System-Embedded Diffusion Bridge Models
Authors: Bartlomiej Sobieski
Matthew Tivnan
Yuang Wang
Siyeop yoon
Pengfei Jin
Dufan Wu
Quanzheng Li
Przemyslaw Biecek
Venue: NeurIPS 2025 poster ()
Keywords: diffusion
generative
inverse
problems
image
translation
restoration
linear
system
bridge
Primary area: deep_learning
TL;DR: We propose a new diffusion bridge method that embeds the linear measurement system into the SDE coefficients, enabling direct use of system information and improving performance across diverse inverse problem benchmarks.

Abstract: Solving inverse problems—recovering signals from incomplete or noisy measurements—is fundamental in science and engineering. Score-based generative models (SGMs) have recently emerged as a powerful framework for this task. Two main paradigms have formed: unsupervised approaches that adapt pretrained generative models to inverse problems, and supervised bridge methods that train stochastic processes conditioned on paired clean and corrupted data. While the former typically assume knowledge of the measurement model, the latter have largely overlooked this structural information. We introduce System-embedded Diffusion Bridge Models (SDBs), a new class of supervised bridge methods that explicitly embed the known linear measurement system into the coefficients of a matrix-valued SDE. This principled integration yields consistent improvements across diverse linear inverse problems and demonstrates robust generalization under system misspecification between training and deployment, offering a promising solution to real-world applications.

System-Embeden Diffusion Bridge Models
Bartlomiej Sobieski .Matthew Tivnan 12.3, Yuang Wang2.3, Singe Yoon 2.3 Pengfei 2 Dufan Wu Quanzheng Li Przemyslaw Biecek 1.4
University of Warsaw, Harvard University,
Massachusetts General Hospital, "Warsaw University of Technology
Abstract
Solving inverse problems-rrecovrring signals from incomplete or noisy mea- surements-is fundamental in science and engineering. Score-based generative models (SGMs) have recently emerged as powerful framework for this task. Two main paradigms have formed: unsupervised approaches that adapt pretrained generative models to inverse problems, and supervised bridge methods that train stochastic processes conditioned on paired clean and corrupted data. While the former typically assume knowledge of the measurement model, the latter have largely overlooked this structural information. We introduce jystem-embedded Diffusion Bridge Models (SDBs), new class of supervised bridge methods that explicitly embed the known linear measurement system into the coefficients a matrix-valued SDE. This principled integration yields consistent improvements across diverse linear inverse problems and demonstrates robust generalization under system misspecification between training and deployment, offering promising solution to real-world applications.
Introduction
Restoring data from corrupted or incomplete measurements a fundamental task in science and engineering (Abbott et al. 2016 Akiyama al. 2019], commonly referred
inverse problem. Its linear formulation plays a central role in practical domains such as signal processing and medical imaging [Bertero et al. 1985 Candes et al. 2006). The emergence of deep learning has significantly advanced this field, enabling major scientific breakthroughs [Ongie et al. 2220]. Since the work of Song et [2021b], the scientific community has increasingly adopted score-based generative models (SGMs), also known as diffusion models, to tackle inverse problems. Two key directions have emerged: one adapts models pretrained for image synthesis conditional generation; the other, often called bridge methods, trains problem-specific models grounded in stochastic differential equations (SDEs), assuming access paired samples of clean data and corresponding measurements.
While pretrained models often assume access to a known linear measurement system, bridge methods have primarily focused developing ceneral-purpose approaches without leveraging such structural information. However, in many real-world settings- as CT or MRI reconstruction- linear measurement process is known priori, and datasets frequently contain paired examples of clean and degraded data.
To address this gap, we propose System-embedded Diffusion Bridge Models (SDBs), novel method that incorporates the system response and noise covariance directly into the coefficients of a matrix- valued SDE. By embedding this measurement-sysse system knowledge, SDB achieves considerable perfor- mance gains across diverse linear systems varying complexity, demonstrated through three distinct instantiations. Furthermore, conduct extensive study of SDB's generalization under system
"orresponning author at b. sobieskieeeee.. Cuv. euw.edu. pl
I 'qqual contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Forward SDB
Pxx, Gydw;
y= y=Ax; Eie
Null space
Range space
Reverse SDE
dx; Fix, C,G. V,, logp/x;)(t C,dw,
A
Figure I: SDB learns a diffusion bridge between pseudoinverse reconstructions and clean samples by embedding the measurement system into the coefficients of the linear SDE, allowing the score model to distinguish between the range and null spaces of the task.
misspecification between training and deployment, showing that SDB consistently outperforms baselines and remains robust in the face of parameter shifts, making it well-suited for real-world deployment.
In addition, our work advocates for more expressive formulations of diffusion processes in generative modeling, opening avenues for future research into system-awes aware and structure-driven stochastic modeling frameworks!
2 Related works
Our work offers a new perspective on constructing task-specific diffusion bridges (supervised), while drawing inspiration from diffusion-based plug-and-play methods (unsupervised) Both directions address the topic of image restoration and inverse problems from different viewpoints. Below, we briefly cover these directions, with much more detailed overview included in the Appendix.
Diffusion models for inverse problems. As shown in the work of Song et al. 2021b pretrained
unconditional diffusion models can easily adapted to conditional synthesis with a simple application of Bayes' Theorem. The resulting likelihood term has been extensively utilized for guided generation
Dhariwal and Nichol 2021], especially for image restoration problems [Daras et al. 2024). These methods range from generic ones [Song al. 202 Chung al. 2022 to those that incorporate additional assumptions and constraints to the generation process, such as DPS [Chung et al. 2023a],
DDNM [Wang et al. 2023 IIGDM (Song al. 2023a DDPG Garber and Tirer 2024] and others
Kawar et al. 2022
Chung et al. 2022 2023a Song et al. 2023a Mardani et al. 2024 Chung et al.
2024
Diffusion bridges. One may also extend the SGM framework to arrive at mappings between arbitrary probability distributions given paired data (Särkkä and Solin 2019 Among many seminal
works in this area [Heng et al. 2021 Sommath et al. 2023 Peluchetti 2022 Delbracto and Milanfar 2023 De Bortoli al. 2021 Liu the Schrödinger Bridge (SB) problem, Luo et al. 2023a (IR-SDE) construct conditional stochastic differential equation (SDE) that maps noisy degradations clean samples, while Yue et al. [2024] (GOUB) generalize it with the Doob's h-transform [Doob 1984), which is also used by Zhou et al. 2024 (DDBM) to generalize the entire SGM framework.
[2023] (12SB) propose simulation- algorithm that solves
Background
Diffusion models. Generative modeling of visual data has advanced rapidly with the introduction of diffusion models (Sohl-Dickstein et al. 2015 Ho et al. 2020 Dhariwal and Nichol 2021), which
We include the source code at httttss/gghhubbooo//oooei///////

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generate images by sequential denoising. Initially formulated as discrete Markov chain with Gaussian kernels, diffusion models were later unified with score matching [Song and Ermon 2019 under the framework of SDEs [Song et al., 2021b), establishing the continuous-time formulation known as SGM.
Let p(x:) denote the probability distribution of x; E R parameterized by time E [0, 1J, where p (xx)) represents the data distribution and p(x1) the prior. In this work, we highlight the often-overlooked linear formulation of SGM. The forward SDE, which maps p(xo) to p(xi), ), is defined as
dx, = Fxx,t + G,dw,,
(1)
where F,x; is a linear drift term with time denendennnn matrix F, E Rddd and G2 is a matrix- valued diffusion coefficient. Following Anderson (1822. the reverse SDE, which maps p(x1) to p(xo), is given by
dx;
[Fx G. Vx, logg(x))) G,dW,,
(2)
where log is the score function and w, the Wiener process with reversed time. The forward-reverse relationship ensures that the marginal distributions p(x:). defined by eq. (1 and
eq. remain consistent.
In standard SGM for image synthesis, p(x1) is typically chosen as an isotropic Gaussian. To generate samples from p(xn). one samples from p(x1) and solves eq. (2) using a time discretization scheme. This requires the unknown score function log p(x.), which is generally approximated by a neural network so (xx, 1) trained with the score- matching loss (Song et al. 2021b]:
[ t) V log x2)1
(3)
where the expectation is taken over Xn ~ (x))), t ~ 2( 1]), and x, p(x, Xx).
The linear formulation in eq. and eq. (2) is special case of the broader SGM framework introduced by [Song et al. [2021b], where drift and diffusion coefficients can be arbitrarily complex functions of x, and t. This linear formulation. however, generalizes the well-established scalar case. Specifically, the time, can be recovered with Ft 3,I and G, = V The Variance-Exploding (VE) SDE, which induces unbounded growth in the variance of p(x1) as t + 1, is obtained with F, = 0 and
Preenncc--rreiiingg (VP) SDE, which maintains constant variance for x; across
G - V def I. While these scalar matrices offer simplicity, they also raise the question of whether more sophisticated designs could yield additional benefits.
de
Inverse problems. For signal sample x E Rdd we define the linear measurement system
E.c.
y
where y E R" is the measurement, A mm is the system response matrix, c (N is the measurement noise with covariance matrix Σ E Rm and p'y x) = [Ax, Σ) is the forward model. We refer to finding the unknown based the provided y as solving linear inverse problem with additive Gaussian noise [arantola 2005). This formulation is central to many practical applications, such signal processing, medical imaging, and remote sensing (Mueller and Siltanen 2012), particularly in the case of noninvertible linear systems, where either m 7 d or A rafk-defi
m. L )
Matrix pseudoinverse. Consider the simplest noiseless (Σ = 0mxm) setting of eq. 44) with invertible A. In this case, the original signal x can be fully recovered by simply using the inverse of d and A is full-rank, which greatly
A, i.e., A y. However, this is only possible when limits the scope of potential inverse problems. In practice, the matrix A is typically not invertible, and noise arises naturally due to, e.g., imperfections in the measurement process.
In the case of invertiibee A, one may utilize its doinverse, denoted as A + It generalizes the notion of standard inverse of square matrices to those of arbitrary shape and rank. Consider the singular value decomposition (SVD) for matrices U, D, V, where U and V are unitary, and D is diagonal with non-negative real entries. Then, its pseudoinverse computed via + VD+ HD+U*. U*, where D is obtained by replacing all of its non-zero singular values with their reciprocals.
inverse, often called the pseu-
UDV* of A
3

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Range-nullspace decomposition. A particularly useful property of the matrix pseudoinverse is its role in the well-knoww range-nullspace decomposition [Strang 2022]. Specifically, any signal x can be decomposed as
x =
Ax + --
(5)
range space
space
where the first term lies in the range (image) of A, and the second in its null space (kernel), i.e., it annihilated by A. This decomposition is particularly relevant in inverse problems: applying A to x eliminates the null space component, while preserving the range component. Consequently, solving an inverse problem can be viewed as denoising in the range space and synthesizing missing information in the null space.
Pseudoinverse reconstruction. A matrix pseudoinverse can also be used to compute the pseu- doinverse reconstruction (PR) & of via A ty. Crucially, this reconstruction always lies in Rd, regardless of the shape of y. This property addresses a common limitation of prior bridge methods, i.e., their default operation in exed-dimensional spaces. When m * d. such methods require additional effort to remain well-defined. For more challenging inverse problems, such as those in medical imaging, PR provides a principled mechanism for resolving this issue.
From a theoretical perspective, assuming access to A also simplifies the relationship between and y [Strang 2.22]. When m < d, the PR is the unique minimizer of ||xx -yllz over all x" E Rd. When > d, it yields the minimum 22-norm solution among those satisfying Ax" y. i.., ming. ||x*|12 subject to Ax' =. These optimality properties further motivate the use of PR when A is known and its pseudoinverse is tractable.
Problem-specific difflusion bridges. Many unsupervised approaches, assuming access to a specific noisy linear system (eq. (4) offer tailored solutions that improve performance under additional
assumptions [Song et al. 2023a Wang et al. 2023 Chung et al. 2023a Garber and Tirer 2024 In
contrast, ofate-of-thh-arr (SOTA) diffusion bridges [Liu al. [023] Luo et al. [0033]] Yue al. 2024]
Zhou et al. 2024) make no assumptions about the underlying system during training, focusing on general mappings between arbitrary distributions. This can be suboptimal when the system (even approximately) known. For instance, in image inpainting, these methods do not distinguish between the known range (unmasked) and the missing null part (masked), resulting in redundant noise in the range space, which is expected to be noiseless. In more complex scenarios, such redundancy could reduce efficiency, while ignoring system-speciin information may hinder generalization. Thus, developing supervised diffusion bridge tailored to specific system remains an important yet unresolved challenge.
4 Method
In this section, we seek to construct a bridge diffusion process that directly incorporates the infor- mation about the assumed measurement system. By that, understand process which maps the PRs to the respective signal samples, ii) synthesizes the missing information directly in the null space and iii) optionally denoises the range space part. We begin by noticing a specific connection between how intermediate steps of a general linear diffusion process are obtained and how noisy linear measurement systems map clean samples to observations. As our key contribution, we propose a specific design of matrix-valued SDEs, which fulfill the initially desired properties.
Connecting systems to SDEs. Following eq. (4), it is clear that y x ~ (Ax, E), i.e., for a given signal x, noisy linear system applies a linear transformation A and adds (possibly correlated) Gaussian noise to sample a measurement. On the other hand, given a clean sample Xo, eq. ! transforms it X2 by applying a cascade of such noisy linear systems, which is noisy linear system itself. That is, X2 Xy (H,,x, Σ,) for some dependent matrices H, Σ,- Hence, one may equivalently arrive at y (or some linear transformation of y) through application of series of noisy linear systems to x instead of just a single one. In the limit, this would imply the existence of a general linear SDE that performs such mapping.
The above considerations suggest that the measurement system could be directly embedded into H, and Eg to arrive at such SDE. While the relationship between these and the drift and diffusion

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coefficients of the corresponding stochastic process is not immediately clear, a recent theoretical result derived by Tivnan et al. [2025] provides convenient way of mapping from one to the other under mild assumptions. We restate it below for clarity.
2025 Assume that x2 Xu evolves according the linear forward process from eq. 1 Then, X, Xo ~ N (H,Xo: Σ,), where
Theorem
Tivnan et al.
). "..,".E....((,,........ ds H
H = exp (F)) =e exp
(6)
where S2g is the Magnus expansion and the approximation becomes equality if. for all s, e [0, t). Fr] = O, i.c., F. and Fs commute. Conversely, given H, and En, the drift and diffusion coefficients can be obtained via:
dH: F. = dt H GGG.
dE. - dt
Therefore, theorem 1 allows one to obtain the linear diffusion process governing the evolution of Xt No defined through H, and Σ,, which in turn makes training and sampling from continuous-time models possible.
Embedding the measurement system. Equipped with the necessary tools, we propose embed the measurement system into the linear diffusion process by using
H. = A+A + (I A+A),
(8a)
Σ, = EA+ + AtA)
(8b)
for as, Bz, "t being the null space drift, null space diffusion and range space diffusion coefficients respectively. We refer to the process resulting from eqs. (8a and (8b) as the Diffusion Bridge Model (SDB).
To better understand the rationale behind this specific design, it is worth considering the range and null space components of the resulting x; separately. Specifically,
A Ax, = A + (Ax, 7 E A Axo + VAA+
(9a)
(I- A)x, - A+xxx + V -
(9b)
where e (N E Emxm), ~ NOs,, Eaxa). The range space part contains the original signal A Axx and the stochastic component A + Ede which directly models the range space noise. If E 0, the true signal fully recovered at every timestep t. The null space part is a mixture of the null component of the true signal at (I - A A)xxxo and null-space space-projected Gaussian noise (, A+ A'c... Notably, SDB links two stochastic processes that evolve simultaneously within the range and null spaces, each modeled with independent noise variables, and e'. Moreover, it evident that proper choices for at, 9g and Y/ lead to a mapping between the PRs at to their respective clean samples at t = 0.
SDE perspective. Applying theorem T to the coefficients from eq. ( leads to the following drift and diffusion coefficients for SDB:
d F: log ar (I A),
(10a)
di
G.G.T dy
EA+ + d33 d dt
23,
log a: (I A+A.
(10b)
dt
dt
With this formulation at hand, we now propose a specific setting of the scalar functions Or and Be, which extend the result of Liu al. 2023) and make direct connection of SDB to the optimal transport (OT) plan [Mikami| 2004].
Theorem 2 For the linear SDE defined in eq. ① with F, and G. given by eqs. 0ia? respectively, let 04 0ē-07 3r af for of = -o (r)drr o? = S. (2( (T) for some non-
10a) and TOb
0²
negative function g(t). Assuming that g + uniformly for all t, the null space part of this SDE
reduces to an OT-ODE:
d(I A A)x ve(xe (x)dd,,
(11)
2000 o, where Ve ve(xe xo) = (limg(0)++
(I A A) -x) xo).
5

