An Unconditional Representation of the Conditional Score in Infinite Dimensional Linear Inverse Problems
Authors: Fabian Schneider
Duc-Lam Duong
Matti Lassas
Maarten V. de Hoop
Tapio Helin
Venue: Accepted by TMLR ()

Abstract: Score-based diffusion models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward mapping to generate a single sample, resulting in significant computational costs for large-scale inverse problems. To address this, we propose an unconditional representation of the conditional score-function (UCoS) tailored to linear inverse problems, which avoids forward model evaluations during sampling by shifting computational effort to an offline training phase. In this phase, a task-dependent score function is learned based on the linear forward operator. Crucially, we show that the conditional score can be derived exactly from a trained (unconditional) score using affine transformations, eliminating the need for conditional score approximations. Our approach is formulated in infinite-dimensional function spaces, making it inherently discretization-invariant. We support this formulation with a rigorous convergence analysis that justifies UCoS beyond any specific discretization. Finally we validate UCoS through high-dimensional computed tomography (CT) and image deblurring experiments, demonstrating both scalability and accuracy.

Published in Transactions on Machine Learning Research (11/2025)
An Unconditional Representation of the Conditional Score in Infinite-Dimensional Linear Inverse Problems
Fabian Schneider
fubion.schneiderSllt.fff
School of Engineering Science Lappeenranta Lahti University of Technology
Vienna University of Technology (TU Wien)
Duc-Lam Duong
duc-lam, duong Batt.
School of Engineering Science
Lappeenranta Lahti University of Technology
Matti Lassas
mattillussa@heessikk...
Department Mathematics ond Statistics
University of Helsinki
Maarten V. de Hoop
mdehoop@rice.com edu
Department of Compulational and Applied Mathemalics
Rice University
Tapio Helin
tapio.helin Alut. fi
School of Engineering Science
Lappeenranta- Lahti University of Technology
Reviewed on OpenReview: https: // opt.//veee....e...emm//e net/ forum? id= r08erhXHPP
Abstract
Score-based diffusson models (SDMs) have emerged as a powerful tool for sampling from the posterior distribution in Bayesian inverse problems. However, existing methods often require multiple evaluations of the forward mapping to generate single sample, resulting in significant. computational costs for large-scale inverse problems. To address this, we propose unconditional representation of the conditional score function (UCoS) tailored to linear inverse problems, which avoids forward model evaluations during sampling by shifting com- putational effort to an oflline training phase. In this phase, a lask-dependent score function is learned based on the linear forward operator. Crucially, we show that the conditional score can be derived exactly from a trained (unconditional) score using affine transformations, eliminating the need for conditional score approximations. Our approach formulated in infnite-dimmessional function spaces, making it inherently dliscrtizatiioniivariiae support this formulation with rigorous convergence analysis that justifies UCoS beyond any specific discretization. Finally we validate UCoS through high-dimensional computed tomography (CT) and image deblurring experiments, demonstrating both scalability and accuracy.
We
1 Introduction
Inverse problerss seek to determine unknown quantities through indirect and noisy measurements, typically leading to ill-posed scenarios. The Bayesian approach to inverse problems frames the task as a quest for information. Blending statistical prior information of the unknown with a likelihood model for the mea- surement data gives rise to posterior distribution, which fully characterizes the unknown conditioned on noisy data Kaipio & Somersalo (2006); Stuart (2010). In severely ill-posed problems, the quality of inference

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is strongly dependent the expressivity of the prior. Traditional hand-crafted priors, such as the total- variation prior, tend not to be expressive enough to characterize complicated structures Sun et al. (2023). Generative models offer a flexible and computationally feasible approach to prior modeling as they offer the possibility of generating new samples after training on a data set characterizing the prior.
