:2311.17922, 2023.
[23] Alex Fang, Gabriel Ilharco, Mitchell Wortsman, Yuhao Wan, Vaishaal Shankar, Achal Dave, and Ludwig Schmidt. Data determines distributional robustness in contrastive language image pre-training (CLIP). In International Conference on Machine Learning, pages 6216-6233 2022.
[24] Chen Fang. Ye Xu, and Daniel N Rockmore. Unbiased metric learning: On the utilization of multiple datasets and web images for softening bias. In Proceedings of the IEEE International Conference on Computer Vision, pages 1657-1664, 2013.
[25] Yuxin Fang. Quan Sun, Xinggang Wang. Tiejun Huang, Xinlong Wang, and Yue Cao. EVA-02: A visual representation for neon genesis. arXiv preprini arXiv:2303.1 / 11331, 2023.
[26] Yuxin Fang, Wen Wang, Binhui Xie, Quan Sun, Ledell Wu, Xinggang Wang, Tiejun Huang, Xinlong Wang, and Yue Cao. EVA: Exploring the limits of masked visual representation learning at scale. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 19358-19369, 2023.
[27] Yaroslav Ganin, Exgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario March, and Victor Lempitsky. Domain-adversaria training of neural networks. Journal of Machine Learning Research, 17(59):1-35, 2016.
[28] Ming Gui, Johannes Schusterbauer, Ulrich Prestel, Pingchuan Ma, Dmytro Kotovenko, Olga Grebenkova, Stefan Andreas Baumann, Vincent Tao Hu, and Bjöm Ommer. Depthfm: Fast generative monocular depth estimation with flow matching. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 3203-3211, 2025.
[29] Ishaan Gulrajani and David Lopez-Paz. In search of lost domain generalization. In International Conference on Learning Representations, 2020.
[30] Wei Huang. Chang Chen, Yong Li, Jiacheng Li, Cheng Li. Fenglong Song, Youliang Yan, and Zhiwei Xiong. Style projected clustering for domain generalized semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3061-3071, 2023.
[31] Zeyi Huang, Andy Zhou, Zijian Ling, Mu Cai, Haohan Wang, and Yong Jae Lec, A sentence speaks a thousand images: Domain generalization through distilling CLIP with language guidance. In Proceedings of the IEEF/CVF International Conference on Computer Vision, pages 1 1685-11695,, 2023.
[32] Christoph Hümmer, Manuel Schwonberg, iangwei Zhong, Hu Cao, Alois Knoll, and Hanno Gottschalk. VI.TSeg: Simple transfer CL.IP-based vision-language representations for domain generalized semantic segmentation. arXiv preprint 2rXiv:2312.020211 2023.
[33] Yuhwan Jeong, Hoonhee Cho, and Kuk-Jin Yoon. Towards robust event-based networks for nighttime via unpaired day-to-night event translation. In European Conference on Computer Vision, pages 286-306, 2024.
[34] Chao Jia, Yinfei Yang. Ye Xia. Yi-Ting Chen, Zarana Parekh, Hieu Pham, Quoc Le. Yun-Hsure Hsuan Sung. Zhen Li, and Tom Duerig. Scaling visual and vision languaee representation learning with noisy text supervision. In International Conference Machine Learning, pages 4904 4916, 2021.
12

---

[35] Sunghwan Kim. Dae-hwan Kim, and Hoseong Kim. Texture learning domain randomization for domain generalized segmentation. Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 677-687, 2023.
[36] Gyuseong Lec, Wooseok Jang. Jinhyeon Kim, Jaewoo Jung, and Seungryong Kim. Domain generalization using large pretrained models with mixture-of caature-of--daae adaatersss. In /EEE/CVF Winter Conference on Applications of Computer Vision, pages 2259-82632 2025.
[37] Sangrok Lee, Jongseong Bae, and Ha Young Kim. Decompose, adjust, compose: Effective normalization by playing with frequency for domain generalization. Proceddigss of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1 11776-11788 2023.
[38] Suhyeon Lee, Hongje Seong, Seongwon Lee, and Euntai Kim. WildNet: Learning domain generalized semantic segmentation from the wild. In Proceedings the IEEE/CVF Conference Computer Vision and Puttern Recognition, pages 9936-9946, 2022.
[39] Byounggyu Lew, Donghyun Son, and Buru Chang, Gradient estimation for unseen domain risk minimiza- tion with pre-trained models. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4436-4446, 2023.
[40] Boyi Li, Kilian Q Weinberger, Serge Belongie, Vladlen Koltun, and Rene Ranftl. Language-driven semantic segmentation. In International Conference on Learning Representations, 2022.
[41] Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M. Hospedales. Deeper, broader and artier domain generalization. In Proceedings of the IEEF/CVF International Conference on Computer Vision, pages 5543-5551, 2017.
[42] Xuanlin Li, Yunhao Fang, Minghua Liu, Zhan I Ling, Zhuowen Tu, and Hao Su. Distilling large vision- language model with out-of-distribution generalizability. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2492-2503, 2023.
[43] Ziyue Li, Kan Ren, Xinyang Jiang, Bo Li, Haipeng Zhang, and Dongsheng Li. Domain generalization using pretrained models without fine-tuning. arXiv preprint arXiv:2203.04600 2022.
[44] Feng Liang, Bichen Wu, Xiaoliang Dai, Kunpeng Li, Yinan Zhao, Hang Zhang, Peizhao Zhang, Peter Vajda, and Diana Marculescu. Open-vocabular semantic segmentation with mask saasteddapted CLIP. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7061-7070. 2023.
[45] Puncet Mangla, Shivam Chandhok, Milan Aggarwal, Vinceth N Balasubramanian, and Balaji Krishna- murthy. INDIGO: intrinsic multimodality for domain generalization. arXiv preprint arXiv:2206.059/ 2022.
[46] Hyeonseob Nam, HyunJae Lee, Jongehan Park, Wonjun Yoon, and Donggeun Yoo. Reducing domain gap by reducing style bias. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pottern Recogmition, pages 8690-8699 2021.
[47] Gerhard Neuhold. Tobias Ollmann, Samuel Rota Bulo, and Peter Kontschieder. The mapillary vistas dataset for semantic understanding of street scenes. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 4990-4999, 2017.
[48] Thao Nguyen, Gabriel Ilharco, Mitchell Wortsman, Scwoong Oh, and Ludwig Schmidt, Quality not quan- tity: On the interaction between dataset design and robustness of CLIP. Advances in Neural Information Processing Systems, 35:21455 15:21455-21469 2022.
[49] Joshua Niemeijer, Manuel Schwonberg, Jan-Aike Termöhlen, Nico M Schmidt, and Tim Fingscheidt. Generalization by adaptation: Diffusion-based domain extension for domain- generalieed semantic segmen- tation. In Proceedings of the JEEE/CVF Winter Conference on Applications of Computer Vision, pages 2830-2840, 2024.
[50] Hongjing Niu, Hanting Li. Feng Zhao, and Bin Li. Domain-unified prompt representations for source-free domain generalization. arXiv preprint arXiv:2209, 14926, 2022.
[51] Maxime Oquab, Timothée Darcet, Théo Moutakanni, Huy Vo. Mare Szafrannecc, Vasil Khalidov, Pierre Fernandez, Daniel Haziza, Francisco Massa, Alaaeddin El-Nouby, et al. DINOv2: Learning robust visual features without supervision. arXiv preprint a7X11:2304.07199, 2023.
13