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We include the proof in the Appendix. To align the dynamics between the range and null space, we oi also use "tt až+o? In practice, we do not set g(l) = 0, but rather keep it at sufficiently low values. Because of the relationship to the SB problem, we term this variant as SDB (SB).
Novel processes. To showcase the versatility of our framework, we introduce two additional variants of SDB that reinterpret the VP and VE diffusion processes of Song et al. [2021b]
SDB (VP): ag t, f vI at. 7t Br. SDB (VE): at 1. maz vt, 7t Vt, where mer >> 1.
In both cases, the original VP or VE process is applied in the null space, while the range space coefficient " is related to 3r to simplify the dynamics. Unlike methods such as DDBM, which symmetrize the variance schedule around 0.5, these variants explicitly control how the original signal is crased in the null space. There, SDB (VP) performs a convex interpolation between the clean input and Gaussian noise, while SDB (VE) converges to isotropic Gaussian N o I) as f - 1. As such, both more closely resemble approaches like IR-SDE that retain non-singular marginals at the endpoint. For simplicity, we parameterize SDB (VP) and SDB (VE) processes with linear scheduling, leaving more sophisticated designs as future work.
Principled posterior sampling. In the asymptotic limit of infinite data and model capacity, it natural to ask whether SDB constitutes an exact probabilistic model of the underlying inverse problem- that is, whether it produces principled posterior samples from p(xly). The following result provides a positive answer under mild conditions on the forward process coefficients, with the proof deferred to the Appendix.
Theorem Let the forward measurement model be p(ylx) A(xx, E). Then, under the SDB dynamics defined by eq. (I. eq. (7), and eq. (8, with time-dependent scalar coefficients an, Sr, and Tt satisfying
o Bg
lim = lim ++1
= 0,
[-1
the corresponding reverse-time SDE generates asymptotically exact samples from the Bayesian posterior distribution p(xly).
Table Comparison of prior SOTA diffusion bridge methods with SDB. Subsequent rows denote the drift (F,) and diffusion (G.) coefficients of the forward process, while f4: Σ, are its respective mean and covariance at timestep Each column follows the original notation of each paper.
Method
12SB
IR-SDE
GOUB
DDBM (VP)
SDB (SB)
x:)
log 1x)
F.
X (A))
A(x)
(0) IiXI Xi)
log A+A)
A'
G.G.
g
dd
28 log (I A+)
of Mer
SNR
(Xi -4
cv(1
A A'Aix
ST A-A
>>
AP(1 - 21)
SNR
Markovian
x
X SNR
of C. tridr,
0: o
0.. 1 9.dr,
uxp (-1 [
at, 9a follow from theorem
Hyperparameters of L. (rrdr
0, = 6.0r
of gldd
= 1 of

---

Algorithm SDB Training
Require: ((xo).p())))))) A, At, E112
-t. Be, Yt. Do
for each iteration do
2:
p(xo)
sample clean data
3: -~ p(t)
sample diffusion timestep sample range-space Gaussian noise > sample null-space Gaussian noise e'
4:
(0,1) E R
5:
e' ((O,I) E Rd
6:
X, + A+ a A A+ A*
forward step compute reconstruction loss for update network parameters
A+A)'' A)c'
7:
L8 (x( 1
8:
+ o VoLg)
9: end for
10: return 0
Algorithm SDB Sampling (Fuler-Maruyame)
Require: A+ 11/2 ,u,, Be, n. H,, F,, Do.
1: 4-1
initialize time set timestep > sample null-space noise pseudoinverse reconstruction initializer
2: A + 1/N
3: é' N(O,I) E
4: + Aty
5: x -I-
6:
for (l,..., N] do
7:
6~~ I, E R
> sample range-space noise b> sample null-space noise
8:
e' I',,( E Rdd
Xt-At f Xx
At [(fiI 2F, (H, xx,) F2x,]
1/
9:
//
dy: di
A+
+
d3.
di de log (gga)) (I A+ A)
23, d
> update
10: 1 1+6 At
D decrement time
11: end for
12: return X
b final sample
Experiments
Baselines. We compare SDB with both supervised bridge methods and unsupervised plug-and-play diffusion-based baselines. For the former, we include 12SB [Liu et al. 2023], IR-SDE [Luo et al. 2023a], GOUB (Yue et al. [004], and DDBM [Zhou et al., 20244) For the latter, we pick DPS [Chung ct al. 2023a IIGDM Song et al. 2023a], and DDNM (Wang et al. 2023), all of which rely on assumption of noisy linear measurement model. Detailed descriptions of all baselines and other technical details are provided in the Appendix. Unlike these methods, which rely on scalar SDEs, SDB introduces more general matrix valued formulation that allows structured control over range and null space components. We summarize the resulting conceptual differences to baseline bridge methods in table [ together with SDB's training and sampling procedures in algorithms 1 and 2 Following standard evaluation practice [Luo et al.. 2023a Yue et al., 2024], we report perceptual scores (FID [Heusel et al. 2017]. LPIPS [Zhang al. 2018 and reconstruction metries (PSNR, SSIM).
Experimental design. To isolate the contribution of SDB diffusion process, we standardize key implementation details across all methods. Specifically: (i) for both supervised and unsupervised approaches, we train score networks from scratch using the training hyperparameters and architecture of Luo et al. [223a]. with 256 training epochs for supervised methods and 512 for unsupervised ones; (ii) each supervised method learns a mapping between signal samples and their PRs, ensuring that SDB does not benefit from favorable parameterization, particularly in settings where m o d; (iii) during evaluation, supervised methods use 100 discretization steps, while unsupervised methods use 200, along with standard Muler-Maruyama solver and their optimal noise schedule (e.g. VP for DDBM). To ensure consistent evaluation, we reimplement methods within unified framework, allowing for the separation of algorithmic advancements from the proposed stochastic process.

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Table 2: Quantitative comparison of SDB with the baselines across four inverse problems. Bold indicates best and underline second-best values for each metric.
Inpaiating- ColebA-HQ
Superesolution- DIV2K
CT Reconstrucion RSNA
MRI Reconstraction Br35H
Method
PID
LPIPS
PSNRT
SSIM PID | LPIPS PSNR
SSIM
FID
LPIPS
PSNR
SSIM
FID
LPIPS
PSNR
SSIM
Unsapervised
DPS
0.213
18.59
0.295
0.549
0.146
0.211
0.530
31,272
0.211
0.665
DDNM
0.112
18.90
0.791
99.21
0.302
17.13
0.513
30.41
0.238
27.226
0.783
42.23
0.196
18.122
0.636
Supervised
12SB
5.56
0.847
27.41
0.889
83.75
0.176
15.21
0.686
24.81
0.107
41.886
0.924
31.34
0.065
28.750
0.849
IR-SDE
4.08
0.031
29.92
0.912
96.22
0.185
23.51
0.003
18.88
0.028
43.438
0.964
30.14
0.065
28.878
0.871
GDUB
3.89
29.89
0.912
98.89
0.178
24. 24.39
0.649
19.90
0.024
43.878
0.967
30.63
0.05K
28.590
0.863
DDBM
6.03
0.047
28.15
0.906
90.16
0.233
25.79
0.720
23.36
0.040
44.415
0.961
32.42
0.074
28.971
0.872
Ours
SDB (VP)
4.90
0.030
30,51
0.914
87.08
0.228
25.91
0.724
15.43
0.019
46.365
0.91
32.88
0.068
20,255
0.881
SDB (VE)
5.97
0,042
32,12
0.944
94.77 73
0.226
15.90
0.718
14.17
0.020
46.325
0.961
13. 13.90
0,0K3
70.098
0.87%
SDB (SR)
4.63
0.8A1
30.40
0.930
81.56
0.226
26.10
0.724
15.02
0
46.672
0.982
29.85
0,053
29,812
0.893
Benchmarks. We evaluate SDB on four inverse problems with varying measurement system complexities, using original images at resolution of 256 x 256. Building on prior work [Luo et al.) 2023a Yue et al. [204]. we first consider inpainting
CelebA-HQ [Karras al. 2018), where A is a masking operator and Σ = 0mxm (noiseless). The measurements consist of the masked original images (m d), which are equivalent to the PRs (A + 1). Additionally, we follow previous studies by examining superresolution on DIV2K [Agustsson and Timofte [017 Timofte et al. 2017). where A represents 4x downsampling operator (d = 4m), implemented through average pooling. For this task, A reconstructs the images using nearest-neighbor interpolation. To evaluate more complex and practical inverse problems, additionally propose two medical imaging tasks: CT reconstruction on the RSNA Intracranial Hemorrhage dataset [Anouk Stein et al. 2019] and MRI reconstruction the Br35H dataset [Merlin 2022], both using 2D axial brain scan slices.
For CT reconstruction, the sinogram y represents line integrals of the object's attenuation coefficient, with the system matrix A describing detector-specific integrations along X-ray trajectories. We implement using its SVD decomposition, A = UDV* To make the problem more realistic and introduce domain- specific artifacts, zero the singular values of D below threshold 3.2 and use a noisy setting with scalar covariance Σ = IfI, where of = 0.0001.
For MRI reconstruction, consists of undersampled Fourier-domain measurements. The system matrix is modeled as = MF, where is the Fourier transform and M is masking matrix. We sample M such that Åι controls the percentage of the lowest frequencies to be kept (a deterministic operation), while \2 specifies the percentage of frequencies to sample from the remaining ones (a stochastic operation). The noisy setting is similarly modeled with scalar covariance matrix Σ = I, where of = 5.0. We pick d = 16 and x2 = 30 as defaults.
5.1 General evaluation
The results summarized in table 2 show that SDB (SB) outperforms all baseline methods across every metric, with the exception of LPIPS on DIV2K. Notably, SDB (SB) demonstrates a clear advantage over 12SB, which shares the most similar stochastic process, highlighting how SDB (SB) effectively leverages additional information from the measurement system. Overall, it also provides empirical justification for the null space OT plan mentioned in theorem 2] Moreover, SDB (SB) displays superior stability, the performance rankings of the baseline bridge methods exhibit more variability across tasks. These advantages are also evident qualitatively (fig.2).
SDB (VE) and SDB (VP) strike different balance between perceptual and reconstruction metrics compared to SDB (SB). In inpainting, they often outperform all other methods, with a noticcable emphasis on reconstruction quality. This trend is MRI reconstruction tasks. In CT reconstruction, their performance is nearly on par with SDB (SB). Although they occasionally underperform relative to baseline bridge methods, this is not unexpected given the nature of their stochastic process. Along with IR-SDE, these methods are unique in not treating p(x1) ) sum of Dirac deltas, which means they start the reverse process from noisy sample. This additional noise may make the problem more challenging, but when directly comparing SDB (VE) and SDB (VP) IR-SDE, the performance improvements become evident.
pronounced in the superresolution and
Finally, under our unified setting, prior bridge methods demonstrate comparable performance with significantly reduced variability compared to what previous works report [Luo et al. 2023a Yue

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y
12SB
IR-SDE
GOUB
DDBM
SDB
Figure 2:
Qualitative comparison of SDB (SB) with the best- pest-performing baselines (bridge methods). Rows depict the results for inpainting, superresolution, CT and MRI reconstruction respectively.
al. 2024 For instance, their PSNR the MRI reconstruction task falls within the range of 590, 28.971]. meaning that performance gains on the order of zs 0.2 can be considered significant this context. Notably, the unsupervised baselines exhibit visibly lower performance compared to bridge methods. While this is expected due to the lack of training with paired data, we note that our setup allocates only twice the training budget to the unconditional models compared to the bridge methods, suggesting that their performance has not yet saturated. We consider improving these models as future work.
5.2 Evaluation under misspecified model
In practice, while the general form of the measurement system in an inverse problem may be known, the specific parameter values often differ between training and deployment. This creates generalization challenge, where the model must learn the underlying dynamics rather than relying on shortcuts. For example, CT reconstruction, a well-trained model should perform stably near the original parameter т. A larger r', which reduces low- wow-frequeccy information, makes the task more difficult. However, generalized model should maintain most of its performance long T! remains reasonably close to r.
This issue is particularly critical for SDB, where the measurement system's parameters embedded directly into the coefficients of its stochastic process. While results section 5.1 indicate performance gains, deploying SDB practice could be risky if it overfits specific parameter values. Therefore, the following experiment tests whether SDB can generalize when evaluated under a misspecified model. For this evaluation, we focus on the west-performing methods from section 5.1 namely the baseline bridge methods and the SB variant of SDB.
Improved generalization. We begin with the MRI reconstruction task, where the system response matrix preserves portion the original signal by keeping A of the frequencies, starting from the lowest. This maintains the general structure of the true image, and as \ is gradually decreased, more information lost, making the task increasingly difficult. This scenario also mirrors real- world situations where a detector captures fewer measurements.
In what follows, we evaluate the checkpoints of bridge methods trained under default setting (A 16, A2 30, oî 5.0) measurements generated from a system with modified parameters.

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LPIPS
PSNRR1
SSIM
000W
Webiic value
2.5
40.66
813
050
666
510
1-A LPIPS (A)
A PSNRR1)
SSIMCS
1.1
Netric value
12.67
0.14
101M
23.0
65
Figure 3: Quantitative comparison of SDB (SB) with other bridge methods in misspecified MRI reconstruction setting. The top row evaluates bridges trained with A 16, o = 5 on data generated from systems with decreasing A The bottom row evaluates performance data with λ 14 and increasing o². Perturbing the original system makes the problem harder in both cases.
Figure 3 (top) shows the performance of the methods as λ is gradually decreased. Notably, the performance gap between SDB and the baseline methods widens as early as λ = 14, and this advantage is sustained even at lower values. SDB continues to better utilize the available measurement information, indicating much greater robustness to reductions in A1-
To make the setting even more challenging, we set Å = 14 and gradually increase the measurement noise variance o2 up to twice its training-time value, simulating a more realistic scenario. Figure (bottom) shows that, even under this perturbation, SDB maintains significant performance advantage over the baseline methods. Unsurprisingly, all methods exhibit greater robustness to changes dz. which is expected due to their denoising nature.
An analogous analysis for the CT reconstruction task is provided in the Appendix, yielding similar observations. Both experiments, closely aligned with the practical applications of inverse prob- lem solvers, underscore the generalization capabilities of SDB across broad range of system perturbations, positioning it as a promising solution for real-world scenarios.
Extended evaluation. Additional experimental results are presented in the Appendix. These include evaluations on more challenging misspecified models, analyses of the influence of ill-conditioned system operators, and detailed assessments of computational complexity and runtime. We also include series of ablation studies. Finally, we outline conceptual extension of SDB to nonlinear measurement systems and demonstrate its empirical advantages over baseline bridge methods in small-scale study. Extending SDB to broader class of problems remains an interesting direction for future work.
Discussion and limitations
SDB offers principled framework for constructing tailored to inverse problems under a linear Gaussian model. By explicitly incorporating information about the measurement system into the generative process, it achieves improved reconstruction quality and exhibits strong generalization under system perturbations. However, the method also comes with natural limitations. Notably, extending SDB nonlinear measurement systems remains open challenge. We provide only a preliminary, proof-of-concept treatment based on local linearization, which is limited differentiable systems. addition, our work adopts simple variance schedules for both the range and null space components; the interplay between these two processes warrants further theoretical and empirical investigation to identify optimal scheduling strategies. From real-world application perspective, tasks such as CT and MRI reconstruction are typically performed in three cimensionss-vvllltiin our method in such settings remains an open direction for future work. Despite these limitations, we view SDB as valuable contribution to the growing literature on diffusion- based inverse problem solvers and a promising foundation for future research.
wtwprrrrmeeettsstttsseeseeeeeeee diffusion bridges
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Acknowledgments
This work was financially supported by the INFOSTRATTEG-IW022200---2 grant funded by the Polish National Centre for Research and Development (NCBiR), the SONATA BIS grant 2019/34/E/ST6/00052 funded by the Polish National Science Centre (NCN), and the NIH grant 5R01HL159183-03.
The computational resources for this work were provided by the Laboratory of Bioinformatics and Computational Genomics and the High Performance Computing Center of the Faculty of Mathematics and Information Science, Warsaw University of Technology. We also gratefully acknowledge Poland's High-perffrmanee Infrastructure PL.rrid ACC Cyfronet AGH for providing computer facilities and support within computational grant no. PLG/2225/018330.
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Appendix for "System-Embeddded Diffusion Bridge Models"
Theoretical results, proofs and derivations
23
Derivation drift and diffusion coefficients for SDB
23
A.2 Proof of theorem 2
23
A.3 Proof of theorem 3
24
Magnus expansion generality
26
Extended related works
26
Diffusion models for inverse problems
26
B.2
Diffusion and Schrödinger bridges.
26
Baselines
27
Extended methodology
28
Network parameterization
28
Nonlinear measurement systems
28
Experimental setup
29
D. Training hyperparameters
29
D.2 Computational requirements
29
Additional experimental results
29
E.1
Evaluation under misspecified model (CT reconstruction)
29
More challenging misspecified settings
30
Motion deblurring with an ill-conditioned system
................................
30
E,4
Contrast recovery with nonlinear system
30
................................
E.5
Hyperparameter analysis
31
E,6
Ablation study on noise schedules
33
7
Computational complexity and overhead of SDB
................................
34
E,8
OT unpaired-data- baselines
35
Extended discussion
35
Advantages of Embedding Measurement System Information
35
Perception- Distortion Tradeoff
35
Broader impact
36
Additional visual results
36
22