This work investigates sampling from the posterior distribution of linear inverse problems using score-based diffusion models (SDMs) [Song al. (2021), which have recently received wide attention in the literature (in
the context of inverse problems,
e.g. Batzolis ct al. (2021): Lim et al. (2025); Hagemann et al. (2025
Graikos et al. (2022): ; Feng et al. 2023 Sun et al. (2023); Pidstrigach et al. (2024); Dey et al. (2024)
Holzschuh ct al. (2023): ;: Dou & Song] (2023): ; Barbano et al. (2025); Cardoso et al. (2023):
: Feng & Bouman
(: Song et al. (2024); Meng & Kabashima (2022); Kveton et al. (2024); Wu et al. (2024a) bb; Baldassari
ct al. (024bb] Yao et al.
) Chen et al. (2025)). ). An SDM consists of two main components: a forward
diffusion process and a reverse generative process. In the forward diffusion, the model gradually transforms
the target distribution into a simpler, tractable distribution with a specific stochastic differential equation (SDE) driven by Gaussian noise. The generative process simulates the time-reversal of the diffusion process with a backwards SDE. This denoising phase relies on the drift, which is computed using the logarithmic gradients (scores) of the diffused data densities. These scores are typically estimated using neural network, allowing efficient simulation of the backwards SDE and, consequently, sample generation from the target distribution. SDMs have demonstrated significant success across a variety of domains, including inverse problems such as medical imaging Dey et al. (2024); Levac et al. 22023 Barbano et al. (2025); Chung et al. (2023b); ; Song et al. (2022].
For inverse problems, posterior sampling with SDMs involves the challenging task of estimating the score function conditioned the measurements. A key difficulty is balancing computational efficiency with scalability while ensuring accurate posterior sampling. Motivated by the limitations of existing approaches, we propose a novel method for conditional sampling, based on an unconditional lask-dependent representation of the conditional score (UCoS) that overcomes these challenges by leveraging an explicit likelihood model, which is often available and predefined in inverse problems. UCoS is particularly well-suited for large-scale imaging tasks that require efficient posterior sampling for varying sets of measurement data.
The existing posterior sampling methods can be broadly categorized into three approaches. The first ap- proach modifies the unconditional reverse diffusion process Lo guide sample trajectories Lowards the posterior, either by adding a correction term to the score Jalal et al. 2222 Chung et al. (2023a); Song et al. (2023)
Adam et al. (2022); Chung& & Ye 2022); |s Levac et al. (2023) or incorporating a data consistency optimization
step Song et al. (2022): Dey et al. [2024] Chung et al. 2202).. The second approach employs gradient- based
Monte Carlo sampling techniques, replacing the prior score with learned score Cardoso et al. (2023) : Sun
let al. ) Feng et al. (2023), with recent extensions to infinite dimensions [aldassari et al. (2024b) ) For
recent review of methods largely following these two approaches (in finite dimensions), we refer to Daras et al. 22024). A key advantage of these approaches is that they allow the use of pre-trained unconditional prior score, accommodating different imaging tasks simultancously without requiring task-specific training. This is particularly useful for handling extensive multipurpose training sets in applications such as image processing. However, such methods suffer from (1) high computational costs due to repeated forward evalua- tions, (2) poor scalability in high--dieensional or large-scale inverse problems, and (3) potential inconsistency in posterior samples due to the lack of rigorous convergence guarantees.
The third approach directly trains conditional score function to perform conditional sampling, learning an amortized version of the conditional score that depends on observations Baldassari et al. (2024ag) y Batzolis
lt al. (2021). This method extends rigorously to mninite-dimensional diffusion models, making it attractive for large-scale inverse problems, as it eliminates the need to evaluate the forward map during sampling. However, it requires extensive training data on the joint distribution to reliably approximate the conditional score function, particularly as the problem dimensionality increases.