---

[52] Byeonghyun Pak, Byeongju Woo, Sunghwan Kim, Dae-hwan Kim, and Hoseong Kim. Textual query- driven mask transformer for domain generalized segmentation. In European Conference on Computer Vision, pages 37-54, 2024.
[53] Xingang Pan, Ping Luo, Jianping Shi, and Xiaoou Tang. Two at once: Enhancing learning and general- ization capacities via IBN-Net. In Proceedings of the European Conference on Computer Vision, pages 464-479, 2018.
[54] Xingang Pan, Xiaohang Zhan, Jianping Shi, Xiaoou Tang, and Ping Luo. Switchable whitening for deep representation learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 1863-1871, 2019.
[55] Stefano Peluchetti. Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling, Journal of Machine Learning Research, 24(374):1 51, 2023.
[56] Duo Peng, Yinjie Lei, Lingqiao Liu, Pingping Zhang, and Jun Liu. Global and local texture randomization for syathetic-to-real semantic segmentation. IEEE Transactions on Image Processing. 30:6394-6608, 2021.
[57] Duo Peng, Yinjie Lei, Munawar Hayat, Yulan Guo, and Wen Li. Semantic-awan aware domain generalized segmentation. In Proceedings of the IEFF/CVF Conference on Computer Vision and Pattern Recognition, pages 2594-2605, 2022.
[58] Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Saenko, and Bo Wang. Moment matching for multi-source domain adaptation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages
2019.
[59] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabricl Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, and Gretehen Krueger. Learning transferable visual models from natural language supervision. In International Conference Machine Learning, pages 8748-8763, 2021.
[60] Yongming Rao. Wenliang Zhao, Guangyi Chen. Yansong Tang, Zheng Zhu, Guan Huang, Jie Zhou, and Jiwen Lu. DenseCLIP: Language-guides dense prediction with context-aware prompting. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 18082-18091, 2022.
[61] Stephan R Richter, Vibhav Vineet, Stefan Roth, and Vladlen Koltun. Playing for data: Ground truth from computer games. In Proceedings of the European Conference on Computer Vision, pages 102-118, 2016.
[62] German Ros, Laura Sellart, Joanna Materzynska, David Vazquez, and Antonio M Lopez. The SYNTHIA dataset: A large collection of synthetic images for semantic segmentation of urban scenes. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1234-3243,22 2016.
[63] Yuge Shi, Jeffrey Seely, Philip Torr, N Siddharth, Awni Hannun, Nicolas Usunier, and Gabriel Synnaeve. Gradient matching for domain generalization. In International Conference on Learning Representations, 2012.
[64] Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion schrödinger bridge matching. Advances in Neural Information Processing Systems, 36:62183-62222 2023.
[65] Yang Shu, Xingzhuo Guo, Jialong Wu, Ximei Wang, Jianmin Wang, and Mingsheng Long. CLIPood: Generalizing CLIP to -uttof-disttibutiinn. International Conference on Machine Learning, pages 31716-31731, 2023.
[66] Vignesh Ram Somnath, Matteo Pariset, Ya-Ping Hsieh, Maria Rodriguez Martinez, Andreas Krause, and Charlotte Bunne. Aligned diffusion Schrödinger bridges. In Uncertainty in Artificial Intelligence, pages 1985-1995, 2023.
[67] Pui Tang, Xiaodong Zhang, Chunze Yang. Haoran Yuan, Jun Sun. Danfeng Shan, and Zijiang James Yang. Unleashing the power of visual foundation models for generalizable semantic segmentation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 20823-20823 2025.
[68] Reihanch Teimouri, Marta Kersten-Oeree and Yiming Xiao. CT-hased brain ventricle segmentation via diffusion Schrädinger bridge without target domain ground truths. In International Conference on Medical Image Computing and Compuder-Asiisted Intervention, pages 135-144, 2024.
[69] Alexander Tong. Nikolay Malkin, Kilian Fatras, Lazar Atanackovic, Yanlei Zhang, Guillaume Huguet, Guy Wolf, and Yoshua Bengio. Simulation- free schrödinger bridges via score and flow matching. In AISTATS, 2024.
14

---

[70] Francisco Vargas, Pierre Thodoroff, Austen Lamacraft, and Neil Lawrence. Solving Schrödinger bridges via maximum likelihood. Entropy, 23(9))1134, 2021.
[71] Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In IEEF Conference on Computer Vision and Pattern Recognition, pages 5385-5394, 2017.
[72] Gefei Wang, Yuling Jiao, Qian Xu, Yang Wang, and Can Yang. Deep generative learning via Schrödinger bridge. In International Conference on Machine Learning, pages 10794-10804, 2021.
[73] Wang, Junyang Chen, Mengzhu Wang, Hao Li, Wei Wang, Houcheng Su, Zhihui Lai, Wei Wang, and Zhenghan Chen. A closer look at classifier adversarial domain generalization. In Proceedings of the st ACM International Conference on Multimedia, pages 280-289, 2023.
[74] Zhixiang Wei, Lin Chen, Yi Jin, Xiaoxiao Ma, Tianle Liu, Pengyang Lin, Ben Wang, Huaian Chen, and Jinjin Zheng. Stronger, fewer, & superior: Harnessing vision foundation models for domain generalized semantic segmentation. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 28619-28630, 2023.
[75] Mitchell Wortsman, Gabriel Ilharco, Jong Wook Kim, Mike Li, Simon Komblith, Rebecca Roclofs, Raphacl Gontijo Lopes, Hannaneh Hajishirzi, Ali Farhadi, Hongseok Namkoong, and Ludwig Schmidt. Robust finettning of zero-shot models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 7959-7971, 2022.
[76] Jingjun Yi. Qi Bi, Hao Zheng, Haolan Zhan, Wei Ji. Yawen Huang. Shaoxin Li. Yuexiang Li, Yefeng Zheng, and Feiyue Huang. Hallucinated style distillation for single domain generalization in medical image segmentation. In International Conference on Medical Image Computing and Cosputer-Asiisted Intervention, pages 438 448, 2024.
[77] Jingjun Yi, Qi Bi, Hao Zheng, Haolan Zhan, Wei Ji, Yawen Huang, Yuexiang Li, and Yefeng Zheng. Learning
tokens for domain generalized semantic segmentation. In Proceedings of the 32nd ACM International Conference on Multimedia, pages 8159-8168, 2024.
[78] Fisher Yu, Wenqi Xian, Yingying Chen, Fangchen Liu, Mike Liao, Vashisht Madhavan, and Trevor Darrell. BDD100K: A diverse driving video database with scalable annotation tooling. arXiv preprint arXiv:1805. 04687, 2(5)66, 2018
[79] Xi Yu, Shinjae Yoo, and Yuewei Lin. CLIPceil: Domain generalization through CLIP via channel refinement and imate-text alignment, Advances in Neural Information Processing Systems, 37:4267-422 2024.
[80] Xiangyu Yue, Yang Zhang, Sicheng Zhao, Alberto Sangiovanni- Vincentelli Kurt Keutzer, and Boqing Gong. Domain randomization and pyramid consistency: Simulation tattatiioootttrtallll to-real generalization without access- ing target domain data. In Proceedings of the IEEE/CVF International Conference on Computer Vision. pages 2100-2110. 2019.
[81] Jiaming Zhang. Xingjun Ma. Xin Wang, Lingyu Qiu, Jiaqi Wang. Yu-Gang Jiang, and Jitao Sang. Ad- versarial prompt tuning for vision-language models. In European Conference on Computer Vision, pages 56-72, 72, 2024.
[82] Xin Zhang, Shixiang Shane Gu, Yutaka Matsuo, and Yusuke Iwasawa. Domain prompt learning for efficiently adapting clip to unseen domains. Transactions of the Japanese Suciety for Artificial Intelligence, B8 1, 2023.
[83] Yuyang Zhao, Zhun Zhong, Na Zhao, Nicu Sehe, and Gim Hee Lee. Style-hallucinatee dual consistency learning for domain generalized semantic segmentation. In Proceedings of the European Conference on Computer Vision, pages 335-552,22 2022.
[84] Yuyang Zhao, Zhun Zhong, Na Zhao, Nicu Sebe, and Gim Hee Lec. hlyle-haluecnntted dual consistency learning: A unified framework for visual domain generalization. International Journal of Computer Vision, 132(3):837-833 2023.
[85] Zhun Zhong, Yuyang Zhao, Gim Hee Lee, and Nicu Sebe, Adversarial style augmentation for domain generalized urban-scene segmentation. In Advances in Neural Information Processing Systems, pages 338-350, 2022.
[86] Kaiyang Zhou, Jingkang Yang, Chen Change Loy, and Ziwei Liu. Learning to prompt for vision- languaeee models. International Journal of Computer Vision, 130(9):2337-2583 23448 2022.
15