---

A Theoretical results, proofs and derivations
A.1 Derivation of drift and diffusion coefficients for SDB
We begin with deriving the drift and diffusion coefficients Fr. G. of SDB (eqs.
10a) and (Obb) by
using theorem ] with the mean and covariance matrices H,, E, (eqs. Sa and (8b)) of forward
process. Recall that H, A+ + ar - A+A), E, = 7 ΣA theorem we obtain
3 A A). Following
A AF,
A A log H. = A + A = 0,
(12a)
dp
dl
(I A A)F,
log 04 A+A),
(12b)
dt d
A+AE
EA
(12c)
dt
d
A A+A
(I A+A)
(12d)
d/
Moreover, note that F. F,, Σ E, due the symmetry of the range and null space projections, and F F,C. = F, since F, only affects the null space and (I- A A) A+ (idempotent). Following theorem 1.
d F,
log a (I - A+A),
(13a)
dt
dy.
G,G, dt
F,E: - E.F.
(13b)
dE,
2F,E,
(13c)
dt d%
A EA* + + +d..
23r d loga (I-A+A)
(13d)
d!
di dl
A.2 Proof of theorem 2
We begin by recalling two propositions of Liu et al. 2023 using our notation, which will serve as the basis of the proof.
Proposition 1 (Analytic posterior given boundary pair) [Liu et al. 2023 The posterior of eq. given some boundary pair (Xu, X1) admits an analytic form:
p(x:
Xo. x1) (1,,, x1), E.),
(14)
where b1 o7+o?
8²
Xo + oē-o?
X1 E,
0. ā.+0?
and oj = -o (T) oj = S,- = (r) with
G. I.
Proposition 2 (Optimal Transport ODE; OT-ODE) [Liu et al. 2023 When (t) O. the SDE
between (Xa, X1) reduces to an ODE:
dx; = (x(x; xo)dl,
(15)
where ve(X;, xo) (2(() (x: xo) whose solution is the posterior mean of eq. 14
of
Consider the null space part of x, given by mean and covariance matrices from eqs. (8a) and (8b)
0f0? नo for of fo (T) of =f; (r)dd,, where g()) is the null space
with नरन. Br
diffusion coefficient given by eq. 10b i.e., g(() = dB. - 2,3 J log an
(I A+AAxxx
A+ +
(I (I- A+A
(16)
23

---

for e ~ (0s, Idxa). Note that the null space part of X1, being the PR of Xu. is zeroed out, i.e., [I A A (I A A+A) axx) This allows us to artificially rewrite x, as A + A)). Hence,
o
of
of of
-1 AT
Xo
Oaxd
A1 A)e. (17)
+ oi
of
oo
of
Equivalently, (I A[x, M((I A)
oftof
07
Xo
+
of
of
0 oo
the posterior mean of SDB (SB) takes the form stated in proposition and proposition applied directly the null space part.
2 can be
As a final remark, we note that, up to this point, the definition of g(t) is interdependent with that of a2 and Sg. It is not immediately clear that defining 0,, Br in theorem fulfills g²
d8. 2 d log ace. We now show that this property is indeed satisfied.
Denote by C of + o? and observe that Bg o. of =C of do? = g² Then,
d
d
o oī
log
0t
log
(18a)
dt
dt
o?
d
(log ot log C)
(18b)
dt
(² of
(18c)
d3
d पهo?
(18d)
di
dt o7 of
d (f - of)
(18c)
dt
d
(Co -o
(18f)
C di
(t)) 2 (g)(()
(18g)
(t) oo))
2
(18h)
By substituting these into the definition of (t). we obtain
di
233 d
d
log
(07(0) - 20 dt
log
(19a)
dt dt dt
=0+ d
2 da
20/ar ( _ day
of
(19b)
ar dt
2 da
=an
2 (1)
oi
(19c)
dl
of
+ of
1
g (t)
C (1)
(19d)
C
(1),
(19e)
which completes the proof.
A.3
Proof of theorem 3
Many diffusion bridge methods, such as 12SB, GOUB, or DDBM, rely on MarkMrvakvinn stochastic processes, where x, depends on both Xy and X₁. In contrast, SDB uses Markovian forward diffusion process, for which p(x,) Xo) is well-defined. More specifically, SDB is a special case of score-based generative models, using the general formulation with matrix-valued drift and diffusion coefficients described in Appendix of [Song et al. 202Tb]. We show below that SDB is principled posterior sampler of p(x(y) with a proof of theorem 3 based on series of lemmas.
24

---

We start by considering the most general practical case with positive semidefinite covariance matrix E in the forward model p(y' x) (A x, E). When Σ is positive definite, we can perform whitening by scaling the measurements and the system matrix as y Σ 'y' and A = Σ A', yielding the equivalent isotropic model p(y x) (Ax, I). In the singular case (i.e.,
positive semidefinite but not positive definite), Σ can be replaced with (Σ+)* to achieve whiteming on the range space of Σ. From this point onward, we drop primes for notational simplicity and proceed in the whitened coordinates. Hence, without loss of generality, we assume an isotropic noise model, which is equivalent to initial whitening of both the measurements and the system response matrix.
Lemma The pseudoinverse reconstruction A A is sufficient statistic for estimation of given That is,
p(xly) p(x|A+y)
This holds because (i) the range space component of the measurements can be exactly recovered by re-applying to y, and (ii) the null space component contains useful information about x.
Lemma 2 A sample from
p(zl)) = NA 31 A+)))
is also sufficient statistic for estimating x given i.e.
p(xly) = pxx|z).
The null-space component of the pseudoinverse reconi is zero; therefore, adding null-space noise preserves sufficiency.
Lemma 3 The SDB forward process the final time step, p(xi (xo = x), is identically distributed to the SDB initializer p(z]x) x) from lemma assuming
0
lim Yo = 1,
lim [+1
B.
Both are conditional Gaussian random vectors with mean A Ax and covariance A --EA++ T + A+
Lemma 4 The trained SDB model is principled sampler of p(xo|x1).
Since SDB is a subset of ccore-based generative models with a standard Markovian forward process. the reverse SDE with learned score network asymptotically samples from p(x,(x)) [Anderson 1982], approaching the exact score with sufficient data and model capacity.
The procedure to generate principled posterior samples from p(xly) is as follows:
1. Compute the pseudoinverse reconstruction A +y.
2. Sample null-space noise:
p(zly) - 3 (I- A+A)).
3. Sample the SDB reverse process:
Xo ~ p(Xp|x1 X1 =2).
By lemmas I and 2 z a sufficient statistic for x. Lemma ensures that initializing the reverse process at valid. Lemma 4 that the trained SDB model samples from p(xo (x1). Combining these results, the procedure produces principled posterior samples from p(xly).
25

---

A.4 Magnus expansion generality
The matrix integral formulation of the reverse-time SDB process holds under the condition that the drift matrices F, commute for all t E [0, 1]. In the proposed SDB formulation (parameterized as in eq. (. this condition is always satisfied: the only time-dependent parameter is the scalar dg,, so the singular vectors of F, remain constant, ensuring commutativity at all times.
This property extends to all scalar diffusion processes, including VP, VE, Flow Matching [Lipman
2.231. Fourier Diffusion Models Tivnan et al. (2025), and Subspace Diffusion Models Jing et al.
et al.
2022]. While diffusion methods with commmmnmt F, matrices are currently not known to
us, theorem remains valid in those cases through the Magnus expansion Kamm et al. [2021).
B Extended related works
B.1 Diffusion models for inverse problems
In recent years, deep neural networks have gained significant attention for solving various inverse
problems from different perspectives [Ongic et al. [2020] Chen et al. 2022]. Solving eq. (4) can be interpreted sampling from the posterior distribution p(x y), and generative models that support conditional generation are naturally suited to this task. Diffusion models, in particular, have emerged as SOTA tools for this purpose, thanks their flexible mathematical formulation and expressive data priors (Daras et al. [2024] Preechakul et al.) 2022, Sobieski and Biecek, 2024]. The standard approach involves extending the diffusion process sample from p(Xg y) at t 0 by applying Bayes' Theorem. The conditional score function can be decomposed as V log p(x, y) Vx, lologg( p(x)) + Vx, "x,logr' p(y xx), where the first term, Vx, lollogpl p(xe), can be approximated using a pretrained score network So (x,,), L). and the second term models the relationship between x, and the measurement y (Song et al. [221b] Dhariwal and Nichol 2021]. This framework has spurred considerable progress in the field, with numerous successful methods emerging [Kawar et al. 2022 Chung et al. 2022 2023a Song et al.) 2023a Mardani et al. 2024 Chung et al.) [2024] Sobieski et al. 2025 Since this approach does not require retraining the score network for each specific problem, we refer to these methods as unsupervised.
Several works leverage the measurement system structure when applying pretrained diffusion models to inverse problems. Wang et al. [2023] (Denoising Diffusion Null-Space Models, DDNM) restrict updates during generation to the null space component of Xt, keeping the range part fixed. Soog et al. 2023a] (Pseudoinverse-Gune Guided Diffusion Models, TIGDM) approximate the likelihood score
via a vector-Jacobian product, where the score network's Jacobian is computed with automatic differentiation, and the vector reflects the range-nullspace decomposition. Garber and Tirer [2024) (Denoising Diffusion Models with Iteratively Preconditioned Guidance, DDPG) propose a method for interpolating between pseudoinverse-bas based and sestt-suares--bseeee conditioning.
B.2 Diffusion and Schrödinger bridges.
Diffusion models, while effective for high-quality image synthesis, are limited by the simplicity of Gaussian priors for p(x1). Recent developments in bridge models (Särkkä and Solin 2019 generalize the diffusion process by allowing p(x;) to be an arbitrary distribution. This especially ipportant in image restoration tasks, where paired samples (Xo, xl)-clean and distorted images-are are available. Bridge models aim to generate samples from the posterior p(Xo x1) by initializing with a sample from p(x1) rather than Gaussian noise. While conditioning the standard score network on x1 is possible approach to achieve posterior sampling, it is often suboptimal [Batzolis et al. 2021].
Several methods have formulated the diffusion process as a stochastic bridge. Heng et al. [222] propose a simulation-based algorithm using a fixed starting and ending point with an approximation of the true score. Liu et al. [2022 extend this with Doob's 's h-transform [Doob] 1984] to bridge distributions. Simulation frre algorithms utilizing the h-transform are presented by /Somath et al.]
[023) and Peluchetti (2022). Delbracio and Milantar [2023] construct Brownian Bridge for direct restoration from x, More recently, Zhou et al. [024]] introduce the DDBM framework, extending the VE and VP processes, while He et al. [202]] study DDBM within the consistency framework
[Song et al., 2023b). /Zheng et al. 2025 link DDBM to Denoising Diffusion Implicit Models (Song
et al. 202
26

---

The Schrödinger Bridge (SB) problem [Schrödinger 1932]. which aligns distributions via constrained forward and reverse SDEs, has also been explored. De Bortoli et [202T] apply Iterative Proportional Fitting (IPF) to solve the SB problem. Liu et al.
2023 propose a tractable class of SBs, leading to simulation-free algorithm (I2SB). CCung et al. 2023b extend with an additional guidance term.
Shi et al. [024] build on IPF and introduce Iterative Markovian Fitting for SB solutions.
In a related approach, Luo al. [023a] derive scalar case of eq. ① that incorporates the start and end points, termed the mean- rever-evvering SDE (IR-SDE). Through specific parameterization, they show that its score function is analytically tractable, connecting to the Ornstein -haieene--hheeeckk (OU) process [Gillespie] 1996]. Yue et al. (2024) extend this work with generalized OU process and Doob's h-transform (GOUB), Further extensions include Luo al. (2023b). who apply IR-SDE in latent spaces, and Welker et al. [2022] and Richter et al. [2023], who adapt similar processes for speech tasks. Recently, Zhu et
2025 unify GOUB and DDBM within the framework of stochastic
optimal control.
B.3 Baselines
B.3.1 Unsupervised
We begin with describing the unsupervised baselines, which rely on solving the following scalar reverse equation:
dx; g log y)]dt + g(t)dw,,
(20)
where f and g represent the drift and diffusion coefficients respectively, while y is the conditioning variable representing measurements in the inverse problem context. Unsupervised diffusion based methods rely on decomposing the score function with Bayes' Theorem through Vx, log p(x, yy) =
log p(x,) Vx. log ply x,) and approximating Vx, log pxx: with pretrained so X2, t), while proposing different approaches for Vx, logog((y x.).
Diffusion Posterior Sampling (DPS, Chung et al. [203aa] ) approximates p(y xx) with p/y & (x, )).
where Xo (x:) E(xo Xx] X, + V log p(x)) is the Tweedie's formula [Robbins) 1992
giving an approximation of the denoised image at timestep L. Similarly, Vx, 1x,logp(x IS also approximated with sy (x, t) in this case.
Pscudoinverse-Guiide Diffusion Model (LIGDM, Song et al. 2023a] proposes to approximate the loglikelihood score with (y Axx)xxxx)))) (0;AA I) A
(xa(X.)
8x, where of I is the measurement
system covariance and r? is time-dependent term that should depend on the data (Appendix A.3.,
Song et al. 2023aa). ). Using automatic differentiation, one can compute 8x.
(x)(xx)) By combining it with
the other terms, the entire approximation can be efficiently computed as a vector-Jacobian product.
In their basic formulation, Denoising Diffusion Null-space Models (DDNM, Wang et al. 2023 rely
on approximating p(x:-1 x) using the following update rule:
va:-113 6x(xx,y y) Var(1 - at-1)
X
X, + O.C,
(21)
1 1-
1 - ã.
where E ~ 10, and a,, Br follow the notation from the original work. This rule utilizes the range-nullspace decomposition via (x, = A+y + (I - A A+)) which preserves the true range space component, while updating the null space part with Tweedie's s estimate.
B.3.2 Supervised
We proceed with a description baseline supervised bridge methods. All of these approaches assume stochastic process conditioned on both the starting and ending point with the following forward and reverse equations:
dx; f XT, ,, T)dt + g(()dwe.
(22a)
dx; =f' X7, t, Ttdt + g())dw,,
(22b)
where f, f' represent general drift coefficients respectively for the forward and reverse equation, g is the diffusion coefficient and xy represents the endpoint for the initial xo.
27