We address the prohibitive sampling cost for large-scale inverse problems by demonstrating that this com- putational overhead can be offloaded to the training phase, thereby improving the efficiency of postcrior sampling. This is achieved by designing a lask-dependentl training phase that depends on the forward map- ping but not on the measurement data. More precisely, we introduce a diffusion- random process whose
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distribution is explicitly dependent on the forward mapping, allowing the score (which we term the task- dependent score) to be estiimaeed using standard methods. We then demonstrate that the conditional score corresponding to the posterior distribution can be recovered from this task-dependent unconditional score through simple affine transformations involving the measurement data. Furthermore, we modify the training procedure to enable evaluation of the conditional score without requiring application of the forward mapping during sampling. In comparison to the conditional method, integrating the explicit likelihood model into the process reduces the dimensionality of the training target. As result, UCoS offers the best of both worlds: like the conditional approach, it avoids forward mapping evaluations during posterior sampling while its score model remains as lightweight as standard unconditional ones. This leads to more favorable scaling of the online sampling cost as the size or dimensionality of the problem grows.
Inverse problems are often about inferring quantities represented by functions such as solutions or parameters to PDEs or integral equations. These models must. be arbitrarily discretized to obtain a finite-dimensional computational model Mueller & Siltanen 2012). In Bayesian inversion, there has been a long-standing effort to develop methods that. are disccretizalion--ideeeennd
Lehtinen al. (1989); Lassas & Siltanen] (2004) Lassas et al. (2009), aligned with the principle to "avoid discretization until the last possible moment" Stuart 2010 This is also critical for the success of SDMs in Bayesian inversion as recent theoretical
studies indicate that the performance guarantees do not always generalize well on increasing dimension
Chen et al. 2023bb : Bortoli (2022) Pidstrigach et al. (2024). Inspired by recent. developments on defining the score-based diffusion framework in infiiiteddimensionall spaces Pidsttricch et al. (2024); Baldassari et al. (2024a), we also extend our method rigorously a separable Hilbert space setting. In particular, in the spirit of Pidstrigach et al. 2024), we perform a convergence analysis of UCOS, establishing a bound between the samples generated by UCoS and the true posterior target measure. Morcover, we conduct numerical experiments of inverse problems related to computerized tomography (CT) and deblurring to illustrate the practical applicability of UCoS.
1.1 Related work
The body of literature on SDMs is growing rapidly. Let us mention that Song et al. (2021)
|| developed a
unified framework combining score-based Hyvärinen & Dayan (2005): : Song & Ermon (2019) and diffusion
Sohl-Dickstein et al. (2015); Ho et al. (2020) models to interpret SDMs as a time-reversal of certain SDEs. Our paper is inspired by the work on conditional SDMs in the context of inverse problems. One line of work seeks modify the unconditional prior score function to generate samples that approximately follow the posterior distribution. Examples of this include projection-basel approaches [Song et al. (2022) Dey et al. (2024), , which project the samples on a subspace that solves the inverse problem during the generaling process; methods based on the gradient of a log-likelihood [Jaaa et al. (2021); Chung et al. (2023a); Levac ct al. (2023) and plug and play approaches Graikos ct al.] (202) which add appropriately chosen constraint- terms to the unconditional score function to steer the process lowards generating desirable samples; and the Doob's h-trransform approach Denker et al.] (2024) which learns a small, auxiliary correction to the unconditional score function. A closely related approach the Monte Carlo guided diffussin framework (MCGDiff) Cardoso et al. (2023), which leverages posterior samples from linear Bayesian inverse problem obtained via sequential Monte Carlo methods using the singular value decomposition (SVD) to guide the training of a diffusion model. The theoretical foundation of MCGDiff however builds upon the availability of the full SVD of the forward operator, which restricts its applicability to mildly ill-posed problems and moderate-dimentt dimensional settings where the SVD can be computed and inverted stably. In contrast, UCoS is designed to operate in a fully matrix-free manner, requiring access only to the forward map and its adjoint.
Another line of work seeks approximate the conditional score function of the posterior distribution directly Saharia et al. (2023): Batzolis et al. (2021); Baldassari et al (2024a). As this approach increases the input dimension of the score function drastically, the training process is more computationally expensive and requires more high quality training samples, which may be restrictive, especially in very high-dimensional problems.