---

[87] Ziqin Zhou, Yinjie Lei. Bowen Zhang, Lingqiao Liu, and Yifan Liu. ZegCLIP: Towards adapting clip for zero-shot semantic segmentation. In Proceedings the IEEE/CVF Conference Computer Vision and Puttern Recognition, pages 11175-11185, 2023.
[88]
Chaoyang Zhu and Long Chen. A survey open-vocabulary detection and segmentation: Past, present, and future. arXiv preprint arXiv:2307.09222 09200, 2023.
[89] Xueyan Zou, Zi-Yi Dou, Jianwei Yang. Zhe Gan, Linjie Li, Chunyuan Li, Xiyang Dai, Harkirat Behl, Jianfeng Wang. Lu Yuan, Nanyun Peng, Lijuan Wang. Yong Jae Lee, and Jianfeng Gao. Generalized decoding for pixel, image, and language. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15116-15127, 2023.
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Technical Appendices and Supplementary Material
Contents
1
Introduction
2
Related work
3
Preliminaries
Methodology
Domain-aware Visual Feature Selection
4.2
Stochastic Cross-Domain Evolution
Stochastic Domain-Agnostie Interpolation
Prediction, Optimization Implementation Details
5
Experiments
Results Domain Generalization in Classification
5.2
Results on Domain Generalized Semantic Segmentation
5.3
Ablation Studies
5.4
Qualitative Results
10
6
Conclusion
10
Theoretical Analysis: Generalization Error Bound
18
Pseudo-code: Schrödinger Bridge-Guided Domain Generalization
20
More Implementation Details
20
More Feature Space Analysis
21
More Visual Prediction Results
21
17

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A Theoretical Analysis: Generalization Error Bound
In this section, we derive a generalization error bound on the unseen target domains of the proposed SBGen, and demonstrate its superiority over the generalization error bound over the VLM baseline.
We start from some key definitions. Let Po and P1 denote the source and the target feature distributions in IR Let Q denote the law of our learned stochastic evolution (Schrödinger Bridge) from Po to Pt. The risk of a classification or segmentation model w.r.t. distribution P can be defined as
(p(h) Eg [ [(h(())))]
(10)
where the task loss function e bounded in [0, 1J. and y denotes the ground truth.
The analysis will be based on the deduction of the empirical error on source domain and the expected error on target domain, defined
R (h) (target risk)
Rp((h) (source risk).
(11)
Lemma 1. Ben-David Transfer Bound. Let Po and P1 be two distributions over common feature
space Z _ RR., corresponding the source and target domains, respectively. Let h 2 + + be hypothesis, and let 8: 3/ y + [0, be bounded loss function. Then, the larget risk of h satisfies:
Rr. (h) S (k((h) + Distancerv (Po. Pi) + Ejuint:
(12)
[((h(z),))) denotes the expected risk under distribution P. Distancerv (Po, Pi) = I IIPo dPi denotes the total variation distance between the dis-
where Re(h) =
tributions, and Ejoint == minwen [Re, (h') + Rp, (h() denotes the joint risk of the optimal shared hypothesis.
Proof. Please refer to 6 for the detailed proof.
Theorem 1. Variation Distance via Schrödinger Bridge. Let Q be the solution to the Schrödinger Bridge problem between distributions Po and P1 over RRC i.e., a path measure such that Qt=o = Po and Qt= = PI. Let TP denote the reference Brownian motion with the same marginals at t = 0 and
t t= 1. Then the total variation distance between Po and P1 is bounded by the KL divergence between and F as:
Distancerv(Po- Pi)
Distancery (Qn Q1) S V (KL(QIP). /
(13)
Proof: We apply Pinsker's 's inequality to the marginals of the SB process:
Distancerv(p.2) S () for all probability measures u,
(14)
Since the Schrödinger Bridge process Q interpolates from Po to P1 over time E [0. 1]. and Qu Po. Q1 = PI. we apply Pinsker's ' inequality to the terminal marginal distributions of Q and
Because Q and IP are path measures with the same support, we have:
Distancerv (Q. Q1) S /
(15)
and by definition Qo Pa Q1 Pl, so: So:
Distancerv(Po.. Pı) S /
(16)
Theorem 2. Generalization Error Bound of Schrödinger Bridge. Let Po and P1 be the source and target feature distributions over RC Let Q9 be the path distribution induced by the Schrödinger Bridge model trained transport 2o Po to ZT Pt, and let be the Brownian reference process. Let hg be the hypothesis (e.g., classifier or segmenter) composed with the SB mapping. Then, the expected target-domain risk is bounded as:
(ho) S Rp, (ho) + AKL || PP) + Ejoint.
(17)
18

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where Rp(h) = (a ((((2),)) y)1 is the expected risk under distribution P, and Ejoint == minwen [Rp, (h') Rp, [ is the optimal joint risk over the hypothesis class.
Proof: From Lemma 1, the basic transfer bound gives:
Rp, (ha) S Rp (ho) + Distancerv (Po, 6, Pi) + Ejoint
(18)
From Theorem 1, we apply Pinsker's inequality to the SB marginals:
DistanceTv(Po,
(19)
Substituting yields:
Rp, (ho) S RPp + (I + Ejoint-
(20)
Theorem 3. Tighter Generalization Bound for Schrödinger Bridge Model. Let Po and P1 be the source and target distributions over R Let Qo denote the Schrödinger Bridge process that evolves samples from Po to P1 with reference prior P. Let Ma RRC - RRC be a deterministic baseline transport (e.g., cosine projection prompt-aligned mapping). and let P. Mø4 Po denote the induced pushforward distribution. Let i be a bounded loss function and hg. he the hypotheses composed with the SB and baseline mappings, respectively. Then the generalization error of the SB model satisfies a strictly tighter upper bound:
R2, (he) I (ha) + [ + Gjoint:
(21)
Rp. (h&) S Rr, (ha) + Distancery (Po, P(t) + Ejoint
(22)
Moreover, since Qo minimizes the entropy-regularized transport cost from Pt, and Ms induces deterministic coupling.
KLL < Distancerv(Po. P.)
(23)
unless Ms itself induces the SB- SB-optimal coupling.
The bound for the SB model is established in Theorem For the deterministic baseline, we consider the mapping 21 (zo) and define pe == Mo Po as the transformed distribution.
Proof:
Using the basic transfer bound (Lemma 1) again:
Rp (l+) < R2, (ha) + Distancerv (Po, PI) + Ejoint-
(24)
In contrast, the SB model produces a path distribution Qo over such that Qr=0= Po. Q1=1 P1. Applying Pinsker's inequality as in Theorem 2, we have:
Distancerv(Po. Pi) s V /LO
(25)
Since the Schrödinger Bridge is known to minimize the KL divergence over all couplings between Po and P1 and the deterministic map Mş induces a coupling & 21) 8(2 Mo(zo)), we have:
(T(QQoIP) < [T[[1I)))
(26)
for any reference coupling R, unless Th itself is the SB-optimal coupling.
Therefore, the divergence and the TV-based generalization bound is strictly tighter under the SB transport.
Corollary 1. Match of the generalization bound between the SB model and the Deterministic Baseline. Under the assumptions of Theorem 3. the generalization bounds of the Schrödinger Bridge model and the deterministic baseline coincide if and only if the SB induced coupling Qo corresponds to deterministic map M satisfying:
@o(2v. 21) 8(21 M (20)) Polzu),
(27)
and this map M pushes Po exactly onto P1, i.e.
M Po P1.
(28)
19