---

Image-to to- Image Schrodinger Bridges (I2SB, Liu et al. (023]) formulate the distribution of xt, given of o.
starting and ending points Xo, XT, as p(x: Xo: XT) - N
( oota?
&f
Xo + ai+a; XT: 5.+0? 1), where
of al. 2020).
1 (T) of -e: (r))d7 and show its equivalence to DDPM posterior sampling HH
Image Restoration SDE (IR-SDE, Luo et al. 2023a is based on formulating the forward and reverse equations as
dx, =0,(x x, x,)dt + (()dw,,
(23a)
dx, =(0:(xT x.) 0² log log + (1)dw,,
(23b)
where 02 -3, 0: Lod dT and À is predefined constant. At = 1, IR-SDE arrives at a Gaussian distribution (with non-zero covariance) centered xy.
Generalized of their framework by incorporating the Doob's 's h-transform Doob 1984.. which pulls the process
Bridges (GOUB, Yue et al. 2024 show IR-SDE as a special case
towards the desired endpoint. Formally, it is defined as h(x,, t, Xy, incorporated into the forward and reverse equations:
Vx. logp(xx X₂) and
20.r
dx,
(O. + (1)
[XT
xi) d g(()dw)
(24a)
-20r
dx,
(0, (1)
1XX xx)- g22 V() log p(; xT) di g())dw),
(24b)
T.r
(t) 2x 00.- = S 0, 02
9 (1) e 20 for a predefined constant
where 0,
20.
Denoising Diffusion Bridge Model, (DDBM. Zhou et al. 2024 also utilize the h-transform, but instead show how to adapt prior unconditional processes, which map images to Gaussian noise, to construct bridge between arbitrary distributions given paired data:
dx, =f((x,, f) g² (t))((x,, xr, T) g(f)dw,,
(25a)
dx; =[f(x,,,) 9² (t) (Vx, logp(x; xT) h(x,, t, xT. T)]] g())dw,
(25b)
where f(x,,t).g(1 follow the original image-to-noise process.
C Extended methodology
C.1 Network parameterization
In scalar setting, various reparameterizztions. of the score function were shown to provide different trade-offs in the final performance [Ho et al.) (2020]. For example, the network could instead predict the added noise directly, which is bijectively obtained from the score. As these reparameterizations related through simple scalar functions, one can choose them freely without additional considerations. In the matrix-valued setting, however, the choice of a parameterization is observe that Vx, logp(xx)
subtle. In this case, - -E, -x, H, H|Eo xx]). Training score-prediction model hence requires access to the inverse of E,, which may in general be costly to obtain. By properly choosing the form of Y and 3,, one can show that G, G, Vx,log p(x,) (fil 2F, (H, x,] x,) for some function ft. i.e., Xo xo-rrddiction model alleviates the need for computing E, Therefore, we treat this parameterization as the default one for SDB.
C.2 Nonlinear measurement systems
While SDB is primarily formulated for linear operators A with global range-nullspace decomposi- tion, it can be extended certain nonlinear inverse problems. For nonlinear, differentiable system response A, we can approximate it locally using a first-order Taylor expansion around some point X
A(x) R A(x)) + J (x xx)x
(26)
where JA (Xo) the Jacobian of at Xp- This linearization induces a local range-nullspace decomposition, enabling the use of SDB nonlinear systems. The procedure is as follows:
28

---

1. For measurement = Ax) + E compute an approximate maximum likelihood estimate of the signal x.
2. Linearize the operator at using eq. 26 and compute the Jacobian JA (x).
3. Treat y = (A(x)x + E1/2 as a linear Gaussian model and apply standard SDB with pseudoinverse and range/nullspace projections.
This approach leverages the Jacobian as local linear approximation of the nonlinear operator, effectively adapting SDB to more general, differentiable systems.
D Experimental setup
D.1 Training hyperparameters
We follow the training procedure proposed by Luo et al. 2023a using the ADAM optimizer Kingma
and Ba 2015 with an initial learning rate of 1 x 10 weight decay, and 32) (0.9, 0.99). multi-step learning rate scheduler is applied, halving the learning rate at the 36th, 60th, 72nd, and 90th epochs, in the original work. All methods size of 8. To ensure fairness across methods, we evaluate each model every 16 epochs and report the performance of the best checkpoint, rather than relying solely on the final
trained using the & loss function with batch
D.2 Computational requirements
All experiments conducted cluster of NVIDIA A100 GPUs, with each method trained using single GPU. The approximate training times for each task are as follows: 4 hours for super-resolution, 24 hours for MRI reconstruction, 48 hours for CT reconstruction, and 72 hours for inpainting.
E
Additional experimental results
E.1
Evaluation under a misspecified model (CT reconstruction)
Figure presents the results of an analogous analysis of performance of bridge methods under misspecified model (here CT reconstruction task), where the first part the experiment considers the default setting of of with increasing while the second part shows the results for 3.6 and increasing of similar manner to section 5.2 SDB variants achieve a clear advantage over other bridge methods when the system's 's parameters are perturbed.
FID (L)
LPIPS (1)
PSNR (1)
SSIM (T)
Meeen veho
CDOS
RD 14)
LPIPS (4)
PSNR (1)
SSIM (1)
1003
Hetti
271
AA05 Coaso
kg:e7
Ingurf
kggin]
Figure 4: Quantitative comparison of SDB (SB) with other bridge methods in misspecified CT reconstruction setting. The top row evaluates bridges trained with = 3.2, of = 0.0001 on data generated from systems with increasing The bottom evaluates performance data with т 3.6 and increasing of. Perturbing the original system makes the problem harder in both cases.
29

---

E.2
More challenging misspecified settings
We investigate more challenging and realistic misspecified scenarios for supervised inverse problems. including adversarial modifications of the forward operator and noise model. We evaluate SDB (SB) against supervised baselines across these settings.
E.2.1 Misspecified operator
We consider two superresolution scenarios: drastically enlarged downsampling kernel (32 x 32) and JPEG compression 30% quality, Both create inputs very different from training-ting time PRs. Table reports results across both tasks, with the task indicated in the first column. SDB (SB) consistently achieves the best performance in 3 out of 4 metrics.
Table 3: Superresolution under more challenging operators (bold indicates best values).
Method
Kernel 32 x 32
JPEG 30%
FID
LPIPS
PSNRT
SSIM
FID
LPIPS.
PSNR1
SSIM
IRSDE
130.47
0.684
17.85
0.381
99.43
0.283
24.81
0.651
12SB
131.33
0.681
17.84
0.381
93.62
0.322
25.12
0.668
DDBM
131.55
0.700
17.83
0.380
92.39
0.373
25.19
0.685
GOUB
134.66
0.690
17.85
0.370
101.08
0.291
24.91
0.662
SDB (SB)
129.87
0.682
17.86
0.390
90.26
0.380
25.41
0.692
E.2.2 Misspecified noise model
We also consider a misspecified noise model, replacing the Gaussian noise with a Poisson distribution to simulate photon-countin physics. For given signal sample measurements as y Poisson(L, Io exp(-Ax)) where Io is the incident photon intensity. We evaluate CT reconstruction, where Poisson noise closely reflects real-world X-ray physics. As shown in table 4 SDB outperforms all supervised baselines across different Lo values, even without explicitly modeling the Poisson distribution.
sampled
Table 4: CT reconstruction with Poisson noise. Bold indicates best values.
Method
1g 10000
1o
5000
Ig
1000
FID.
LPIPSL
PSNR
SSIM!
FID1
LPIPSL
PSNR1
SSIM1
FIDL
LPIPS.I
PSNRT
SSIM
IRSDE
17.29
0.0256
44.17
8.9681
17.86
0.0299
43.88
0.9665
21.12
0.0678
41.84
0.9506
DDBM
20.69
0.0402
44.11
0.9622
20.57
0.0406
43.93
0.9613
21.75
0.0575
42.53
0.9525
(SB)
We further consider MRI reconstruction under Poisson noise an adversarial setting, where no natural process typically modeled with Poisson noise, making highly out-of-distribution. Table shows that SDB again outperforms other methods in 3 out of metrics while almost matching the best SSIM.
E.3 Motion deblurring with an ill-conditioned system
To assess the practicality of SDB in ill-oondtiooned systems, we consider a motion deblurring task 128 128 flower images from the Flowers 102 dataset (Nilsback and Zisserman 2008 The system matrix modeled as a block Toeplitz matrix with Toeplitz blocks (BTTB), implemented via 2D convolutions in the pixel domain. For the pseudoinverse approximation, Tikhonov regularization (Wiener filtering) in the frequency domain. Although this differs from the exact pseudoinverse, SDB matches the best performance in LPIPS and clearly surpasses all supervised baselines in the other three metrics, demonstrating robustness to ill-conditioned operators (table
apply zero-order
E.4 Contrast recovery with nonlinear system
In the following, provide an initial study concerning the application of SDB to nonlinear systems.
30

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Table 5: MRI reconstruction with Poisson noise (bold indicates best values).
Method
Io 20.0
To
16.0
FID.
LPIPS|
PSNR
SSIMT
FID.
LPIPS_]
PSNR
SSIM
DDBM
40.2161
0.2647
21.9509
0.5572
39.0941
0.2360
22.6901
0.5919
GOUB
39.8257
0.2821
21.4963
0.5246
39.3272
0.2530
22.2050
0.5577
IRSDE
39.8516
0.2706
21.5772
0,5383
39.4095
0.2445
22.2666
0,5722
12SB
40.0352
0.2767
21.4514
0.5219
39.0571
0.2492
22.1715
0.5546
SDB (SB)
39.7303
0.2513
22.0684
0.5569
38.9677
0.2245
22.8237
0.5891
Method
Io
12.0
1o
8.0
FID|
LPIPS|
PSNR
SSIM1
FIDJ
LPIPS
PSNR1
SSIM
DDBM
38,2960
0.2036
23.6083
0.6345
36.6739
0.1653
24.7777
0.6873
GOUB
38.4142
0.2196
23.0794
0.5991
36.9538
0.1794
24.2220
0.6524
IRSDE
38.4208
0.2141
23.1378
0.6139
37.0698
0.1769
24.2971
0.6680
12SB
38.1304
0,2168
23.0778
0.5957
37.1202
0.1782
24.2578
0.6492
SDB (SB)
38.1484
0.1936
23.7615
0.6293
36.6346
0.1569
24.9707
0.6811
Table 6: Motion deblurring performance Flowers dataset using BTTB system matrix with Tikhonov-regularized pseudoinverse. Bold indicates best values.
Method
FID
LPIPS
PSNRT
SSIM
GOUB
17.03
0.053
23.84
0.813
IRSDE
16.27
0.033
26.76
0.842
DDBM
16.00
0.040
29.74
0,892
I2SB
15.62
0.025
29.29
0.874
SDB (SB)
14.89
0.025
30.39
0.903
In practice, the initial estimate the true signal x mentioned in appendix C.2 few iterations of gradient descent
can be obtained via
min
|A y12..
providing a reasonable starting point for linearization. Steps and (appendix C.2 involve computing the Jacobian at the given point and using it to obtain the pseudoinverse reconstruction (PR) and range- nullspace projections for SDB. These operations can be implemented efficiently without explicitly storing the Jacobian, using Jacobian- vector products (JVPs) gradient-based operations available in autodifferentiation frameworks such as Py Torch, or by exploiting an analytic expression for the Jacobian when available.
To illustrate the applicability of SDB nonlinear systems, we perform a proof-of- -ffnccept- benchmark on CIFAR10 Krizhevsky et al., 2009) using a nonlinear contrast operator
A = o okk(x xx),
where a is the sigmoid function, &o controls contrast strength, and a is contrast bias. We apply the linearization procedure described in appendix C.2] step uses 5 iterations of gradient descent to obtain an initial guess, while steps 2 and 3 use an analytic Jacobian to compute the pseudoinverse reconstruction and projections.
As shown in table 7 SDB (SB) outperforms all supervised baselines across all metrics in this nonlinear contrast recovery task. This demonstrates that, under nonlinear system responses, SDB can effectively leverage the locally linearized operator to produce high-fidelity reconstructions, achieving both superior perceptual quality (FID, LPIPS) and reconstruction accuracy (PSNR, SSIM). We emphasize that this represents only an initial proof-of-coneept study, and future work should investigate the applicability of SDB nonlinear systems more thoroughly.
E.5 Hyperparameter analysis
We conduct an ablation study to analyze the sensitivity of SDB to its hyperparameters. Following prior works [Zhou et al. 2024), stochastic process is undefined at f = 1, requiring sampling to
31

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FID (4)
LPIPS
0,296
60
0.8
0.01
59.92
54.34
39.93
36.23
36.30
36.40
36.51
0.01
0.830
0.775
0.190
0.129
0.129
0.130
0.130
.55
0.7
0.001
58.21
52.63
35.28
31.67
31.75
31.85
31.92
50
0.001
0.704
0.683
0.088
0.054
0.055
0.055
0.056
0.6
0.0001
31.45
4 0.0001
0.5
45
5c-05
58.25
31.45
5c-05
-10
1c-05
38.27
32.47
$8.87
31.32
31.45
33.51
51.37
1c-05
0.874
0.886
0.082
0.034
0.053
0.058
0.038
0.2
35
0.1
5c-06
58.26
$2.47
34.87
31.31
11.46
11.51
31.57
5e-06
0.874
0.860
0.082
0.054
0.055
0.058
0.058
0.0001
0.001
0.01
0.05
0.06
0.07
0.08
0.0001
0.001
0.01
0.05
0.07
0.08
PSNR
SSIM (1)
0.01
5.41
10.59
24.78
28.52
28.42
28.34
28.26
25
0.01
0.029
0.105
0.664
0.742
0.741
0.739
0.738
0.001
7.13
12.18
26.82
28.77
29.85
29.14
29.45
0.001
0.032
0.225
0.848
0.887
0.886
0.885
0.884
-00
0.0001
7.45
0.0001
0.859
15
5e-05
0.255
100
1e:05
7.49
12.30
27.04
29.81
29.69
29:38
29.49
1e:05
0.040
0.256
0.860
0.894
0.895
0.892
0.891
5c-06
7.49
12.50
27.04
29.81
29.69
25.58
29.49
5c-06
0.040
0.256
0.860
0.894
0.893
0.892
0.891
0.0001
0.001
0.01
0.05
0.06
0.07
0.08
0.0001
0.001
0.01
0.05
0.06
0.07
0.08
FID (4)
LPIPS (4)
Valuz
Value
PSNR T
SSIM
0.50
Value
Value
Figure 5: Top: Performance heatmaps for MRI reconstruction under varying (e1, using the optimal (bg, by) configuration. The red box highlights the I.PIPS-optimll setting. Bottom: Boxplots showing the best achievable performance across 64 (bo, b1) pairs on the superresolution task, where (E1, E2) are tuned separately for each (bg, b1) pair.
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Table 7: Nonlinear contrast reconstruction on CIFARIO. Bold indicates best values.
Method
FID.
LPIPS!
PSNR
SSIM
I2SB
2.97
0.00103
37.53
0.9906
GOUB
4.60
0.00182
32.54
0.9743
DDBM
1.92
0.00016
42.74
0.9963
IRSDE
4.35
0.00140
33.53
0.9789
SDB (SB)
1.31
0.00015
44.48
0.9977
begin / 61 for some 61 >> Similarly, we terminate sampling at 62, where 62 > 0. For SDB (VP) and SDB (VE), e1 and e2 are the only hyperparameters. In contrast, SDB (SB) includes an additional design choice: the shape of the null-space diffusion coefficient g(t). We parameterize it as
(t) l.gu.a((() + Le>u.s.(1-
where ((() = (vbp + [(vb) vbn) which is a continuous version of the coefficient proposed by Liu et al. 2023]. This introduces additional hyperparameters, bo, O. In the remainder of this section, we focus on SDB (SB), noting that we observed qualitatively similar trends for the VP
and VE variants.
We first examine how the choice of é and 62 influences the optimal configuration of (bo, b1) for (top)
SDB (SB) in the MRI reconstruction task. Figure (top) shows the performance trend when varying (c1,62) 62). We observe a simple and stable relationship with performance, where 6 has a stronger influence than Eg. Moreover, a broad plateau emerges, allowing for easy tuning. We also highlight the best configuration in terms of LPIPS, which lies close to the optimal Pareto frontier in this setting.
Next, we analyze the stability of SDB (SB) across different (bo, b1) configurations, Since these values determine the shape of the variance function, the optimal 61,62 may differ across (bo, by) pairs. Figure 5 (bottom) presents boxplots of the best performance for 64 configurations of (bo. b1) on the superresolution task. For each pair, we use grid search to identify the best e1, 62 values, leveraging the small size of the DIV2K dataset. The results reveal strong stability, with performance concentrated around the best observed value and only a few outliers.
E.6 Ablation study on noise schedules
We evaluate the effect of different schedules for the VP and VE variants of SDB, considering linear, quadratic, and cosine versions on the superresolution task. Using our notation:
a2
12
(quadratic VP), at = cos
(2)
IT
(cosine VP),
B1 0 m 1 (quadratic VE), Bg = max sin
T (51)
(cosine VE).
We also assume 7/6
σ 3 for VE.
Table 8: Ablation results for SDB (VP and VE) with different noise schedules. Lower FID and LPIPS indicate better perceptual quality; higher PSNR and SSIM indicate better reconstruction fidelity. Bold indicates the best value for each metric within VP VE.
SDB (VP)
SDB (VE)
Schedule
FID 1
LPIPS |
PSNR t
SSIM f
FID
LPIPS
PSNR
SSIM
Linear
87.08
0.228
25.91
0.724
94.73
0.226
25.90
0.718
Quadratic
75.25
0.174
24.77
0.680
76.88
0.159
25.29
0.678
Cosine 77.99
0.160
24.75
0.674
76.14
0.217
26.17
0.732
The results show a clear trade-off between perceptual quality (FID, LPIPS) and reconstruction fidelity (PSNR, SSIM). Linear schedules favor distortion metrics, while quadratic and cosine schedules improve perceptual metrics at the expense of reconstruction accuracy. In particular, Cosine SDB (VE) achieves the best values on three out of four metries, demonstrating that exploring alternative schedules can further improve performance.
33