The theory of infiite-dimennionaal SDMs has been initiated only very recently. Hagemann et al. (2025) modifies the training phase of diffusion models to enable simultancous training on multiple discretization levels of functions and proves the consistency of their method. Lim et al. (2025) also generalizes the trained
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model over multiple discretization levels proposing to generate samples with the annealed Langevin algorithm in infinite dimensions. Piddttrgaac et al. (2024) was the first to formulate the SDM directly in infinite- dimensional space, demonstrating that the formulation is well-posed and providing theoretical guarantees. Our work is closely connccted to Baldassari et al. (2024a), where the authors introduce the conditional score in an infiniteedimensionall setting. Moreover, they provide a set. of conditions to be salisfied ensure robustness of the generative method and prove that the conditional score can be estimated via a conditional denoising score matching objective in infinite dimensions.
1.2 Main contribution
This work presents novel and scalable framework for posterior sampling in high-dimensional and infinite- dimensional inverse problems. The key insight that posterior samples generated via conditional SDMs be computed without any evaluation of the forward operator during sampling. Instead, without introducing any errors, the computational effort can be shifted to the offline task of training the unconditional score of a specific diffusion-like random process. This Toundational principle generalizes to other infinite-dimensional diffusion models, beyond the Ornstein- Uhlenbeck process studied here. More precisely,
In Theorem 3.7 we establish an identity for the conditional score connecting it task fask-dependent unconditional score through affine transformations depending on the forward mapping and the mea- surement data. The thcorem extends this general principle to an infiiiteddimensional setting.
In Theorem 1.1 we derive an error estimate for the generative process following the ideas suggested by Pidstrigach ct al. (024)). In particular, the error estimate explicitly underscores the contribution of the loss function employed in training.
We numerically explore our method in Section 5 demonstrating that task-dependent training is effective in practice and that online posterior sampling can be performed without evaluations the forward operator.
2 Background
2.1 Score-based diffusion models in infinite dimensions
Finite-dimensional score-based diffusion models Score-based diffusion models (SDMs) are state-of- the-art machine learning generative models (Song et al.] (2021)) that learn data distribution through gradual denoising of normally distributed random variable. A diffusion process diffuses image Xo Po from == to ( = T via the SDE (Ornstein-Uhleeneeck process, OU)
dX, = - x,dt + dWW,
where the marginal densities are denoted by Pt, in particular, for large T, pr is close to the Gaussian densily N/0,, T). The marginal densities are then reversed using the backward SDE
Yddl + V. dY: = 2
ddW,
such that Yr ~ Po-
Infinite-dimennsioaal score-based diffusion models Let us now review the unconditional SDMs in infinite dimensions proposed by Pidstrigach et al. (2024). Let μ be the target distribution, supported on a soparable Hilbert space (H, ( ()) Let (X7) stand for the infiitte-dimensional diffusion process for a continuous time variable L [0, 7] satisfying the following infnite-diimnnsionll SDE
1 dx( ===-x X C
(1)
where C is fixed trace class, positive-dffnite, symmetric covariance operator C: H + H and B, a Wiener process H with covariance IT, see Appendix A.2 We assume that the process X initialized with μ.