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Algorithm Schrödinger Bridge-Guided Domain Generalization
Require: Source images {ar J class text queries (2e) vision encoder &, text encoder T, time horizon I', noise scale E, number of steps L Ensure: Learned drift model Ud. prediction decoder ?>
1: Initialize U4. D
2: for each training iteration do
3: Sample mini-batch {2;, Vi ) from source domain
4: ### Domain- Visual Feature Selection ####
5:
Extract dense visual features: F; = [xx;)
6:
Encode class queries: Yr: = =
7:
Compute similarity scores Sh.w.c = (Fh,w, 9r)
8:
Select top-k features: Fs <- query-guided selection from for each feature vector 20 Fs do
9:
10:
Initialize 2t f- 20 for / = 1 L do
11:
12:
t t
13:
Sample noise & ~ I),
14:
### Stochastic Cross Domain Evolution & Domain-Agnostic Interpolation ###
15:
Update: 24 - 2t + Mo 1) At + /2e A/g
16:
end for
17:
Store final evolved feature 2r
18:
end for
19:
### Prediction Head ###
20:
Predict: Yels Useg D({=T), (g.3)
21:
Compute task losses Ssup
Estimate SB divergence (e.g.. via score matching or IPFP): CsB
23: Update parameters via V (Suup + À (L(Q))P))
end for
In this case, the KL divergence collapses to:
KL(Ce (P) = Distancerv 2 (Po. Pı),
(29)
and the generalization bounds for both models are equal:
Rp, (ha) = Rp, (ha).
(30)
We conclude this section by the following remark. The proposed SBGen, Schrödinger Bridge guided framework, not only provides a principled dynamic interpolation between source and target distributions but also holds a strietly tighter generalization error upper bound compared to the deterministic baseline.
B Pseudo-code: Schrödinger Bridge-Guided Domain Generalization
pseudo-code implementation of the proposed SBGen is given in Algorithm 1
C More Implementation Details
Following prior work [52], the same training configuration set for all types of pre-trained foundation models (e.g., CLIP, DINOv2, and EVA02), and for both domain generalization in classification and semantic segmentation.
In all the experiments, the images are cropped and resized into 512x512 pixels. The batch size set 16, with an AdamW optimizer. The initial learning rate set 1 x 10-5 5 for all the wynthetic-to-reat settings, and is set to be 1 x 10-4 for all the real-to-real settings. The learning rate backbone further scaled by 0.1. The training does not terminate after 20,000 iterations. Following [52]. a linear warm-up is applied after 1500 iterations, followed by linear decay. Some common data augmentation techniques, namely, random scaling, random cropping, random flipping, color jittering, and rare class sampling, also used.
20

---

Photo
Art
Green
Sketch
Figure 5: t-SNE visualization. Feature embedding is extracted before the decoder. Left: EVA02 baseline; Right: ours.
Domain generalization in classification. For the classification task, the image encoder and the text encoder T use the pre-trained CLIP in align with the prior DG methods. The task-specific decoder D is linear layer followed by Softmax layer.
Domain generalization in segmentation. Following prior domain generalized semantic segmen- tation methods (52 67), the default image encoder & and the text encoder T use the pre-trained EVA-02 [25]. The image encoder & can also be switched to CLIP, SAM and DINOv2 in our ex- periments. The task task-specific decoder D integrates the pixel decoder of the Mask2Former model [4]
D
More Feature Space Analysis
Fig.4 in the main text inspects whether the proposed SBGen can improve the generalization ability over the baseline, the task of domain generalized semantic segmentation (DGSS). In the supplementary material, we further inspect whether the proposed SBGen can improve the generalization ability over the baseline domain generalization in the classification task.
Specifically, extract the feature of each sample from the PACS dataset before the decoder and concatenate it into feature vector. Then, we display the feature vector of each sample regardless of the domain identity by t-SNE visualization. Feature vectors from the Photo, Art Painting, Cartoon and Sketch domains are colored in red, yellow, green and blue, respectively.
The feature space of the original baseline and the proposed SBGen is visualized in the left and right of Fig. 5 respectively. In each cluster that shares the same semantic category, the samples from uniformly distributed by the proposed SBGen, indicating its effectiveness
different domains to mitigate the domain gap.
E More Visual Prediction Results
Fig. shows more results under G -> B, M, setting. The segmentation results show that the proposed SBGen shows better pixel-wise prediction than the compared DGSS methods, especially in terms of the completeness of objects.
21

---

Unseen images
WildNet
CMrmere
FADA
tqdm
Our
FOR
STOP
STTP
ST
Figure 6: Visual segmentation results on unseen target domains under the G B, M, setting. The
proposed SBGen is compared with WildNet [38] CMFormer [9) FADA ( and tqdm [52].
22

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Please refer to our LLM policy (https://neuri)) /tter:///eerrrpp.cc///oooerrerrr /cC////feeeeeeeee/222////.. for what should or should not be described.
29