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E.7
Computational complexity and overhead of SDB
We analyze the computational complexity and runtime overhead of SDB compared to other supervised baselines. First, we discuss the asymptotic complexity, followed by empirical measurements.
Complexity
For the general case where the pseudoinverse can be computed analytically using a stored matrix A, two cases arise: m 2 d and m < d. The pseudoinverse can be computed as
A ('TA'-A" AT or A A AT(ATT)=
respectively, The resulting asymptotic complexities summarized in table 9 Computing the PR has the same complexity (in the sense) as computing the pseudoinverse itself.
Table Asymptotic computational complexity for pseudoinverse computation and range/null space projections.
Operation Type
m N d (mdl dd) ((md2) (dLm)
m < d
Pseudoinverse
O m²) O(m²d)
Range Projection A
Null Projection (I A A)
((dm²)
In practice, we do explicitly construct A+ Instead, we implement the application of A, AT, A+, and their combinations to obtain projections and PRs. For each inverse problem, these imple- mentations differ. For example, in inpainting, A corresponds element-wise masking, whereas in CT reconstruction, truncated SVD is used to implement A efficiently, typically resulting in lower empirical complexity.
Empirical Overhead
We measured runtime in seconds for the most computationally intensive operations on the CT reconstruction task. The evaluation was conducted over four batches of size eight. We report three types of operations:
Forward Step Without Network: sampling from p(x: xo).
Forward Step With Network: Forward Step Without Network plus forward pass through the neural network,
Reverse Step With Network: sampling from p(x, X2,X1) using the neural network, corresponding to single discrete reverse-time step.
The empirical runtimes are summarized in table 10
Table 10: Runtime SDB and baseline supervised methods for the reconstruction task (seconds per batch of size 8).
Method
Forward Step Without Network
Forward Step With Network
Reverse Step With Network
SDB (SB)
0.0062
0.0885
0.5981 0.5851
12SB
0.0003
0.0821
IR-SDE
0.0003
0.0832
0.5895
GOUB
0,0008
0.0830
0,5863
DDBM
0.001 I
0.0825
0.5880
While Forward Step With Network shows small increase in computation time for SDB compared with other methods, this overhead negligible practice. The relative cost of the neural network forward pass dominates runtime, and matrix-base operations are efficiently parallelized the NVIDIA A 100 GPU used for all experiments. Consequently, SDB runtime during inference is nearly identical to that of other supervised methods.
34

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E.8 OT unpaired-datee umpaired-data-baseed baselines
We provide an additional small-scale comparison with recent OT-based unpaired-data methods (UNSB by Kim et al. 2024] and SBF De Bortoli et al. (024) on the MRI reconstruction task, as shown in table Following the original training procedures, we trained the models to map between the distributions of PRs and true signal observations. Since these methods lack direct supervision from paired data, their performance is naturally inferior to that of supervised approaches.
Table 11: MRI reconstruction results across all supervised and unpaired-data- FID and LPIPS indicate better perceptual quality; higher PSNR and SSIM indicate better reconstruc- tion fidelity.
methods. Lower
Unpaired-data-bused
Supervised
Metric
UNSB
SBF
12SB
IR-SDE
GOUB
DDBM
SDB (VP)
SDB (VE)
SDB (SB)
FID L
35.91
43.63
31.54
30.14
30.63
32.42
32.88
33.90
29.85
LPIPS
0.237
0.290
0.065
0.065
0.058
0.074
0.068
0.083
0.053
PSNR
18.72
18.46
28.75
28.88
28.59
28.97
29.26
29.10
29.81
SSIM
0.483
0.291
0.849
0.871
0.863
0.872
0.881
0.876
0,893
F
Extended discussion
F1 Advantages of Embedding Measurement System Information
Incorporating measurement system information into the generative process offers several conceptual and practical advantages. While earlier approaches that relied on hand-crafted priors often provided limited gains, the proposed SDB framework does not depend on such priors. Instead, SDB embeds the measurement system directly into the model dynamics, which naturally decomposes the inverse problem into two orthogonal components with distinct roles: the range space and the null space. This decomposition provides structured way to separate reconstruction from synthesis without manually injecting prior knowledge.
By embedding the measurement operator, SDB defines a more efficient stochastic path from the ob- served measurements the clean image, akin other diffusion bridge models that refine conditional diffusion by employing more informed reverse-process initializations. Within this framework, the neural network simultancously learns two complementary tasks: (i) an optional denoising task in the range spacc-typically simpler and often directly constrained by the and (ii) a synthesis task in the null space, which accounts for missing unobserved components.
In noise-free scenarios, this structure allows exact preservation of range-space content, ensuring that information already supported by the measurements remains unaltered. Under noisy conditions, the model benefits from the fact that the range space already encodes the correct underlying structure, requiring only localized refinement. Consequently, the range space acts structurally informative prior that guides the null-space synthesis. For example, in generative inpainting tasks-wherd the unmasked region is noise-free nearly so-SDB avoids unnecessary corruption of the observed region and concentrates its modeling capacity on generating the missing content.
F.2 Perception- Distortinn Tradeoff
The trade-off between perception and distortion mentioned by Blau and Michaeli 2018 is an important topic in the context of this work. Thanks to specific formalization, where perception is measured as a divergence between the true distribution and the one induced by the model, while distortion is quantified through the expected reconstruction error, one may hypothesize about the behavior of a given algorithm from a theoretical point of view. Following the formulation in Blau and Michaeli 2018), since perception and distortion are related through Pareto front, one can draw more principled conclusions from quantitative results.
The range-nullspace decomposition provided by the assumed system response matrix also enables deeper insights. In comparison to other methods, SDB's advantage lies in its differential treatment of the range and null spaces: it must only (optionally) denoise the former, while new content is
35

---

synthesized in the latter. This dual treatment leads to improved performance from the outset. In the noise-free case, the range space component is left untouched, resulting in zero distortion, while in the noisy case it undergoes milder transformations to reach the final result, which intuitively should also reduce distortion. Under noise-free scenario, is also easier to reason about perception: if the model synthesizes null-space content that perfectly aligns with the ground truth image, it achieves zero distortion at an inevitable cost in perception. Conversely, if it synthesizes null-space content that is fully in-distribution but deviates from the true signal, perception is maximized at the cost of distortion.
From theoretical perspective, in the limit, SDB samples from the true conditional distribution p(x y). which suggests that the method is inclined toward perfect perception. However, since we implicitly assume non- conee null space, SDB will always lack some information required to perfectly reconstruct the true x, mirroring the trade-off considered by Blau and Michaeli] (2018).
In practice, we observe a clear trend SDB outperforming all other methods in three out of four metrics in the super-resolution task (table 2 while obtaining slightly worse results in terms of LPIPS. Following Blau and Michaeli (2018). this observation leads to an interesting insight. Better FID, representing divergence between distributions, indicates that SDB achieves better perceptual results. However, it is also superior in terms of PSNR and SSIM, which measure reconstruction accuracy (distortion). Based on Blau and Michaeli 2018 one could argue that, in this task, the methods operate near the optimal performance. Since LPIPS does not quantify distributional divergence but instead measures perceptual distortion the space of neural network representations, it illustrates how specific type of semantic reconstruction must be sacrificed to move closer to the Pareto front. Moreover, the ablation study on noise schedulers for the VP and VE variants SDB shows how better LPIPS values can be recovered, providing another manifestation of this trade-off.
G Broader impact
By incorporating measurement system parameters into its SDE, SDB offers positive societal impacts by enabling more accurate and efficient reconstructions in critical applications such medical imaging, remote sensing, and scientific inverse problems, thereby supporting improved diagnostics, sustainability, and broader accessibility. However, it also poses potential negative societal risks, in- cluding misuse for surveillance or de-anonymization, amplification of bias from flawed measurement models, and overreliance on plausible but incorrect outputs in high-stakes domains. Additionally, the method could be repurposed for generating deceptive content from limited data. However, SDB and prior bridge methods have not been demonstrated industry-scale deployment levels, and in practice, they are typically suited for problem-specific applications rather than large-scale settings with vast amounts of data. This limits their potential for misuse in broader societal contexts.
H Additional visual results
We provide additional qualitative samples for inpainting (fig. 6), superresolution (fig. MRI reconstruction (fig. and CT reconstruction (fig. (i
36

---

x
12SB
IR-SDE
GOUB
DDBM
SDB (VP)
SDB (VE)
SDB (SB)
Figure 6: Qualitative comparison SDB variants with the best-performing baselines (bridge meth- ods). Rows depict the results for inpainting.
*
I29B
IR-SDE
GOUB
DDBM
SDB (VP)
SDB (VE)
SDB (SB)
Figure 7: Qualitative comparison SDB variants with the best-performing baselines (bridge meth- ods). Rows depict the results for superresolution.
37

---

X
12SB
IR-SDE
GOUB
DDBM
SDB (VP)
SDB (VE)
SDB (SB)
Figure 8: Qualitative comparison SDB variants with the best-performing baselines (bridge meth-
ods). Rows depict the results for MRI reconstruction.
*
12SB
IR-SDE
GOUB
DDBM
SDB (VP)
SDB (VE)
SDB (SB)
Figure 9: Qualitative comparison SDB variants with the best-performing baselines (bridge meth- ods). Rows depict the results for CT reconstruction.
38