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i.e. X" μ.. Notice carefully that embed the initial condition μ to the notation X t 2 0, as later analyze the interplay between different initializations. Here and in what follows, we assume that the initial conditions and the driving process are independent. We adopt the formal definition of the score functions by Pidstrigach et al. (2024):
Definition 2.1. Define the (weighted, unconditional) score function s(x, lip) for x E H as
-- (s "XX = =x))
sxxtt;p) 1
(2)
Remark 2.2. Assume that H R and that the distribution μ admits a density po with respect to the Lebesgue measure. For any 0 the random variable X obtained through SDE in Equation | has a density Pr. The unconditional score s(z,t;p) given in Equation 2 then satisfies s(x, t;ptt:) = CV log (z).. In other words, in finite dimensions, Definition 2.1 reduces to the common score identified as the log-gradient of the density, scaled by the covariance matrix C E Raxa
Now assume that for T > 0,
sup E (XX t; 4))l < co, 6001
(3)
It is shown in Pidstrigach et al. (2024) that given Y ~ L(x+) the following SDE
(- 1
-- s(Y), dt Cl/2BB
dY, =
(4)
is the time-reversal of Equation ! where B, is a different Wiener process on H. The process Y is independent of the past increments of B₁, but not of the future ones, see Anderson 1982
Training and sampling In both finite- and infinite dimensionll diffusion models, the score function is learned by a neural network so(-,<;)) : H x 10.t] - H such that so(z, t; x s(x, t; μ1) (where in finite dimension H = RR" we interpret s(x, (1) as V Vlog(())). A common technique to enable empirical learning of the score function is conditional denoising score matching, introduced in Vincent (2011). This method has been shown to work in the conditional Batzolis et al. (2021) and infnite-dimmnsionnal setting [Baldasari] lt al.] (2024a) for the forward SDE given by the OU process. Training performed via stochastic gradient descent. by minimizing score-matching objective (loss function),
SM(sa) 18,, 2(x;)
A(t) ((s)))((((( (se -
s(x, m)ll
over some appropriate class of neural network. Here X [6. 7] - R is a positive weighted function and < T a truncation that avoids mumerical instability for small times. In general, the score function s is intractable, such that a denoising score-matching objective is introduced,
(SM = (100:1
(10( (; - (1 (1- e 120
It can be shown, that the two objectives equal up to a constant that does not depend on the parameter A. After training a score so(.,t) i) for all time i E [0, T), we can then generate samples from μ by running the backward SDE Equation 4 with the trained score so instead of s. The solution of the backward SDE can then be numerically solved using traditional methods such Euler Maruyaaa approximations.
2.2 Bayesian inverse problems and conditional score
In this paper, we focus on the setting, where the score corresponding the reverse drift is conditioned on observations. We consider linear inverse problem
y
Az + +
(6)
where E H is the unknown in some separable Hilbert space H, y E R" stands for the measurement and A H Rm is bounded linear operator. The random noise modeled by centered Gaussian
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distribution -V), E R=xm We adopt Bayesian approach to inverse problems Stuart (0010) and assume to have some prior knowledge of the distribution of æ before any measurement is made. This knowledge is encoded in a given prior μe, defined as a probability measure on H. This approach gives rise to a posterior that is absolutely continuous with respect to the prior and its Radon-Niin -olon-Nikody derivative is given by, jtalmost everywhere,
du dy
(x) = ( exp (-2 A -uli
Z(y) / exp
(-2 ||Ar u11 u(dx).
In this context, Baldassari et al. 2024a) defines the conditional infiniteddimensional score as follows: Definition 2.3. Define the (weighted) conditional score function s(x, 1; p²) for H as
8(2, (0, 1 (
(x- = x, =/==)) x))
Remark 2.4. Note the dependency of the conditional score the measurement In the same spirit of Equation Baldassari et al. (2024a) assume the uniform boundedness in time of the expected square norm of s(x, t; 1 for all y E IR to prove the well-posedness of the corresponding time-reversal SDE, given by Equation with replaced by nD and s(x, by s(x, (.))
3 UCoS: theory and implementation
In this section, we establish the theoretical foundation for our method by showing that the conditional score associated with the posterior distribution in the linear inverse problem in Equation @ can can be expressed
affine transformation of unconditional score derived from a modified diffusion process. This key observation allows us to shift the complexity of posterior sampling to an offline training phase, where only the prior distribution and forward model are used.