Reviews:
[{"review_id": "LUsqs56eaW", "reviewer": "Reviewer_HFke", "summary": "The paper proposes a domain alignment framework based on Schrödinger Bridge methodologies, named SBGen, for visual domain generalization. By utilizing domain-invariant textual queries, the framework recognizes domain-aware features in the visual domain and, through a stochastic process in the feature space, transports them to domain-agnostic text embedding space. As a result, source domain features are transformed into generalized representations. The framework is supported by state-of-the-art performance on two different domain generalization tasks, namely image classification and segmentation.", "questions": "* Could the authors provide more information on which optimal transport solving methodology was used to model the domain alignment procedure?\n* How costly is the bridge modeling for training the models? Please quantify total training time and FLOPs so the scalability of the framework can be gauged.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* The idea of stochastically modeling the feature space between text and image is novel and provides sufficient space for more generalized representations to occur, thoroughly supported by the extensive experimentation.\n* The paper is supported by sufficient mathematical explanation and each proposed module has a clear purpose into solving a particular problem within the task or within the framework. Ablation studies also support each module effectively and timestep ablation supports the role of the Schrödinger Bridge.\n\nWeaknesses:\n* Technical details on which methodology for Schrödinger Bridges is utilized for modeling the transport are missing. These types of problems typically require certain solving strategies, such as Conditional Flow Matching [1] or Iterative Proportinal Fitting [2], some details on which methods are utilized or if different methods have a different result would strengthen the work.\n\n[1] Tong, Alexander, et al. \"Simulation-free schr\\\" odinger bridges via score and flow matching.\" arXiv preprint arXiv:2307.03672 (2023).\n\n[2] De Bortoli, Valentin, et al. \"Diffusion schrödinger bridge with applications to score-based generative modeling.\" Advances in Neural Information Processing Systems 34 (2021): 17695-17709.", "was_revised": true, "final_justification": "I will retain my rating of 5, given that the metholodogy has sufficient improvements on the baselines and also touches a novel idea in generalization and modeling. Authors have explained that target data is not used, which I think is significant as well.", "rebuttal": "We appreciate the reviewer for the positive comments on the technical novelties, extensive experiments and module design. The below is a point-by-point response to clarify the optimal transport method and the computational cost.\n\n**W1 \\& Q1**: Details on the optimal transport solving method.\n\n**R**: Thanks for your insightful question. Concretely, our method is an entropy-regularized optimal transport method by solving the stochastic differential equation, which is different from existing methods. The steps are described as follows.\n\nFirst, we initialize the image feature $\\boldsymbol{z}_0$ by the image encoder and the proposed Domain-aware Visual Feature Selection component, which is used as the starting point of the optimal transport. \n\nThen, the image feature $\\boldsymbol{z}\\_{t}$ of each step of the stochastic evolution is updated by the time embedding $\\gamma(t)$ and the drift term, computed as\n$$ \\text{drift} = \\text{mlp}( \\boldsymbol{z}\\_{t} + \\boldsymbol{q}\\_{c} + \\gamma(t))$$\nwhere $\\boldsymbol{q}\\_c$ is the text embedding, and $\\text{mlp}$ denotes the drift network.\n\nThen, in every time step $t$, $\\boldsymbol{z}\\_{t}$ is updated by the stochastic differential equation.\nSpecifically, based on the noise term (generated by a normal distribution $\\mathcal{N}(0, I)$ to mimic the random permutation) and the drift term $\\text{drift}$, it is computed as\n$$ z_{t+1} = z_{t} + \\text{drift} \\times dt + \\sqrt{2 \\epsilon dt} \\times \\mathcal{N}(0, I)$$\nwhere $dt$ is the time interval, and $\\epsilon$ is a regularized parameter to control the intensity of the noise.\n\nThen, we compute the $K-L$ divergence of the drift noise's sum square, which will be later used as a regularization to smooth the transport path. It is computed as\n$$ \\text{KL term} = \\frac{ (\\text{drift}^2) \\cdot dt }{4 \\epsilon}$$\n\nAfter $T$ time steps, the final image feature $z_T$ is aligned with the text embedding $\\boldsymbol{q}_c$ by the mean square error (MSE) loss, given by\n$$ \\text{MSE loss} = \\text{MSE}( z_T, \\boldsymbol{q}_c ) $$\n\nTheoretically, the final error regularized optimal transport (EROT) loss $\\text{EROT loss}$ is a weighted sum of the MSE loss and the $K-L$ loss, given by\n$$ \\text{EROT loss} = \\text{MSE loss} + \\epsilon \\times \\text{KL loss}, $$\nwhere the $K-L$ loss is an average of all the $T$ time steps, given by\n$$ \\text{KL loss} = \\frac{1}{T} \\sum_{t=1}^{T} \\text{KL term}_t $$\n\nNotice that, the computation of the $K-L$ loss is approximated in practical, where the approximation details are mentioned in L208-209 in the main text. \n\nWe will enrich these details in the revised main text and supplementary material.\n\nTo further check if different methods have a different result, we further compare the proposed method with the method in [1] and another commonly-used method, namely, Sinkhorn.\nThe attached results show that these methods achieve a very similar result. The proposed method shows a slight improvement, which may be explained that it is more tailored for the alignment between image and text embeddings. \n\n| Method | G->C | G->B | G->M | Avg. | C->B | C->M | Avg. |\n| ------ | ----- | ------- | ------ | ------ | ----- | ------- | ------ |\n| Baseline | 68.88 | 59.18 | 70.10 | 66.05 | 64.72 | 76.15 | 70.44 |\n| [1] | 70.73 | 61.05 | 70.64 | 67.47 | 64.81 | 76.57 | 70.69 |\n| Sinkhorn | 69.62 | 60.18 | 70.57 | 66.79 | 65.29 | 76.48 | 70.89 |\n| Ours | **71.24** | **62.26** | **71.91** | **68.74** | **66.03** | **77.90** | **71.97** |\n\nWe will add these results and discussions in the revised version.\n\nSince [2] does not have publicly available source code, we will cite and discuss it in the related work.\n\n\n**Q2**: Clarify the computational cost.\n\n**R:** Per your suggestion, we've accordingly compared the proposed method with the baseline in terms of the training time, GFLOPs and model size under the DGSS experimental setting.\nThe attached table shows that although the proposed method achieves an acceptable trade-off between computational cost and performance improvement over the baseline. \nSpecifically, the increase of GPU hour is 0.2 hours, the parameter number increase is 1.58 million, and the model size increase is 0.01GB. The GPU hour refers to the A100 GPU hardware.\n\n| Method | Training Time | Parameter Number | Model Size | GFLOPs | G->C | G->B | G->M | Avg. |\n| ------ | ----- | ------- | ------ | ------ | ----- | ------- | ------ | ----- | \n| Baseline | 79.0 GPU hours |\t788.59M | 5.60GB | ~4.73 | 68.88 | 59.18 | 70.10 | 66.05 |\n| Ours | 79.2 GPU hours\t| 790.17M | 5.61GB | ~4.74 | **71.24** | **62.26** | **71.91** | **68.74** |\n\nShould you have further comments or questions, we are happy to address during the author-reviewer discussion stage."}, {"review_id": "bpk3cP9shX", "reviewer": "Reviewer_FGsB", "summary": "This paper tackles the challenge of domain generalization (DG) in computer vision by leveraging vision-language models (VLMs) for improved robustness to unseen domains.  SBGen is introduced as a novel framework that formulates cross-modal alignment as a Schrödinger Bridge (SB) problem – essentially an entropy-regularized optimal transport between the source-domain image feature distribution and the target textual semantic distribution.  The approach consists of three key stages: (1) Text-guided domain-aware feature selection, which isolates the most semantically relevant image tokens per class by selecting top-$k$ spatial features that best match the class’s text embedding.  This focuses the model on domain-invariant content and removes background or domain-specific noise.  (2) Stochastic cross-domain evolution, which learns a time-dependent drift function to gradually transform the selected image features towards the text embedding distribution via a stochastic differential equation (SDE) – effectively simulating the Schrödinger Bridge dynamics between modalities.  (3) Stochastic domain-agnostic interpolation, which explicitly models and utilizes the continuous trajectory of intermediate features produced by the SDE, rather than only the end-points, to ensure a smooth semantic evolution and robust alignment.  After this evolution, the evolved features (now closer to domain-agnostic semantics) are injected back into the full image feature map and used for prediction by the task head.", "questions": "How did you choose the number of top-$k$ features for the domain-aware selection stage, and how sensitive is the performance to this choice?", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\nThe paper introduces a novel perspective on domain generalization by using the Schrödinger Bridge formulation to connect image and text feature distributions. This is a fresh idea that extends beyond conventional static alignment.\n\nWeaknesses:\n\n1. The SBGen pipeline adds non-trivial complexity to the training process. It requires simulating a multi-step SDE for each batch, which involves multiple forward passes through the drift network and adding noise. \n\n2. Some design choices in SBGen are not exhaustively explored in the paper. For instance, the number of top-$k$ features selected per class (domain-aware selection) is presumably a hyperparameter – the method’s performance could depend on this, especially for segmentation where $k$ might decide how much of each object is covered by the stochastic alignment. The paper does not mention how $k$ was chosen or tuned. Am I missed?\n\n3. The evaluation, while broad, still focuses on certain standard scenarios. For example, in segmentation, they tried single-source generalization (one source domain at a time). An interesting scenario could be multi-source DG for segmentation. SBGen in principle could handle multiple source domains by just pooling them into $P_S$, but whether that offers further gains or poses challenges (like needing to balance multiple domain biases) wasn’t explored.\n\n4. The ablation studies provided are thorough for segmentation, but the paper does not explicitly show ablation results for the classification benchmarks. It would strengthen the paper to know if the three components (DFS, SCE, SDI) similarly contribute on classification accuracy.", "was_revised": true, "final_justification": "I will maintain my score. My primary reason is that the contribution of this line of work to the community appears limited, particularly since many generalization issues can be addressed by scaling model size and training data. Moreover, it remains unclear whether the proposed method is applicable at larger scales.", "rebuttal": "We appreciate the reviewer for acknowledging its novelties over existing methods to align the text and image features. The below is a point-by-point response to clarify the computational cost, hyper-parameter selection and to provide more experimental analysis.