Reviews:
[{"review_id": "oWGS1w62xX", "reviewer": "Reviewer_unSw", "summary": "The paper presents a supervised framework for solving linear inverse problems using score-based generative models. The core motivation is to embedding known information about the linear measurement system into the diffusion SDEs, which contrasts with previous diffusion bridge methods that often remain agnostic to this structural information. Experimental results demonstrate the effectiveness of the proposed method.", "questions": "Please refer to the section of Weaknesses.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The writing is clear and the motivations are promising.\n\n2. The construction of the system-embedded SDEs is reasonable and novel.\n\n3. Experimental results demonstrate the effectiveness of embedding measurement-system into diffusion SDEs.\n\nWeaknesses:\n\n1. The method relies on pseudo-inverse of the forward measurement matrix A. In the experiments the author use SVD to compute the pseudo-inverse.  However, for some linear measurement system, SVD is hard, e.g., motion deburing [R1]. This could limits the practicality of the proposed method.\n\n2. I believe that the conventional datasets for diffusion-based solvers (ImageNet, FFHQ) [R1，R2] should be included to better demonstrate the effectiveness of the proposed method.\n\n3. It is not immediately clear how the proposed model ensures sampling from the correct conditional distribution $p(x|y)$. The authors should provide further clarification or theoretical justification on this point.\n\nReference:\n\n[R1] Chung, Hyungjin, et al. \"Diffusion posterior sampling for general noisy inverse problems.\" arXiv preprint arXiv:2209.14687 (2022).\n\n[R2] Zhang, Bingliang, et al. \"Improving diffusion inverse problem solving with decoupled noise annealing.\" Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.", "was_revised": true, "final_justification": "The authors have clarified the concerns in the responses, including the proof demonstrating that the proposed method can achieve principled posterior sampling. Thus, I raise my rating.", "rebuttal": "We thank the reviewer for the insightful feedback and for appreciating our work. To address the noted weaknesses, we provide separate comments below.\n\n## (W1.) Limited practicality due to SVD computation\n\nWe appreciate this observation. In the main manuscript, we aimed to provide an intuitive view of the pseudoinverse via its relationship with SVD, where non-zero singular values are replaced with their reciprocals. However, SDB does not require explicit SVD or matrix storage. Instead, we rely on how $\\mathbf{A}$, $\\mathbf{A}^\\top$, and $\\mathbf{A}^+$ act on inputs. For instance, in the superresolution task, we do not store the downsampling matrix but apply $\\mathbf{A}$ via 2D average pooling. This allows for implementing $\\mathbf{A}^+$ as nearest-neighbor up-sampling, which is much more efficient than storing the dense SVD.\n\n### Motion deblurring task\n\nWe agree that numerical instabilities may affect SDB’s practicality, including SVD if used. To address this, we consider motion deblurring on 128×128 Flowers102 images [1]. The system matrix has a block Toeplitz structure with Toeplitz blocks (BTTB), implemented using 2D convolutions. For pseudoinverse approximation, we apply zero-order Tikhonov regularization (Wiener filtering) in the frequency domain. While approximate, SDB matches the best LPIPS and clearly outperforms all supervised baselines on the remaining metrics, as shown below.\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|GOUB|17.03|0.053|23.84|0.813|\n|IRSDE|16.27|0.033|26.76|0.842|\n|DDBM|16.00|0.040|29.74|0.892|\n|I2SB|15.62|**0.025**|29.29|0.874|\n|SDB (SB)|**14.89**|**0.025**|**30.39**|**0.903**|\n\n### Further analysis of stability\n\nWe note that CT reconstruction is the only case where we use a precomputed SVD of the system matrix. In this setting, truncated SVD serves as a regularization technique by discarding small singular values, typically tied to high-frequency components and noise. In our paper, the parameter $\\tau$ controls this regularization. During training, we set $\\tau=3.2$ to emulate a cut-off filter, zeroing out the lowest singular values which typically cause stability issues. Nonetheless, as Table 2 in the main paper shows, SDB consistently outperforms all baselines, indicating robustness to such regularization. This is further supported by the misspecified CT setting (Appendix E.1, Figure 4), where increasing $\\tau$ leads SDB to maintain or even expand its advantage.\n\n### Summary\n\nWe hope the two discussed experiments clarify concerns about SVD-related computations and their influence on SDB’s performance.\n\n## (W2.) Conventional datasets for diffusion-based solvers\n\nWe thank the reviewer for this suggestion. Regarding FFHQ [2], we find it highly similar to the CelebA-HQ dataset used in our work. Both feature high-resolution, centered human faces, with FFHQ being about twice as large (70,000 samples). As for ImageNet [3], we fully agree it would be a strong testbed for diffusion bridges. However, its scale makes training from scratch extremely costly. Even recent SOTA bridge methods for both paired (I2SB [4], IR-SDE, GOUB, DBBM) and unpaired (UNSB [5], SBF [6]) data rarely use ImageNet. I2SB is a notable exception — despite NVIDIA support and initialization from a pretrained unconditional ImageNet diffusion model, its authors recommend continued training with 2 nodes of 8×32GB V100 GPUs, highlighting the significant computational burden.\n\n## (W3.) Correct sampling from $p(x|y)$\n\nThank you for this comment. To begin, it is important to clarify that many other diffusion bridge methods such as I2SB, GOUB, or DDBM, use a non-Markovian stochastic process, where $\\mathbf{x}_t$ is a function of both $\\mathbf{x}_0$ and $\\mathbf{x}_1$, whereas our method uses a Markovian process, that is, a standard forward diffusion process, where $p(\\mathbf{x}_t|\\mathbf{x}_0)$ is well-defined. More specifically, our method is a special case of score-based generative models, using the general formulation with matrix-valued drift and diffusion coefficients in Appendix A of [8]. With this in mind, we provide below a formal theorem and proof showing that SDB is a principled posterior sampler of $p(\\mathbf{x}|\\mathbf{y})$. \n\n### Theorem 3\nGiven a forward measurement model $p(\\mathbf{y}|\\mathbf{x}) = \\mathcal{N}(\\mathbf{A} \\mathbf{x} , \\Sigma)$, one can generate principled posterior samples from $p(\\mathbf{x}|\\mathbf{y})$, using the reverse-time stochastic process of SDB, where the forward stochastic process as defined by Eqs. (1), (7), (8a), and (8b), parameterized by the time-dependent scalar coefficients, $\\gamma_t$, $\\alpha_t$, and $\\beta_t$, satisfy $\\lim_{t\\rightarrow 1} \\gamma_t = 1$ and $\\lim_{t\\rightarrow 1} \\frac{\\alpha_t^2}{\\beta_t} = 0$. Here \"principled\" means the training and sampling methodologies result in an asymptotically exact probabilistic model in the limit of infinite training data and infinite model capacity.  \n\n### Lemma 3.1 \nThe pseudoinverse reconstruction $\\mathbf{A}^{+}\\mathbf{y}$ is a sufficient statistic for estimation of $\\mathbf{x}$ given $\\mathbf{y}$. That is, $p(\\mathbf{x} | \\mathbf{y})= p(\\mathbf{x} | \\mathbf{A}^{+} \\mathbf{y})$. This is true because 1) the range space component of the measurements can be exactly recovered by re-applying $\\mathbf{A}$ to $\\mathbf{A}^{+} \\mathbf{y}$, and 2) the null space component contains no useful information about $\\mathbf{x}$. \n\n### Lemma 3.2 \nA sample from $p(\\mathbf{z}|\\mathbf{y}) = \\mathcal{N}(\\mathbf{A}^{+}\\mathbf{y},  \\beta_1 (\\mathbf{I} - \\mathbf{A}^{+} \\mathbf{A}))$ is also a sufficient statistic for estimation of $\\mathbf{x}$ given $\\mathbf{y}$. That is, $p(\\mathbf{x} | \\mathbf{y})= p(\\mathbf{x} | \\mathbf{z})$. This is true because the null-space component of the pseudoinverse reconstruction is zero. Therefore, there is no information in the null-space component that is useful for estimation of $\\mathbf{x}$ and so the additive null-space noise preserves sufficiency for estimation of $p(\\mathbf{x}|\\mathbf{y})$.   \n\n### Lemma 3.3\nThe SDB forward stochastic process at the final time step, $p(\\mathbf{x}_1|\\mathbf{x}_0=\\mathbf{x})$,  as defined by (1), (7), (8a), and (8b) is identically distributed to the SDB initializer, $p(\\mathbf{z}|\\mathbf{x})$  from Lemma 3.2., assuming \n\n$\\lim_{t\\rightarrow 1} \\gamma_t = 1$ and $\\lim_{t\\rightarrow 1} \\frac{\\alpha_t^2}{\\beta_t} = 0$.  \n\nThis is true because one can show they are both conditional Gaussian random vectors with the same mean, $\\mathbf{A}^{+} \\mathbf{A} \\mathbf{x}$, and same covariance, $\\mathbf{A}^{+} \\Sigma {\\mathbf{A}^{+}}^T +  \\beta_1 (\\mathbf{I} - \\mathbf{A}^{+} \\mathbf{A})$. \n\n### Lemma 3.4 \nThe trained SDB model is a principled sampler of $p(\\mathbf{x}_0 | \\mathbf{x}_1)$.  This is valid because the SDB is a subset of score-based generative models using a standard Markovian forward diffusion stochastic process.  According to Anderson theorem [7] and the general framework in Appendix A of [8], the reverse SDE in [8] using a learned score estimator in (3) results in a principled sampler of $p(\\mathbf{x}_0|\\mathbf{x}_1)$. To be clear, the score network is not the exact score function, but the methodology is \"principled\" because it approaches the exact score with large amounts of training data and many parameters in the score network.  \n\n### Proof of Theorem 3:\nThe following procedure results in principled posterior samples of $p(\\mathbf{x}|\\mathbf{y})$: \n1) Compute the pseudoinverse reconstruction $\\mathbf{A}^{+}\\mathbf{y}$\n2) Add null-space noise to sample $p(\\mathbf{z}|\\mathbf{y}) = \\mathcal{N}(\\mathbf{z}; \\mathbf{A}^{+}\\mathbf{y},  \\beta_1 (\\mathbf{I} - \\mathbf{A}^{+} \\mathbf{A}))$\n3) Sample the SDB reverse process $p(\\mathbf{x}_0=\\mathbf{x}  | \\mathbf{x}_1 = \\mathbf{z})$\n\nAccording to Lemma 3.1 and 3.2, step 2 results in a sufficient statistic, $\\mathbf{z}$, for estimation of $p(\\mathbf{x}|\\mathbf{y})$. According to Lemma 3.3, the initialization $\\mathbf{x}_1=\\mathbf{z}$ is valid because the SDB initializer, $p(\\mathbf{z}|\\mathbf{x})$, is identically distributed to the forward process, $p(\\mathbf{x}_1 | \\mathbf{x}_0=\\mathbf{x})$, assuming $\\mathbf{x}_0$ is sampled from $p(\\mathbf{x})$ during training. According to Lemma 3.4, the trained SDB model is a principled posterior sampler of $p(\\mathbf{x}_0=\\mathbf{x} | \\mathbf{x}_1=\\mathbf{z})$. Therefore, the result will be principled posterior samples of $p(\\mathbf{x}|\\mathbf{y})$.\n\n## Summary\n\nWe hope that the provided comments clarify the mentioned issues. We are open to further discussion if any further questions arise.\n\n---\n\n[1] Nilsback, M-E. and Zisserman, A., \"Automated flower classification over a large number of classes\",  Proceedings of the Indian Conference on Computer Vision, Graphics and Image Processing, 2008.\n\n[2] Zhang, B., et al. \"Improving diffusion inverse problem solving with decoupled noise annealing\", Proceedings of the Computer Vision and Pattern Recognition Conference. 2025.\n\n[3] Chung, H., et al. \"Diffusion posterior sampling for general noisy inverse problems\", International Conference on Learning Representations, 2023.\n\n[4] Liu, G.H., et al., \"I2SB: Image-to-Image Schrödinger Bridge\", International Conference on Machine Learning, 2023.\n\n[5] Kim, B., et al., \"Unpaired Image-to-Image Translation via Neural Schrödinger Bridge\", International Conference on Learning Representations, 2024.\n\n[6] De Bortoli, V., et al., \"Schrödinger Bridge Flow for Unpaired Data Translation\", Neural Information Processing Systems, 2024.\n\n[7] Anderson, Brian DO. \"Reverse-time diffusion equation models.\" Stochastic Processes and their Applications 12.3 (1982): 313-326.\n\n[8] Song, Yang, et al. \"Score-based generative modeling through stochastic differential equations.\" arXiv preprint arXiv:2011.13456 (2020)."}, {"review_id": "qz7LUp8CON", "reviewer": "Reviewer_Tpm3", "summary": "This work proposes System-embedded Diffusion Bridge Models (SDBs) that embed the known linear measurement system into the coefficients of the SDE process for solving linear inverse problems. Experiments show the effectiveness of the proposed method.", "questions": "1. The experiments, while covering standard benchmarks, are all 2D, moderately sized, and with relatively controlled corruptions. It would strengthen the work to see results on more diverse and challenging real-world scenarios (e.g., more realistic noise, higher noise levels, extreme undersampling, or out-of-distribution corruptions).", "limitations": "The authors have provided the limitations of their work.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strength:**\n1. This work provides a theoretically grounded method for embedding measurement system into the inverse process, filling the gap of prior bridge methods;\n2. The proposed SDB is shown to be robust when the measurement system differs between training and deployment;\n3. Experiments shown the effectiveness of the proposed method across diverse domains, including natural images and medical images;\n\n**Weaknesses:**\n1. While the paper includes some evaluations for system parameter changes, these are controlled perturbations. It would be valuable to explore settings where the operator A or noise model are misspecified in more adversarial or realistic ways (e.g., using a wrong noise distribution, or a blurred operator);\n2. As the proposed SDB requires paired data for supervised training, which may not always be available, especially for rare measurements or privacy-sensitive domains. It does not address scenarios where only unpaired data is accessible, limiting applicability compared to plug-and-play or unsupervised diffusion approaches.\n3. There is little ablation on key design choices such as the choice of noise schedules, the effect of imperfect A, or the benefit of each SDB component.\n4. Some theoretically motivated variants (SDB-SB, SDB-VP, SDB-VE) are included, but the empirical results do not fully disentangle when each variant is preferable. For instance, SDB-SB generally performs best, but it is not entirely clear under what circumstances SDB-VP/VE might be better or whether their variance schedules are optimal for the tasks at hand.", "was_revised": true, "final_justification": "Considering the authors' rebuttal that solved most of my concerns, I keep my positive score.", "rebuttal": "We thank the reviewer for the insightful comments and appreciation of our work. We address the mentioned weaknesses and questions below.\n\n## (W1.) More challenging misspecified settings\n\nWe agree with the reviewer that, although the system parameter changes we presented correspond to realistic scenarios, they represent controlled perturbations. To address this, we evaluate four additional settings that are either more adversarial or more reflective of practical degradations.\n\n### Misspecified operator\n\nFirst, we evaluate all supervised methods under a drastically modified operator in the superresolution task. We increase the downsampling kernel size from the default $4\\times4$ to $32\\times32$, which yields inputs sharing very little mid- and high-frequency content with the training-time pseudoinverse reconstructions (PRs). While the performance gap between methods naturally shrinks due to the increased difficulty, SDB remains the best-performing method in 3 out of 4 metrics.\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|IRSDE|130.47|0.684|17.85|0.381|\n|I2SB|131.33|**0.681**|17.84|0.381|\n|DDBM|131.55|0.700|17.83|0.380|\n|GOUB|134.66|0.690|17.85|0.370|\n|SDB (SB)|**129.87**|0.682|**17.86**|**0.390**|\n\nNext, we replace the downsampling operator with JPEG compression preserving only 30% of the original image quality, simulating high-frequency loss via a markedly different mechanism. The results replicate our original ranking, with SDB (SB) outperforming all others in 3 of 4 metrics.\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|IRSDE|99.43|**0.283**|24.81|0.651|\n|GOUB|101.08|0.291|24.91|0.662|\n|I2SB|93.62|0.322|25.12|0.668|\n|DDBM|92.39|0.373|25.19|0.685|\n|SDB (SB)|**90.26**|0.380|**25.41**|**0.692**|\n\n### Misspecified noise model\n\nWe also consider a misspecified noise setting, replacing the Gaussian noise with Poisson noise to simulate photon counting physics. Specifically, given a signal sample $\\mathbf{x}$, measurements $\\mathbf{y}$ are drawn from $\\text{Poisson}(\\mathbf{I}_0 \\cdot \\exp(-\\mathbf{A}\\mathbf{x}))$, where $\\mathbf{I}_0$ is the incident photon intensity. We evaluate on the CT reconstruction task, where this noise model closely reflects X-ray acquisition physics. Despite never explicitly observing Poisson noise and relying only on its Gaussian approximation in the limit, SDB outperforms all supervised baselines across different intensity values.\n\n|Method|$\\mathbf{I}_0=10000$||| |$\\mathbf{I}_0=5000$||| |$\\mathbf{I}_0=1000$||| |\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n| |FID↓|LPIPS↓|PSNR↑|SSIM↑|FID↓|LPIPS↓|PSNR↑|SSIM↑|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|GOUB|17.78|0.0230|44.85|0.9717|17.83|0.0240|44.59|0.9705|20.52|0.0491|42.79|0.9588|\n|IRSDE|17.29|0.0256|44.17|0.9681|17.86|0.0299|43.88|0.9665|21.12|0.0678|41.84|0.9506|\n|DDBM|20.69|0.0402|44.11|0.9622|20.57|0.0406|43.93|0.9613|21.75|0.0575|42.53|0.9525|\n|I2SB|21.91|0.1014|41.68|0.9242|21.82|0.1018|41.54|0.9231|22.43|0.1170|40.44|0.9130|\n|SDB (SB)|**14.03**|**0.0193**|**46.05**|**0.9799**|**14.61**|**0.0203**|**45.70**|**0.9785**|**19.80**|**0.0461**|**43.44**|**0.9657**|\n\nTo propose a more adversarial setting, we also replace the Gaussian noise with Poisson noise in the MRI reconstruction task, where Poisson noise is typically unnatural, thus imposing a highly out-of-distribution challenge. SDB again outperforms all methods in 3 of 4 metrics across all intensities, while nearly matching the best SSIM.