3.1 UCoS in finite-dimensiones space
To build intuition, we begin by illustrating the idea in the finite-dimensional setting H = R and assume C / for simplicity. Consider the posterior n in Equation densities of the prior μ and X are absolutely continuous w.r.t. Lebesgue measure and denote them by Po and qe (-lv),, respectively. The transitional probability for the multivariate OU process is given by pe(z)20) N(ee 1/2rg, (1 e-))C) and thus
7 diffusse by Equation D Suppose the probability
(x(l)) / (0(2))))((<(<)d>
Bm
1 / Po(2)exp (-1 (yy)
|1Az [1
vIl? pe(x)z) dz
(8)
complete square
wr(z. v)
Z(y) / Po(z) exp
(-111== -- m,(x, (((...))1 dz,
where we completed the square w.r.t. the product of likelihood and transitional probability Pe- Here, the quantities E,, my mx(x. y) and w( satisfy
-
- A
Σ, = e
me(x,y) 9) = E, TT
+
and
et/2 e-2/2
1
1/2 Aze lool())
wr(x, 9) & exp
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where C (c' - 1)AA
I
Rmxm. Consequently, we have that
1
Vx lologe(z/ =
Vx
y e/2 Ar
+ log ((po NN (m,
2
C₁
(9)
1 V
y e/2 Ar
||: c.
2 + (V.m (2, y))² [ log (po (0, (2,()))).
2
The identity in Equation can be formally interpreted as expressing qt for a given time as a mixture of the prior Po and a Gaussian distribution with time-dependent covariance E, modulo transformations with wi and my- What more, the conditional score s(;p²) corresponding to qe in Equation 9 can be expressed transformation of the score & of this mixture, $(2; t4) V, log (pu N/ Σ.)). This idea enables to approximate the posterior score offline up to affine transformations. More precisely, we have the following identity:
s(x, p²) = X(t) (5 (mma y) t: + + 1)AT (y - e/2 sx2Ax)
(10)
x(t) (m y) t: + m)(x,)) e²12x)
with
X(t) = ce
Remark 3.1. The principled idea of transforming from task-dependent unconditional score to the conditional score outlined above generalizes to other linear diffusion models beyond the specific Ornstein- Ueriinn-Uhlebcckk process Equation studied here. This raises the important question of how to design an efficient. underlying diffusion model to balance the computational effort further in a desirable way, c.g. by temporal or spectral weighting of the diffusion. This consideration is beyond the scope of this paper.
The result of Equation 10 motivates a new way to approximate the score function in a task-dependent manner. Notice first that the term my m,(2,y) depends on the forward map A in a non-trivial and time- dependent way. Therefore, even training of can be performed oflline and utilized as an approximation Equation [0] one would still need evaluate A during the sample generation. In what follows, we propose an approximation scheme for the conditional score that circumvents this issue. Notice that the dependence of m, on A is mostly through multiplication with Σ, Hence may learn the function given by
r(;4;)) = 5(Ee,tin)) E.
instead of 8. By substituting r into Equation 10 we obtain the final identity
s(x, p²) X(t) (r(Ea(x:)) - e222x)
with & given by
A A(t)..
We can compute and save the term A Ar-ly to memory using a single adjoint evaluation. For each time step > 0, we add the time-dependent sccond term to obtain & without the need for any further forward evaluations during sampling.
Remark 3.2. Assume that the prior pu (N,00 is Gaussian. Then r is given by
1 r(c, p.) e - c A
C-
and a bounded linear mapping for any > 0. In particular, r Rn R + R" depends only on a spatial variable Ç and a temporal variable but not on the measurement y. This is in stark contrast to the conditional approach [Balassari et al. 20244), where the conditional score
8: (, s(x, (;)) R2 x 1m R
(scc Definition 2.3 depends on both spatial variables x and y. The added dependence on y increases the input dimensionality from n n+1 to n m++m+3 1, which substantially raises the required sample complexity and the size of the neural network required to achieve given error tolerance.
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3.2 Theoretical foundations of UCoS in function space
We will now make the transformations above precise in infinite-dimensionaal separable Hilbert space H. First, we define random process {Xt Jt=o such that its distribution is given by the mixture model
x X Zz.
X ~ p and Z N10, Ex)
(11)
with E, given by
Σ = (e' - -1)- (e' 1, AC
(12)
and
C -c-cc-1)ACA 1)ACA + Rmxm.
(13)
Notice that Σ, coincides with its formulation Section fiite-dimennionall case, see Lemma B.6 The next lemma ensures that D, is valid covariance operator.
Lemma 3.3. For t 0, the operator E, H H is trace class, self-adjoint and positive definite.