\n\n**Q1**: Computational Cost.\n\n**R**: Thanks for your insightful comments. We definitely agree with the reviewer that, as the proposed method simulates a multi-step SDE. To inspect its impact, we compute the training time, GFLOPs and model size between the proposed method and the baseline. \nThe outcomes in the attached table show that, the proposed method only moderately increases the computation complexity but leads to a clear performance improvement, achieving a good trade-off between them.  \nSpecifically, the increase of GPU hour is 0.2 hours, the parameter number increase is 1.58 million, and the model size increase is 0.01GB. The GPU hour refers to the A100 GPU hardware.\n\n| Method | Training Time | Parameter Number | Model Size |  G->C | G->B | G->M | Avg. |\n| ------ | ----- | ------- | ------ | ----- | ------- | ------ | ----- | \n| Baseline | 79.0 GPU hours |\t788.59M\t| 5.60GB | 68.88 | 59.18 | 70.10 | 66.05 |\n| Ours | 79.2 GPU hours\t| 790.17M | 5.61GB | **71.24** | **62.26** | **71.91** | **68.74** |\n\n**Q2**: Sensitivity of hyper-parameter $k$.\n\n**R**: We are sorry for the formatting of the supplementary material, which is less clear than intended.\nSpecifically, we've conducted an experiment to test how the selection of $K$ impacts the performance of domain generalized semantic segmentation.\nThe attached table shows that, a too small $K$ (\\textit{e.g.}, 0.1) may not be able to select sufficient visual features that are domain-specific for the alignment between the domain-agnostic class text and the domain-specific visual features, which may under-fit the generalization representation. \nA too large $K$ (\\textit{e.g.}, 0.4 and 0.5) may introduce more visual features that are not domain-specific in the representation learning, which may lead to over-fit and result in a slight performance drop. \n\n| $K$ | G->C | G->B | G->G | Avg. |\n| ------ | ----- | ------- | ------ | ----- | \n| 0.1 | 70.57 | 60.81 | 70.90 | 67.43 |\n| 0.2 | 71.04 | 61.58 | 71.59 | 68.07 | \n| 0.3 | **71.24** | **62.26** | **71.91** | **68.74** |\n| 0.4 | 71.15 | 62.04 | 71.37 | 68.19 |\n| 0.5 | 70.92 | 61.75 | 71.16 | 67.94 |\n\n**Q3**: Performance on multi-source DG for segmentation.\n\n**R**: Thanks for your valuable suggestion. We've accordingly added an experiment setting of domain generalized semantic segmentation, where two synthetic domains are jointly used for training. The attached results~show that, compared with the cases when simply using either one of the source domains, SBGen leads to further gains by pooling them together.\n\n| Method | Source Domain | ->C | ->B | ->M | Avg. |\n| ------ | ----- | ------- | ------ | ----- | ----- | \n| Baseline | G | 68.88 | 59.18 | 70.10 | 66.05 |\n| Baseline | S | 57.99 | 52.43 | 54.87 | 55.10 |\n| Baseline | G+S | **68.06** | **60.57** | **70.95** | **66.53** |\n| Ours | G | 71.24 | 62.26 | 71.91 | 68.74 |\n| Ours | S | 59.84 | 54.69 | 57.02 | 57.18 |\n| Ours | G+S | **72.81** | **64.15** | **73.28** | **70.08** |\n\n**Q4**: Ablation Study on Classification.\n\n**R**: Per your suggestion, we've accordingly added an ablation study on each component, tested on the classification benchmarks.\nAs attached, each of the three components shows a similar contribution on classification accuracy.\n\n| DFS | SCE | SDI | PACS | VLCS | OfficeHome | DomainNet | TerraInc | Avg. |\n| ------ | ----- | ------- | ------ | ----- | ----- | ------ | ----- | ----- | \n| $\\times$ | $\\times$ | $\\times$ | 95.5 | 82.4 | 86.5 | 59.6 | 58.3 | 76.5 |\n| $\\checkmark$ | $\\times$ | $\\times$ | 96.3 | 83.9 | 87.4 | 61.8 | 59.5 | 77.8 |\n| $\\checkmark$ | $\\checkmark$ | $\\times$ | 96.9 | 85.2 | 88.6 | 63.3 | 61.2 | 79.0 |\n| $\\checkmark$ | $\\times$ | $\\checkmark$ | 97.1 | 86.0 | 89.5 | 63.9 | 62.6 | 79.8 |\n| $\\checkmark$ | $\\checkmark$ | $\\checkmark$ | **97.4** | **86.7** | **89.9** | **64.4** | **63.5** | **80.4** |\n\nShould you have further comments or questions, we are happy to address during the author-reviewer discussion stage."}, {"review_id": "JpIL57GPTA", "reviewer": "Reviewer_WcC6", "summary": "This paper proposes a method for domain generalization by learning a Schrödinger Bridge between visual and textual embeddings. The approach is designed to work for both classification and segmentation tasks. The authors also provide a theoretical analysis of the generalization error bound, supported by supplementary materials.", "questions": "See Weaknesses.", "limitations": "See Weaknesses.", "rating": 2, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "*Strengths*\n\n- The paper presents extensive experimental results across multiple tasks.\n\n-  I have reviewed the supplementary materials and find that the main theoretical analysis regarding the generalization error bound appears to be technically sound.\n\n*Weaknesses*\n\nSeveral serious concerns remain about the problem setting and experimental details, which require further clarification. \n\n- The paper seems to conflate the concepts of robustness and generalization, which are related but distinct; the authors are encouraged to use these terms more precisely and consistently throughout the paper.\n\n- As shown in Figure 2, the target domain images seem to be involved in the Schrödinger Bridge learning process. **If this figure accurately reflects the methodology, it contradicts the standard domain generalization setting where target domain data should not be accessible during training.** This potential data leakage casts serious doubt on the reported improvements. If, on the other hand, the figure is inaccurate, it reflects a lack of clarity and attention to detail, suggesting the paper requires major revision to reach publication quality.\n\n- The notational system is quite disorganized. For instance, the symbol $T$ is used to denote at least three different concepts: the deterministic transport map (e.g., Line 108), the target domain in $P^T$, and the number of steps or time horizon (e.g., Line 186). Furthermore, Figure 2 uses discrete time indices “$t = 1, 2, ..., T$”, whereas the rest of the paper repeatedly describes $t \\in [0, 1]$ as a continuous interval. This inconsistency is confusing and undermines the clarity of the presentation. A thorough proofreading and cleanup of the notation is strongly recommended.\n\n- In the checklist, the authors claim that limitations are discussed in the conclusion. However, I could not find any explicit discussion of the method’s limitations in that section.\n\n- Around line 208, the KL loss is computed as an accumulation from step 0 to $T-1$. This iterative formulation likely incurs significant computational overhead. However, the paper does not provide any analysis or discussion regarding its computational cost or efficiency. Including such analysis would strengthen the work.\n\n- The proposed method involves stochastic processes and may be sensitive to random seed choices. Reporting results averaged over multiple seeds or showing variance across runs would help assess the method's stability.\n\n- Around line 291, the authors state that “as shown in Fig. 4, the samples from three unseen target domains are more uniformly distributed and aligned closer to the source domain by the proposed method.” However, from my perspective, the visualization alone does not convincingly support this claim. This might be due to the fact that segmentation datasets often contain diverse object types within a single image, making the domain gap difficult to visualize meaningfully. It is possible that visualizations based on embeddings from classification tasks might better illustrate the claimed alignment effect.\n\n- In Supplementary Algorithm 1, the loss term $\\mathcal{L}_{SB}$ is used but never explicitly defined.\n\n\n\nOverall, though theoretical analysis is provided, the paper lacks careful writing, and significant revisions are needed. Therefore, I recommend rejection at this stage.", "was_revised": true, "final_justification": "As I have addressed, the current version of this paper may need major revisions regarding those flaws and require another round of review. I will retain my rating.", "rebuttal": "We appreciate the reviewer for the positive comments on the theoretical analysis and extensive experiments. The below is a point-by-point response to clarify the notations, experimental settings and other presentation issues.\n\n**Q1**: Confusion between *robustness* and *generalization*.\n\n**R**: We are sorry for the confusion. We will replace the use of *robustness* to *generalization* throughout the paper for consistency.\n\n**Q2**: Clarity of Fig.2. Does it follow the domain generalization assumption?\n\n**R**： We are terribly sorry for the presentation of Fig.2, which is less clear than intended to properly demonstrate the domain generalization setting.\nTo clarify, as detailed in the experimental section, the data from the target domain(s) is only involved in the inference stage for testing, not fed into the model in the training stage.It follows the basic assumption of the domain generalization. We will use: \n\n- put the data from the source/target domain into a box of solid/dashed line, respectively; \n\n- use the solid/dashed arrows of the data propogation to represent the input in the training/inference stage, respectively, where only source/target domain is involved.\n\n**Q3**： Notation system. $T$ has been defined three times.\n\n**R**: Many thanks for your careful checking, and we are sorry for the confusion in the notation system. \nFor clarity and for correctness:\n\n- We will only use $T$ to denote the time indices.\n\n- Instead, we will use $\\mathcal{M}$ to denote the deterministic transport map to distinguish. \n\n- We will replace $P^T$ by $P^U$ to denote the distribution of the unseen target domain.\n\n- We will remove the continuous interval ($t \\in [0,1]$) and only keep the discrete interval for consistency. \n\n**Q4**: Missing explicit limitation discussion.\n\n**R**: Thanks for your careful checking and we are sorry for the ignorance. We will add the following text to explicitly discuss the limitation of the proposed method, written as:\n\n*However, the proposed SBGen requires a simulation of multiple-step stochastic differential equation (SDE) for each batch, which additionally adds multiple forward passes and increases the complexity over the baseline. Still, it exhibits a good trade-off between the complexity and the clear performance improvement on unseen target domains.*\n\n**Q5**: Computational cost analysis.