\n\n|Method|$\\mathbf{I}_0=20.0$| | | |$\\mathbf{I}_0=16.0$| | | |$\\mathbf{I}_0=12.0$| | | |$\\mathbf{I}_0=8.0$| | | |\n|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|-|\n| |FID↓|LPIPS↓|PSNR↑|SSIM↑|FID↓|LPIPS↓|PSNR↑|SSIM↑|FID↓|LPIPS↓|PSNR↑|SSIM↑|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|DDBM|40.22|0.265|21.95|**0.557**|39.09|0.236|22.69|**0.592**|38.30|0.204|23.61|**0.635**|36.67|0.165|24.78|**0.687**|\n|GOUB|39.83|0.282|21.50|0.525|39.33|0.253|22.21|0.558|38.41|0.220|23.08|0.599|36.95|0.179|24.22|0.652|\n|IRSDE|39.85|0.271|21.58|0.538|39.41|0.244|22.27|0.572|38.42|0.214|23.14|0.614|37.07|0.177|24.30|0.668|\n|I2SB|40.04|0.277|21.45|0.522|39.06|0.249|22.17|0.555|38.13|0.217|23.08|0.596|37.12|0.178|24.26|0.649|\n|SDB (SB)|**39.73**|**0.251**|**22.07**|**0.557**|**38.97**|**0.225**|**22.82**|0.589|**38.15**|**0.194**|**23.76**|0.629|**36.63**|**0.157**|**24.97**|0.681|\n\n## (W2.) Limitation to paired data\n\nWe thank the reviewer for highlighting this important limitation. We agree that unpaired data scenarios are critical in many practical applications, where paired-data methods, including SDB, are not currently applicable. This limitation fundamentally distinguishes paired-data supervised approaches from plug-and-play or unsupervised methods, which trade off some performance for broader applicability. We will include a discussion of this limitation and potential directions in the updated manuscript.\n\n## (W3.) Additional ablation studies\n\nThe original evaluation of misspecified measurement systems was broad, covering two parameters for both the CT and MRI tasks (see Figure 3 for MRI and Figure 4 (Appendix) for CT). In general, we ensured the evaluation focused entirely on i) benefits of replacing scalar SDEs with matrix-valued ones, and ii) differentiating the effect of noise scheduling. To isolate these factors, all baselines (both un- and supervised) were retrained from scratch using the same architecture, training procedures, and optimization hyperparameters—varying only the SDE formulation.\n\nRegarding noise schedule design, Appendix E.2 includes a broad hyperparameter analysis for SDB (SB). Since SB schedule fixes a parameterization, we verify which \"shape\" performs best by grid-searching over $b_0$, $b_1$, and truncation points $\\epsilon_1$, $\\epsilon_2$.\n\n### Ablation studies with linear, quadratic and cosine scheduling\n\nWe agree the schedule choice for VP and VE variants was unaddressed. We thus test linear, quadratic, and cosine schedulings. Using our notation: $\\alpha_t = 1 - t^2$ (quadratic SDB (VP)), $\\alpha_t = \\cos\\left(\\frac{\\pi}{2} t\\right)$ (cosine SDB (VP)), $\\beta_t = \\sigma_{\\max} t^2$ (quadratic SDB (VE)), and $\\beta_t = \\sigma_{\\max} \\sin\\left(\\frac{\\pi}{2} t\\right)$ (cosine SDB (VE)). Other coefficients follow lines 202–203 of our submission, assuming $\\gamma_t = \\sigma_{\\max}^{-1} \\beta_t$ for SDB (VE).\n\n#### SDB (VP)\n\n|Schedule|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|Linear|87.08|0.228|**25.91**|**0.724**|\n|Quadratic|**75.25**|0.174|24.77|0.680|\n|Cosine|77.99|**0.160**|24.75|0.674|\n\n#### SDB (VE)\n\n|Schedule|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|Linear|94.73|0.226|25.90|0.718|\n|Quadratic|76.88|**0.159**|25.29|0.678|\n|Cosine|**76.14**|0.217|**26.17**|**0.732**|\n\nMore sophisticated schedulers yield a trade-off between perception (FID, LPIPS) and distortion (PSNR, SSIM). Linear favors distortion; quadratic and cosine sacrifice distortion for perceptual quality. Interestingly, with Cosine SDB (VE), we set a new state-of-the-art on 3 of 4 metrics when compared with the initial baselines, showing that deeper exploration of SDB dynamics is promising.\n\n## (W4.) Preference over SDB variants\n\nWe thank the reviewer for noting the need to clarify preference among SDB variants. SDB (SB) is preferred, based on theory (Theorem 2) and experiments (Table 2). For VP and VE, we suggest prioritizing VP for performance, supported by its ranking over VE in Table 2. Visual inspection indicates VE yields increased diversity, matching literature on unconditional diffusion models.\n\nThe ablation above indicates a cosine schedule for VE and linear for VP is preferred. Appendix E.2’s ablation over SB shapes confirms that, for a given reconstruction task, choosing an optimal schedule is a complex and challenging problem.\n\n## (Q1.) More diverse and real-world scenarios\n\nWe hope that the new results on extreme misspecifications above already showcase SDB’s robustness. We add two new benchmarks for other real-world scenarios.\n\n### Motion blur\n\nTo test SDB on an ill-conditioned, rank-deficient system, we benchmark motion blur on 128×128 Flowers102 images [1], per Reviewer unSw’s suggestion. The system matrix is a block Toeplitz with Toeplitz blocks (BTTB), implemented as 2D convolutions. For pseudoinverse, zero-order Tikhonov regularization (Wiener filtering) is applied in frequency domain. Though it must handle the approximate pseudoinverse application, SDB matches the best LPIPS and outperforms supervised baselines on other metrics, as shown below:\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|GOUB|17.03|0.053|23.84|0.813|\n|IRSDE|16.27|0.033|26.76|0.842|\n|DDBM|16.00|0.040|29.74|0.892|\n|I2SB|15.62|**0.025**|29.29|0.874|\n|SDB (SB)|**14.89**|**0.025**|**30.39**|**0.903**|\n\n### Nonlinear contrast\n\nTo show SDB’s applicability to nonlinear systems, we benchmark CIFAR10 [2] with a nonlinear contrast system $\\mathbf{A}(\\mathbf{x}) = \\sigma(k(\\mathbf{x} - \\alpha))$, where $\\sigma$ is sigmoid, $k$ controls contrast, and $\\alpha$ is bias. We apply the linearization method from response **(W2., Q4.)**. The table shows SDB once again outperforming supervised baselines:\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|I2SB|2.97|0.00103|37.53|0.9906|\n|GOUB|4.60|0.00182|32.54|0.9743|\n|DDBM|1.92|0.00016|42.74|0.9963|\n|IRSDE|4.35|0.00140|33.53|0.9789|\n|SDB (SB)|**1.31**|**0.00015**|**44.48**|**0.9977**|\n\n## Summary\n\nWe hope these comments address the reviewer’s concerns and are open to further discussion.\n\n---\n\n[1] Nilsback, M-E. and Zisserman, A., \"Automated flower classification over a large number of classes\", Proceedings of the Indian Conference on Computer Vision, Graphics and Image Processing, 2008.\n\n[2] Krizhevsky, A., \"Learning Multiple Layers of Features from Tiny Images\", 2009."}, {"review_id": "UDlgF0FRPC", "reviewer": "Reviewer_FGzg", "summary": "This paper introduces System-embedded Diffusion Bridge Models (SDBs), a framework for supervised diffusion bridge methods that explicitly embeds the known linear measurement system into the coefficients of matrix-valued stochastic differential equations. The key contribution lies in leveraging the range-nullspace decomposition of linear operators to design specialized diffusion processes that handle range and null space components differently. The authors propose three variants (SDB-SB, SDB-VP, SDB-VE) and demonstrate improvements over existing bridge methods (I2SB, IR-SDE, GOUB, DDBM) across four inverse problems: image inpainting, super-resolution, CT reconstruction, and MRI reconstruction.", "questions": "I would like to know following the cons:\n\n1. Thm1 applicability. The Magnus expansion approximation becomes exact when drift matrices commute. How restrictive is this condition for the proposed SDB variants? Do the specific choices of parameters in equations (8a-8b) satisfy this condition?\n2. Pseudoinverse stability. For rank-deficient measurement matrices A, how does numerical stability of the pseudoinverse computation affect SDB performance? Have the authors considered regularized pseudoinverses for ill-conditioned systems?\n3. What is the computational complexity of the matrix-valued SDE compared to scalar formulations? How does this scale with problem dimension d and measurement dimension m?\n4. The authors mention local linearization as a potential workaround for nonlinear measurement systems. Can you provide more details on how this would work in practice and its limitations? e.g., failure cases\n5. The current formulation treats range and null spaces independently. Could there be benefits to introducing coupling terms between these spaces?", "limitations": "please refer to the questions and cons.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "pros:\n1. The principled embedding of measurement systems into matrix-valued SDE coefficients represents a theoretical advancement beyond scalar SDEs. The connection to optimal transport theory through Theorem 2 provides solid mathematical foundation.\n2. The paper evaluates across diverse inverse problems with varying system complexities, from simple inpainting to complex medical imaging tasks (CT/MRI reconstruction), demonstrating broad applicability.\n3. The systematic evaluation under system misspecification (Section 5.2) addresses a critical practical concern, showing SDB's superior generalization when training and deployment parameters differ.\n4.  The range-nullspace decomposition provides intuitive understanding of how SDB handles known (range) and missing (null space) information differently, which is conceptually appealing for inverse problems.\n4. SDB consistently outperforms baselines across metrics, with particularly notable improvements in medical imaging tasks where the gains are substantial.\n\ncons:\n1. While Theorem 1 from Tivnan et al. [2025] provides the foundation, the paper lacks theory analysis of the novel matrix-valued formulation. The conditions under which the Magnus expansion approximation becomes equality (commuting matrices) are restrictive and not proper discussed.\n2. The framework is limited to linear Gaussian measurement models. While the authors acknowledge this limitation, many real-world inverse problems involve nonlinear degradations that cannot be addressed by this approach.\n3. The paper does not analyze the computational overhead of the matrix-valued SDE formulation compared to scalar methods. The range-nullspace decomposition and pseudoinverse computations may introduce significant computational costs and bias.\n4. While the paper compares against recent bridge methods, it lacks comparison with state-of-the-art unsupervised methods like DPS, πGDM variants, or recent optimal transport approaches that don't require paired training data.\n5. The authors use simple linear scheduling for range and null space coefficients, missing opportunities for more sophisticated designs that could better leverage the dual-process structure.", "was_revised": true, "final_justification": "I thank the authors for their detailed response and effort on the results. they had addressed my concerns. hope authors can revise it in the final version accordingly. I maintain my positive rating.", "rebuttal": "We thank the reviewer for their insightful feedback. Below we address the noted weaknesses and questions.\n\n## (W1., Q1.) Applicability of Theorem 1, restrictiveness of the Magnus expansion\n\nAs noted, the matrix integral form holds if the drift matrices $F_t$ commute for all $t\\in[0,1]$. This condition is always satisfied for SDB under Eq. (10a): the only time-dependent term is scalar $\\alpha_t$, so $F_t$ has constant singular vectors and thus commutes for all $t$. This applies to all scalar diffusion processes (e.g., VP, VE, Flow Matching OT [1], etc.), Fourier Diffusion [2], and Subspace Diffusion [4] models. We are not aware of any methods with non-commuting $F_t$, but even if present, Theorem 1 still applies via the Magnus expansion [3].\n\n## (Q2.) Influence of pseudoinverse stability\n\nWe address pseudoinverse stability through both the CT reconstruction task (with a misspecified setting from the main manuscript) and a new motion deblurring benchmark using Tikhonov regularization.\n\n### Truncated SVD in CT reconstruction\n\nThe system matrix models line integrals in CT, with SVD precomputed and used in all operations. Truncated SVD is standard for regularization—zeroing out low singular values linked to noise/high frequencies. The $\\tau$ threshold controls its strength; during training we set $\\tau=3.2$ to simulate low-dose scenarios. As shown in Table 2, SDB consistently outperforms all baselines, showing robustness to regularization. These findings are reinforced in the misspecified setting (Appendix E.1, Fig. 4), where increasing $\\tau$ only strengthens SDB’s relative performance.\n\n### Tikhonov regularization in motion deblurring\n\nTo explore a distinct ill-posed case, we include a benchmark on 128×128 flower images (Flowers102 dataset [5]), following Reviewer unSw’s suggestion. The system matrix is a BTTB (block Toeplitz with Toeplitz blocks) from 2D convolutions. We apply the pseudoinverse via zero-order Tikhonov regularization (Wiener filtering) in the frequency domain whenever needed (e.g., range projection). Despite approximation error, SDB matches the best LPIPS and surpasses all supervised baselines in FID, PSNR, and SSIM:\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|GOUB|17.03|0.053|23.84|0.813|\n|IRSDE|16.27|0.033|26.76|0.842|\n|DDBM|16.00|0.040|29.74|0.892|\n|I2SB|15.62|**0.025**|29.29|0.874|\n|SDB (SB)|**14.89**|**0.025**|**30.39**|**0.903**|\n\n### Summary\n\nThese results demonstrate SDB’s resilience to pseudoinverse instability. We’re happy to include further details (e.g., on Flowers102 training) in the main text or Appendix.\n\n## (W2., Q4.) Application of SDB to nonlinear systems via local linearization\n\nWe outline how SDB applies to nonlinear systems via local linearization, then demonstrate this with a proof-of-concept experiment on nonlinear contrast in CIFAR10. Although SDB was not originally designed for nonlinear problems, the initial results demonstrate strong effectiveness.\n\n### Theory\n\nSDB assumes a global range-nullspace decomposition of $\\mathbf{A}$, valid in linear cases but not nonlinear. For a differentiable nonlinear $\\mathbf{A}$, we approximate it via a first-order Taylor expansion:\n\n$$\\mathbf{A}(\\mathbf{x}) \\approx \\mathbf{A}(\\mathbf{x}_0) + \\mathbf{J}_A(\\mathbf{x}_0)(\\mathbf{x} - \\mathbf{x}_0) \\tag{1}$$\n\nwith $\\mathbf{x}_0$ near $\\mathbf{x}$. The Jacobian $\\mathbf{J}_A$ defines a local linearization of $\\mathbf{A}$, enabling pseudoinverse and range/nullspace projections for SDB in a global sense, but at a natural cost of inaccuracies. Theoretically, the full procedure is:\n1. Given $\\mathbf{y} = \\mathbf{A}(\\mathbf{x}) + \\boldsymbol{\\Sigma}^{1/2}\\epsilon$, estimate $\\mathbf{x}$ with $\\hat{\\mathbf{x}}$ via maximum likelihood.\n2. Linearize $\\mathbf{A}$ at $\\hat{\\mathbf{x}}$, get $\\mathbf{J}_A(\\hat{\\mathbf{x}})$.\n3. Approximate $\\mathbf{y} \\approx \\mathbf{J}_A(\\hat{\\mathbf{x}})\\mathbf{x} + \\boldsymbol{\\Sigma}^{1/2}\\epsilon$, which forms a linear Gaussian model suitable for SDB.\n\n### Practice\n\nIn practice, step 1 uses a few gradient descent steps on $\\min_{\\hat{\\mathbf{x}}} || \\mathbf{A}(\\hat{\\mathbf{x}}) - \\mathbf{y} ||_2$ for an initial guess. Steps 2–3 use Jacobian-vector products (JVPs) or gradients—easily available via autodiff (e.g., PyTorch)—or analytic Jacobians if possible.\n\nTo test this, we benchmark nonlinear contrast on CIFAR10 [8], with $\\mathbf{A}(\\mathbf{x}) = \\sigma(k(\\mathbf{x} - \\alpha))$, where $\\sigma$ is sigmoid, $k$ controls contrast, and $\\alpha$ is bias. We apply our linearization steps: step 1 uses 5 gradient descent steps, and steps 2–3 use the analytic Jacobian.\n\nSDB outperforms all baselines under this nonlinear setup:\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|I2SB|2.97|0.00103|37.53|0.9906|\n|GOUB|4.60|0.00182|32.54|0.9743|\n|DDBM|1.92|0.00016|42.74|0.9963|\n|IRSDE|4.35|0.00140|33.53|0.9789|\n|SDB (SB)|**1.31**|**0.00015**|**44.48**|**0.9977**|\n\n### Summary\n\nThis provides early evidence that SDB can handle nonlinear problems, which we view as a promising direction for future work.\n\n## (W3., Q3.) Computational overhead and complexity of SDB\n\nWe first discuss the asymptotic complexity, then show that SDB’s overhead is negligible in practice.\n\n### Complexity\n\nIf $\\mathbf{A}$ is stored in memory and $\\mathbf{A}^+$ is computed analytically, we distinguish two cases: $m \\geq d$ and $m < d$. Then, $\\mathbf{A}^+ = (\\mathbf{A}^\\top \\mathbf{A})^{-1} \\mathbf{A}^\\top$ or $\\mathbf{A}^+ = \\mathbf{A}^\\top (\\mathbf{A} \\mathbf{A}^\\top)^{-1}$ respectively. The resulting complexities:\n\n|Operation|$m \\geq d$|$m < d$|\n|-|-|-|\n|Pseudoinverse|$O(md^2 + d^3)$|$O(m^2d + m^3)$|\n|Range Projection $\\mathbf{A}\\mathbf{A}^+$|$O(md^2)$|$O(m^2d)$|\n|Null Projection $(\\mathbf{I} - \\mathbf{A}^+\\mathbf{A})$|$O(d^2m)$|$O(dm^2)$|\n\nIn practice, we don’t form $\\mathbf{A}^+$ explicitly, but use its actions via matrix-vector operations. These vary by task: e.g., inpainting uses a binary mask, CT uses truncated SVD. Actual runtime is often lower than theoretical bounds.