Remark 3.4. While the mixture Equation 11 is well-defined for any > O, we observe that the process X, is no longer diffusive in the sense that it may stay dependent of X as grows. Indeed, we notice the dual behavior from two cases:
if = I, have E, C(C -1)-1)
for = 0 we have Σ, = (c' - -IC.
This indicates different asymptotics for the variance of X depending on the singular values of A. order improve intuition on the correlation structure of X,, we depict the diagonal of Σ, for different values of and the inverse problems related to inpainting in Figure | and CT imaging in Figure A detailed description of these problems can be found in Section 5 We further note studying SDE corresponding to X or its tive-seversal is beyond the scope of this paper.
Pad Vakae
7ort Valtee
Rod
Fox Wat
Figure 1: Diagonal of Σ, for inpainting problem. Time-steps from left to right: t = 0.05, t 0.1, t = 0.5 and = 0.9.
0.340
0.054
22
0L144
1.
30
1.00
00342
1.4
0.09
30
Figure 2: Diagonal of Σ, for CT imaging problem. Time-steps from left to right: t= 0.05, /. 0.1. = 0.5 and f = 0.9.

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Definition 3.5. We define the (weighted, unconditional) score function of the process in Equation 11 by
((2 = (2
(14)
Lemma 3.6. Assume that H = R= and our prior measure admits density given by Po. Moreover, let X be defined by Equation 11 Then for every l 0, the random variable X₁² admils a density pc(x) satisfying
$(2,t: p)) = E,V, log (z))
Let us fix some / > 0 and define the operator Rg . + H through
R = (c' - 1)I + (e' 1))CA
(15)
By Lemma B.1 Rg is bijective. Define the affine transformatton of $ by
r(.,;;p) = 8(Re,t: + Re:
(16)
Let proceed to the main result of this section.
Theorem 3.7. Let He be the Cartron-Martin space of C (see Appendir A and assume that the prior satisfies p(H)) = 1. Then the for conditional score function s(x, t; t2 ) it holds that
(1,1;,1) x(1) (()()(((())))) (E(E((2))) - -()() e/
(17)
for (I,2) H x Inm a.e. in Y) and > 0, where
X(1) = (e e -/22))1
CCTT + A(1xx.
CATT-y
(18)
Special case of Gaussian prior Let now study the score corresponding to the special case of a Gaussian prior measure. This enables us to derive an explicit formula which can give insights, in particular, regarding the regularity of the score function.
Lemma 3.8. Let μ = (O, So) be Gaussian measure and suppose the covariance operator C satisfies the assumptions of Theorem 3.7 Then there exists a covariance operator on H such that Σ,(E, So) H + H can be well-defined as linear and bounded operator for t O. Moreover, in that case
8(2,4;1) -E, (<E++5S) S0)
3.3 Matrix-free implementation
Building on the theoretical framework established in the previous section, we now describe how UCoS can be implemented efficiently in practice. A key strength of our approach that it remains entirely matrix-free: it requires only the ability to evaluate the forward operator A and its adjoint A without ever forming inverting these operators explicitly. In discretized setting, this corresponds to avoiding computing and saving the matrix representing A, which can be expensive in terms of computational time and memory for large scale problems. This property enables UCoS to scale to high-dimensional and ill-posed inverse problems where standard representations (e.g., via SVD or matrix factorizations) become computationally infeasible. In what follows, we outline the two-phase implementation pipeline: an offline training phase in which a task-dependent score approximation is learned, and an online sampling phase where posterior samples are generated using only the trained network and a precomputed observation-dependeet shift.
3.3.1 Offline phase: learning the task-dependent score
the setting of forward process X Let us define continuous-time score matching objective (loss function) and the denoising score-matching objective
The next lemma adapts the training procedure discussed in Section 2.1
SM(ro) = E [>(1}2 [3 151) + Ts - ro(R, 1,, (;,

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DSM(r)) = A(t)² (R, ( 34, t; - T0ll
with the expectations taken w.r.t. U (Øo, 2+) 2 (Xx X²'). We make an assumption regarding the boundedness of the second moment of the conditional score uniformly in time following Baldassari et al. (0024a).