\n\n**R**: Per your suggestion, we've accordingly compared the proposed method with the baseline in terms of the training time, parameter number and model size under the DGSS experimental setting. The attached table~shows that although the proposed method achieves an acceptable trade-off between computational cost and performance improvement over the baseline. Specifically, the increase of GPU hour is 0.2 hours, the parameter number increase is 1.58 million, and the model size increase is 0.01GB. The GPU hour refers to the A100 GPU hardware. \n\n| Method | Training Time | Parameter Number | Model Size |  G->C | G->B | G->M | Avg. |\n| ------ | ----- | ------- | ------ | ----- | ------- | ------ | ----- | \n| Baseline | 79.0 GPU hours | 788.59M | 5.60GB | 68.88 | 59.18 | 70.10 | 66.05 |\n| Ours | 79.2 GPU hours | 790.17M | 5.61GB | **71.24** | **62.26** | **71.91** | **68.74** |\n\n**Q6**: Stability, ideally justified by the variance of multiple runs.\n\n**R**: Many thanks for your valuable suggestion. \nWe would like to kindly remind the reviewer that, the result is an average of three independent runs for domain generalized segmentation. We therefore report the standard deviation of these three runs from both the proposed method and the baseline.\nAs shown below, the proposed method shows stability across independent runs. \n\n| Method | G->C | G->B | G->M | C->B | C->M |\n| ------ | ----- | ------- | ------ | ----- | ------- |\n| Baseline | 68.88$\\pm$0.78 | 59.18$\\pm$0.62 | 70.10$\\pm$0.56 | 64.72$\\pm$0.32 | 76.15$\\pm$0.67 |\n| Ours | **71.24**$\\pm$0.76 | **62.26**$\\pm$0.45 | **71.91**$\\pm$0.58 | **66.03**$\\pm$0.42 | **77.90**$\\pm$0.46 |\n\n**Q7**: t-SNE visualization in classification is more meaningful than segmentation.\n\n**R**: Thanks for your valuable suggestion. Unfortunately this year NeurIPS does not allow any PDF attachment or external link.\nAfter consulting the program chair, we describe the visualization method and the result.\nThe classification experiments are conducted when PACS is used as the unseen target domain.\nThe features from the last layer of the encoder are extracted and projected into the latent space for t-SNE visualization. \nIt is observed that the per-class samples from our method are more uniformly distributed than the baseline.\nThe updated figure will be incorporated into the revised version. \n\nShould you have further comments or questions, we are happy to address during the author-reviewer discussion stage."}, {"review_id": "tQuQyiM8nt", "reviewer": "Reviewer_3EAi", "summary": "This paper addresses the limitations of static alignment strategies in vision-language domain generalization by proposing a novel framework, SBGen, which models the alignment between domain-specific visual features and domain-agnostic textual semantics as a stochastic evolution process via a Schrödinger Bridge formulation. The method leverages a time-indexed stochastic differential equation to progressively transform visual features toward semantic anchors. The proposed framework is evaluated on standard domain generalization benchmarks for both classification and segmentation tasks and achieves consistent improvements over strong baselines. While the approach is theoretically well-grounded and empirically effective, the experimental validation could be further strengthened by isolating the contribution of the stochastic evolution process.", "questions": "1. Can the authors include an ablation study that replaces the stochastic evolution with a static alignment baseline, such as direct cosine matching or prompt learning without time-indexed dynamics? This would clarify the contribution of the stochastic modeling to the overall performance.\n2. Could the authors provide an analysis of the stochastic evolution paths, such as visualizations of intermediate feature distributions or quantitative metrics like Wasserstein distance over time? This would help assess whether the feature trajectories are smooth, stable, and semantically meaningful.\n3. Given that the training process involves stochastic modeling, is it possible to extend the inference stage to quantify uncertainty, for example through sampling multiple evolution paths? Such an addition could enhance the robustness of predictions under domain shift.", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "The motivation of the paper is clear and well-justified. The authors identify a key limitation in existing vision-language DG methods—the reliance on static alignment—which may fail under severe distribution shifts. In response, the paper proposes modeling the alignment as a stochastic semantic evolution, providing a more flexible and theoretically grounded solution. The experimental evaluation is extensive, covering a wide range of datasets and tasks. The method achieves strong empirical results, outperforming state-of-the-art baselines in both image classification and semantic segmentation. The paper also includes ablations on hyperparameters and feature selection, adding robustness to the empirical claims. However, a significant limitation lies in the lack of direct ablation isolating the impact of the proposed stochastic evolution process. While the ablations adjust loss balancing factors and feature selection, they do not explicitly compare the stochastic evolution against a purely static alignment baseline. This omission makes it difficult to rigorously assess how much of the observed improvement is due to the dynamic modeling itself. Additionally, the stochastic process is not analyzed in terms of stability, convergence, or intermediate behavior, and inference remains deterministic without leveraging uncertainty modeling, leaving some aspects of the framework under-explored.", "was_revised": true, "final_justification": "The authors provided clear, direct, and convincing responses. Based on their effective rebuttal and my continued recognition of the paper's core strengths, I will maintain my initial positive assessment and rating.", "rebuttal": "We appreciate the reviewer for the positive comments on the clear motivation, technical novelties, extensive experiments and strong performance. The below is a point-by-point response to provide more ablation studies, more detailed analysis on stochastic evolution paths and uncertainty modeling.\n\n**Q1**: More ablation studies on static alignment baselines, such as direct cosine matching or prompt learning without time-indexed dynamics.\n\n**R**: Per your suggestion, we've added the experiments when replacing the stochastic evolution with two static alignment methods, namely, direct cosine matching (DCM) and without time-indexed dynamics (w.o. TID).\nThe experiments are conducted when using GTA as the source domain.\nThe results in the attached table~show the stochastic evolution clearly outperforms both two methods, indicating its contribution to the overall performance.\n\n| Method | G->C | G->B | G->M | Avg. |\n| ------- | ------ | ----- | ------- | ------ | \n| Baseline | 68.88 | 59.18 | 70.10 | 66.05 |\n| DCM | 69.78 | 60.92 | 70.84 | 67.18 |\n| w.o. TID | 70.01 | 61.16 | 71.13 | 67.43 |\n| Ours | **71.24** | **62.26** | **71.91** | **68.74** |\n\n**Q2**: More analysis of the stochastic evolution paths, such as visualizations of intermediate feature distributions or quantitative metrics like Wasserstein distance over time?\n\n**R**: Many thanks for your valuable suggestion. \nUnfortunately, this year NeurIPS does not allow the submission of visualization in 1-page PDF or external links.\nAfter consulting the program chair, we are allowed to describe the visualization method and the outcome in text.\nWe extract the intermediate feature embeddings from both the text embedding and image embedding over different time steps during the stochastic evolution.\nWe then display them in the feature space by t-SNE visualization. \nIt is observed that the intermediate features from the text embedding and the image embedding are gradually more uniformly mixed, indicating it is stable and domain meaningful.\nWe also compute the Wasserstein distance over time. The results attached below~show that the Wasserstein distance metric is becoming smaller over time, indicating the feature trajectories are converging, smooth and stable.\nBoth results are promised to be added in the revised manuscript. \n\n| Time $T$ | 1 | 2 | 3 | 4 | 5 | \n| ------- | ------ | ----- | ------- | ------ | ------ | \n| Distance | 1.3367 | 1.3046 | 1.2751 | 1.2409 | 1.2210 |\n\n**Q3**: Is it possible to extend the inference stage to quantify uncertainty, for example through sampling multiple evolution paths? Such an addition could enhance the robustness of predictions under domain shift.\n\n**R**: To extend the robustness of inference under domain shift, we leverage the inherent stochasticity of the Schrödinger Bridge evolution to model uncertainty in the prediction stage. Specifically, we simulate multiple independent realizations of the stochastic process, forming a Monte Carlo ensemble of evolved features.\n\n\n\n**Monte Carlo Evolution Paths**.  Given the initial visual tokens  $\\{\\boldsymbol{z} _{0}\\} \\sim P _{0}$, we simulate $M$ independent evolution paths by sampling distinct noise trajectories $\\{\\xi _{n}^{(m)}\\} _{n=0}^{T-1}$, where for each path we have $m = 1, \\ldots, M$. The terminal state of each path is computed via:\n\n$$\n\\boldsymbol{z} _{t _T} ^{(m)} = \\boldsymbol{z} _{t _0} + \\sum _{n=0}^{T-1} f _\\theta(z _{t _n}^{(m)}, t _n) \\Delta t + \\sum _{n=0}^{T-1} \\sqrt{2\\varepsilon \\Delta t} \\xi _n^{(m)},\n$$\nwhere each $\\boldsymbol{z} _{t_n}^{(m)}$ is recursively computed from its own noise realization.\n\n\n**Prediction Mean and Uncertainty**.  We perform prediction using the decoder $\\mathcal{D}$ over each sample path:\n\n$$ \\hat{y}^{(m)} = \\mathcal{D}(\\mathcal{F}'^{(m)}, \\{\\boldsymbol{q}_{c}\\}), $$\n\nand estimate the predictive mean and variance:\n\n$$ \\bar{y} = \\frac{1}{M} \\sum_{m=1}^{M} \\hat{y}^{(m)}, $$\n\n$$ \\mathbb{V}(y) = \\frac{1}{M} \\sum_{m=1}^M (\\hat{y}^{(m)} - \\bar{y})^2.$$\n\nThe variance $\\mathbb{V}(y)$ quantifies the epistemic uncertainty, providing a confidence-aware prediction strategy across domains.\n\nAs a result, the variance $\\mathbb{V}(y)$ is used to report the uncertainty score.\nThe lower variance refers to more confident predictions and less uncertainty across domains.\nWe report the uncertainty score on the domain generalized semantic segmentation, when we conduct 5, 10, 15, 20, 25 and 30 Monte Carlo runs.\nThe attached table~shows that the proposed method does not show clearly higher variance over the baseline, indicating it is robust to domain shift. \n\n| Run Number | 5 | 10 | 15 | 20 | 25 | 30 |\n| ------- | ------ | ----- | ------- | ------ | ------ |  ------ | \n| Baseline | 4.9439e-05 | 3.7132e-05 | 2.8410e-05 | 1.5496e-05 | 8.1127e-06 | 7.6321e-06 |\n| Ours (full) | 4.0271e-05 | 3.5851e-05 | 2.3389e-05 | 1.1906e-05 | 7.1032e-06 | 6.7117e-06 |\n\nShould you have further comments or questions, we are happy to address during the author-reviewer discussion stage."}]