\n\n### Overhead\n\nWe report runtime (in seconds) on CT reconstruction—the most demanding setting—averaged over 4 batches (size 8, A100 GPU). Three steps are shown: **Forward Step Without Network** (sampling $p(\\mathbf{x}_t|\\mathbf{x}_0)$), **Forward With Network** (adds NN forward pass), and **Reverse With Network** (a single reverse-time step).\n\n|Method|Forward w/o Net (s)|Forward w/ Net (s)|Reverse w/ Net (s)|\n|-|-|-|-|\n|SDB (SB)|0.0062|0.0885|0.5981|\n|I2SB|0.0003|0.0821|0.5851|\n|IRSDE|0.0003|0.0832|0.5895|\n|GOUB|0.0008|0.0830|0.5863|\n|DDBM|0.0011|0.0825|0.5880|\n\nWhile projections increase **Forward w/o Net** time, the **Forward w/ Net** and **Reverse** steps are dominated by network inference. Thanks to CUDA parallelism on A100 GPUs, SDB’s overall runtime closely matches that of other methods.\n\n## (W4.) Comparison with SOTA baselines\n\nWe note that both DPS and $\\Pi\\text{GDM}$ are already included in Table 2 of the main manuscript.\n\n### Two new optimal transport baselines (unpaired)\n\nWe thank the reviewer for suggesting unpaired OT methods. While typically absent from bridge literature, we include two: Unpaired Neural Schrödinger Bridge (UNSB) [6] and Schrödinger Bridge Flow (SBF) [7]. We trained both on MRI reconstruction (between PRs and true signals). Though unsupervised, they perform competitively:\n\n|Method|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|UNSB|35.91|0.237|18.72|0.483|\n|SBF|43.63|0.290|18.46|0.291|\n\nDue to time limits, we couldn’t run all datasets but will include full results in the paper.\n\n## (W5.) More sophisticated coefficient design\nThe SB variant of SDB already uses nonlinear coefficient scheduling; Appendix E.2 includes extensive ablations on its shape and truncation.\n\n### New ablation: linear, quadratic, cosine scheduling\nFor VP and VE variants, we now evaluate quadratic and cosine schedules on superresolution. Using our paper’s notation:  \n- VP: $\\alpha_t=1-t^2$ (quadratic), $\\alpha_t=\\cos(\\frac{\\pi}{2}t)$ (cosine)  \n- VE: $\\beta_t=\\sigma_{max}t^2$ (quadratic), $\\beta_t=\\sigma_{max}\\sin(\\frac{\\pi}{2}t)$ (cosine)  \nwith $\\gamma_t = \\sigma_{max}^{-1}\\beta_t$.\n\n#### SDB (VP)\n|Schedule|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|Linear|87.08|0.228|**25.91**|**0.724**|\n|Quadratic|**75.25**|0.174|24.77|0.680|\n|Cosine|77.99|**0.160**|24.75|0.674|\n\n#### SDB (VE)\n|Schedule|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|Linear|94.73|0.226|25.90|0.718|\n|Quadratic|76.88|**0.159**|25.29|0.678|\n|Cosine|**76.14**|0.217|**26.17**|**0.732**|\n\nSophisticated schedules shift the perception–distortion trade-off. Linear favors distortion metrics; quadratic and cosine improve perceptual ones. Cosine SDB (VE) outperforms all methods in 3 of 4 metrics, setting a new SOTA. These results suggest deeper study of SDB’s dynamics is worthwhile.\n## (Q5.) Coupling range and null spaces\nWe appreciate the reviewer’s observation. While range and null spaces are orthogonal, their contributions to the SDE are separately defined by $\\alpha_t, \\beta_t, \\gamma_t$, but can be functionally related, as in SDB (VP) and SDB (VE). SDB estimates the score via $\\mathrm{E}[\\mathbf{x}_0|\\mathbf{x}_t]$, so the network inputs and outputs span both subspaces and implicitly learn the coupling, as the reviewer suggests—e.g., in inpainting, unmasked data guides masked estimation.\n## Summary\nWe hope these responses address all concerns and remain open to further questions.\n\n---\n[1] Lipman et al., \"Flow Matching for Generative Modeling\", ICLR 2023.  \n[2] Tivnan et al., \"Fourier Diffusion Models\", IEEE TMI, 2025.  \n[3] Kamm et al., \"Stochastic Magnus Expansion and SPDEs\", J. Sci. Comput., 2021.  \n[4] Bowen et al., \"Subspace Diffusion Generative Models\", ECCV 2022.  \n[5] Nilsback & Zisserman, \"Automated Flower Classification\", ICVGIP 2008.  \n[6] Kim et al., \"Neural Schrödinger Bridge\", ICLR 2024.  \n[7] De Bortoli et al., \"Schrödinger Bridge Flow\", NeurIPS 2024.  \n[8] Krizhevsky, \"Learning Features from Tiny Images\", 2009."}, {"review_id": "ORBRxydbZ0", "reviewer": "Reviewer_JgTN", "summary": "This work identifies that many existing supervised methods for inverse problem solving ignore the a priori knowledge of the degradation operator. Motivated by this findings, the authors develop a new type of Schrödinger bridge model that takes the degradation operator into account for choosing the form of the SDE applied in training. In several experiments, the authors demonstrate the superiority of their approach over previous methods.", "questions": "Please see weaknesses.\nAdditionally:\n* Could the proposed method be applied such that several degradations are handled by a single model?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* The paper’s motivation is clearly established, and the writing is easy to follow. I find that system-embedded diffusion bridges are an insightful extension to Schrödinger bridge models, making the most of the a priori knowledge of the inverse problem.\n* The evaluations are sound, and the results shown in the paper are encouraging. Moreover, SDB’s generalization capabilities are non-trivial.\n\nWeaknesses:\n* While the empirical results clearly show an improvement, I believe the manuscript lacks a theoretical analysis, hypothesis or discussion for why methods that embed the measurement information into the sampling scheme are desired. Many fields in deep learning have shown that hand-crafted priors may offer only limited benefits over systems that purely model the training data, for fully converged networks with sufficient data. The authors should take extra care to demonstrate that the increase in performance is solely driven by the measurement-embedding technique, and disentangle all other differences.\n* I believe that a discussion of the Perception-Distortion Tradeoff [1] is warranted for the comparisons made in this work.\n* Minor: I believe the paper would be easier to follow if a training and sampling algorithm was included.\n\n[1] Blau, Yochai, and Tomer Michaeli. \"The perception-distortion tradeoff.\" Proceedings of the IEEE conference on computer vision and pattern recognition. 2018.", "was_revised": true, "final_justification": "I believe this an insightful paper with a novel approach and should be accepted. While the concerns I have raised have only been partially addressed by the rebuttal, I believe that the paper is of more than sufficient quality for acceptance.", "rebuttal": "We are grateful to the reviewer for the insightful feedback and such appreciation of our work. We aim to address the concerns with the comments below.\n\n## (W1.) Discussion on embedding the system information\n\nThank you for pointing out the lack of discussion on why methods that incorporate measurement system information should be preferred. While some earlier approaches based on hand-crafted priors offered limited benefits, SDB does not rely on such priors. Instead, it embeds the measurement system directly, which disentangles the problem into two orthogonal components with distinct characteristics—range and null spaces—without injecting hand-crafted information. This embedding defines a more efficient stochastic path from measurements to clean images, similar in motivation to other diffusion bridge models, which improve upon standard conditional diffusion by using more informed reverse-process initializations. In SDB, the neural network gains principled access to an (optional) denoising task in the range space—typically simpler—and a more difficult synthesis task in the null space. In noise-free settings, this enables exact preservation of the range space content; in noisy settings, the model is guided by the fact that the range space already contains the correct structure, requiring only refinement. In both cases, the range space provides a structurally informative prior for null space synthesis. For example, in generative inpainting, where the unmasked region has low or zero noise, SDB avoids unnecessary corruption of that region and allows the model to focus its capacity on generating only the missing content.\n\nWe agree that making claims about the influence of the novel matrix-valued parameterization must be supported with a careful disentanglement of other factors. In the paper, we attempted to do that by\ni) retraining all methods (both supervised and unsupervised) from scratch ii) using the same network architecture and iii) training procedure, including optimizers, learning rate schedulers and their respective hyperparameters. Moreover, we provide a general implementation that unifies all of the methods within a single framework, supporting reproducibility and fairer comparisons, e.g., in terms of runtime. We also conducted a broad hyperparameter analysis for the SB variant in Appendix E.2. and include additional ablations regarding VP and VE variants to further disentangle their particular influence on performance.\n\n### Ablation study on linear, quadratic and cosine noise schedules\n\nUsing the parameterization introduced in our paper, we select $\\alpha_t = 1 - t^2$ for the quadratic scheduler and $\\alpha_t = \\cos\\left(\\frac{\\pi}{2} t\\right)$ for the cosine scheduler for the VP variant. In the variance-exploding (VE) case, we let $\\beta_t = \\sigma_{\\text{max}} t^2$ for the quadratic schedule, and $\\beta_t = \\sigma_{\\text{max}} \\sin\\left(\\frac{\\pi}{2} t\\right)$ for the cosine version. The remaining terms are derived as in lines 202–203 of our submission, assuming the standard relationship $\\gamma_t = \\sigma_{\\text{max}}^{-1} \\beta_t$ in the VE setting.\n\n#### SDB (VP)\n\n|Scheduler|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|Linear|87.08|0.228|**25.91**|**0.724**|\n|Quadratic|**75.25**|0.174|24.77|0.680|\n|Cosine|77.99|**0.160**|24.75|0.674|\n\n#### SDB (VE)\n\n|Scheduler|FID↓|LPIPS↓|PSNR↑|SSIM↑|\n|-|-|-|-|-|\n|Linear|94.73|0.226|25.90|0.718|\n|Quadratic|76.88|**0.159**|25.29|0.678|\n|Cosine|**76.14**|0.217|**26.17**|**0.732**|\n\nThese findings reveal that the use of more advanced scheduling strategies alters the balance between perceptual quality (FID, LPIPS) and reconstruction accuracy (PSNR, SSIM). The linear scheduler tends to favor distortion-based metrics, whereas both quadratic and cosine variants emphasize perceptual improvements, even at the expense of some distortion. Notably, the cosine SDB (VE) configuration surpasses all previous baselines and SDB setups in three out of four metrics. This underlines the potential value of deeper investigations into the dynamics of SDB’s underlying stochastic process.\n\n## (W2.) Discussion of the Perception-Distortion Tradeoff\n\nWe agree that the trade-off between perception and distortion mentioned by [1] is an important avenue for discussion in our context. Thanks to a specific formalization, where perception is measured as a divergence between the true distribution and the one induced by the model, while distortion is quantified through expected reconstruction error, one may hypothesize about the behavior of a given algorithm from a theoretical point of view. Following the paper, since perception and distortion are related through a Pareto front, one can draw more principled conclusions from the quantitative results.\n\nIn fact, the range-nullspace decomposition provided by the assumed system response matrix also allows for obtaining deeper insights. In comparison to other methods, SDB's advantage is that it treats the range and null spaces differently -- it must only (optionally) denoise the former, while new content is only synthesized in the latter. This dual treatment should provide improved performance from the outset -- in the noise-free case, the range space component is left untouched, leading to its zeroed distortion, while in the noisy case it undergoes milder transformations to arrive at the final result, which intuitively should also lower the distortion. Under a noise-free scenario, it is also easier to reason about perception -- if the model synthesizes null-space content that is perfectly aligned with the ground truth image, it will achieve zero distortion at an inevitable cost in perception. On the other hand, if it always synthesizes null-space content that is fully in-distribution, but deviates from the true signal, perception is maximized at the cost of distortion.\n\nOne may also consider more theoretical aspects -- as shown in the response to Reviewer unSw (**(W3.)**), in the limit, SDB samples from the true conditional distribution $p(\\mathbf{x}\\mid\\mathbf{y})$, which suggests that the method is inclined toward perfect perception. On the other hand, since we implicitly assume a non-empty null space, SDB will always lack some information to perfectly reconstruct the true $\\mathbf{x}$, mirroring the trade-off considered in [1].\n\nIn practice, we observe an evident trend of SDB outperforming all other methods in 3 out of 4 metrics in the superresolution task (Table 2 in the main paper), while obtaining slightly worse results in terms of LPIPS. Following [1], this observation leads to an interesting insight. Better FID, representing a form divergence between distributions, means that SDB obtains better perceptual result. However, it is also superior in terms of PSNR and SSIM, which measure reconstruction accuracy (distortion). Based on [1], one could argue that, in this task, the methods are close to the optimal performance -- since LPIPS does not quantify distributional divergence, but instead measures perceptual distortion in the space of neural network representations, it shows how a specific type of \"reconstruction\" (in this case: semantic, human-understandable) must be sacrificed to obtain results closer to the Pareto front. Moreover, the above ablation study on the noise schedulers for VP and VE variants of SDB shows how better values of LPIPS can be recovered, providing another form of trade-off.\n\nWe would be happy to include such extended discussion in the updated version of our work.\n## (W3.) Training and sampling algorithms\nDue to space constraints, these algorithms were included in Appendix C.3 of the original submission. We would be happy to move them into the main text in a revised version of the paper.\n\n## (Q1.) Several degradations handled by a single model\nThank you for raising this point. Indeed, SDB can handle multiple degradations within a single model. This is supported by the fact that a cascade of multiple linear systems with additive Gaussian noise can always be modelled by an equivalent total linear system and total additive Gaussian noise. For example, in the MRI reconstruction task, the measurement system combines a Fourier transform followed by a spatial-frequency-domain masking operator. As Table 2 shows, SDB outperforms all baselines for that task. A formal theorem and proof to support this claim is provided below.\n\n### Theorem\nLet the random variables be defined recursively:\n- $ \\mathbf{y}_1 \\sim \\mathcal{N}(\\mathbf{A}_1 \\mathbf{x}, \\boldsymbol{\\Sigma}_1) $\n- $ \\mathbf{y}_2 \\sim \\mathcal{N}(\\mathbf{A}_2 \\mathbf{y}_1, \\boldsymbol{\\Sigma}_2) $\n- $\\ldots$\n- $ \\mathbf{y}_k \\sim \\mathcal{N}(\\mathbf{A}_k \\mathbf{y}ₖ₋₁, \\boldsymbol{\\Sigma}_k) $\n\nThen the marginal distribution of $ \\mathbf{y}_k $ given $ \\mathbf{x} $ is:\n\n$\\mathbf{y}_k \\sim \\mathcal{N} \\big(\\mathbf{A}_k \\cdots \\mathbf{A}_1 \\mathbf{x}, \\boldsymbol{\\Omega}_k \\big),$\n\nwhere the covariance satisfies\n\n$\\boldsymbol{\\Omega}_1 = \\boldsymbol{\\Sigma}_1,$\n\n$\\boldsymbol{\\Omega}_j = \\mathbf{A}_j \\boldsymbol{\\Omega}ⱼ₋₁ \\mathbf{A}_j^\\top + \\boldsymbol{\\Sigma}_j,$\n\nfor $j=2,\\ldots,k$.\n\n### Proof\nBy induction on $k$. For $k=1$, the claim holds by definition. Assume it holds for $j$, then\n\n$\\mathbf{y}ⱼ₊₁ = \\mathbf{A}ⱼ₊₁ \\mathbf{y}_j + \\boldsymbol{\\epsilon}ⱼ₊₁,$ \n\n$\\boldsymbol{\\epsilon}ⱼ₊₁ \\sim \\mathcal{N}(0, \\boldsymbol{\\Sigma}ⱼ₊₁).$\n\nSince $\\mathbf{y}_j$ is Gaussian and independent of $\\boldsymbol{\\epsilon}ⱼ₊₁$,\n\n$$\n\\mathbf{y}ⱼ₊₁ \\sim \\mathcal{N}\\left(\\mathbf{A}ⱼ₊₁ \\mathbb{E}[\\mathbf{y}_j], \\mathbf{A}ⱼ₊₁ \\boldsymbol{\\Omega}_j \\mathbf{A}ⱼ₊₁^\\top + \\boldsymbol{\\Sigma}ⱼ₊₁\\right),\n$$\n\nand the mean and covariance match the induction step, completing the proof.\n\n## Summary\n\nWe hope these clarifications address the concerns and are open to further discussion regarding any other issues.\n\n---\n\n[1] Blau, Yochai, and Tomer Michaeli. \"The perception-distortion tradeoff.\" Proceedings of the IEEE conference on computer vision and pattern recognition. 2018."}]

Author rebuttal:

Meta-review: This paper introduces System-embedded Diffusion Bridge Models (SDBs), a framework for supervised diffusion bridge methods that explicitly embeds the known linear measurement system into the coefficients of matrix-valued stochastic differential equations. The key contribution lies in leveraging the range-nullspace decomposition of linear operators to design specialized diffusion processes that handle range and null space components differently. The authors propose three variants (SDB-SB, SDB-VP, SDB-VE) and demonstrate improvements over existing bridge methods (I2SB, IR-SDE, GOUB, DDBM) across four inverse problems: image inpainting, super-resolution, CT reconstruction, and MRI reconstruction.


During the rebuttal, the authors provided additional clarifications and results to address the reviewers remarks ( comparison with SoTA baselines, results on more challenging misspecified settings (reviewer Tpm3), ablation studies influence of pseudo-inverse stability (FGzg) ).  The responses and promised clarifications have addressed the reviewers points and should be included in a revised version.
Decision: Accept (poster)