Author rebuttal:

Meta-review: The paper proposes an approach for visual domain generalization called learning cross-modal Schrodinger bridge. The idea is to leverage textual features to identify domain-specific features and then transport them to domain-agnostic text embedding space via a stochastic process. This provides source domain features that are generalizable.
Among the main strengths identified by the reviewers were:
- idea of stochastically transporting and modeling is interesting
- method is explained well with theoretical analyses and empirical validations
- novel idea of using Schrodinger bridges, not explored in DG literature before
- well-justified idea with clear experimental results

Some of the important concerns from the reviewers were:

- Missing technical details on exact methodology for Schrödinger Bridges 
- non-trivial complexity of Schrodinger bridge learning
- issues with notations and main figure presentation
- limited contribution of the work, as many generalization issues can be handled with scaling data and/or model
- ablation study on replacing the dynamic Schrodinger component with static alignment one

In the post-rebuttal discussion phase, two reviewers remain satisfied with the author's responses. However, two reviewers stayed negative by mentioning aspects of data and model scaling and paper presentation issues. The authors do provided further experiments to address scaling issue and how to address the presentation issues. AC believes that the method is novel and is well-supported by both theoretical and experimental validation (as acknowledged by all reviewers), the remaining concerns are of minor nature and should not warrant rejection. The presentation issues are limited to use of a symbol at few places and clarification in a figure. Therefore, the decision is to accept the paper.
Decision: Accept (poster)