Assumption 3.9. The prior a has bounded second moment, IE Exx=||XX|l oo, and
sup Byww, E (x)"" [Is( (=) <c. tE|6,]]
Notice that Assumption 3.9 does not require us to sample from Y ~ Ty or to compute the transform mz(2, y). Soe Lemma B.4 in the appendix for detailed justification. Lemma 3.10. Let Assumption
3.9 hold. It follows that
SM(ra) DSM((ro) + V,
where the constant V < is independent of 0.
The result follows from repeating the arguments in Baldassari al. (2024a). Truncation / 2 å guarantees that complications with singularity at t + O are avoided, see Appendix B.4 for details.
By the definition of the process X/", we have X X + Zi in distribution, with Z₁ N(, Ex). Using this identity, we observe that R7 -r, conditioned on Tn is Gaussian with covariance
Cov(R
x6) = E (RT e c- -
C
by the identity
R 1
followed from Lemma B.1 and Lemma B.6 (i). Now we can directly generate the desired quantily by observing that
R R1 = - I+CAT Eo
C1/2 21 1/
Vel
in distribution, where (2o. ~ C(Xg Xx+) and 21, 22 ~ N(O. I) iidd In consequence, the training phase requires application the forward operator A, adjoint A", the covariance of the noising process and the precision matrix of the observational noise r-1 We note that T in the case degenerative noise distribution, p-1 can be replaced by the pseudoinverse.
3.3.2 Online phase: posterior sampling without forward map evaluations
The offline training, detailed above, will yield an (e.g. FNO) approximation ~ independent of any measurement data. Once this is done, sampling from the posterior is performed by simulating a reverse-time
diffusion process using the transformed score s, defined as (taking into account Theorem 3.7
s(x, (1 A(t) (re(Ea(z,0).tti) e e-bx)
where
6(x,y) CAT A(t)ar
Note that after collecting a new measurement vector y, the term CA computed once before the generative process and remains constant throughout it. In consequence, the generative process does not require evaluation of any operators in addition to ra.
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4 Convergence analysis
The aim of this section is to conduct rigorously the theoretical convergence of the proposed method. More precisely, we establish quantitative bound of the error term that indicates how far the samples generated by UCoS lie from the true posterior target measure. In particular, we quantify how this error term depends on different types of numerical approximations. Let us fix 0 and E (0,T). Consider a partition {6 = t I S in π} of [8. with mesh size At == min(t; 4++ S S N N- 13 and define [t] = max [4/1 < <is n, 4 < t). For clarity, we reverse time in Equation and let WH" = YT for y a.e be the ideal solution satisfying
dW?" = ---2 Wt
-T- -1:00) W+ - <(We", T 1:
dl + C2/2
(19)
initiated at we = (1==((())=tt-777 TT/2 for 2 C),,C) and Vy ~ n Observe that Equation 19 is a 'standard' SDE where W is independent of the future increments of B Moreover, let correspond the numerical approximation to Wh satisfying
dVF" = 2 Vu - A(T - L(J) (ro (Er b), [0]:m) -TTT)--------AA -W dt + CB
and initiated at uo e Here, the two stochastic processes share the same Wiener process B₁ and initialization 2. In what follows, we consider a discrete-time loss given by
ELOSS = EE- 2 ((Et) ((E |
+ uy ro 2 [t]: [t]: H²
(20)
We ready to state the main result of this section.
Theorem 4.1. Let Assumption 3.9 and the assumptions of Theorem 3.7 hold. Assume further that s( t: and rol:, () are Lipschila continuuus with Lipschilz consland L.()) E ([8, T)). Then,
By E (eer.r- 2(Wp wa lor or- s M (ENUM + eLOSs EINIT + 8)
where eross is given in Equation 20 and
EINIT - IX ENUM S O(At),
and the constant M depends on the quantities
Eu-UiOTT B,y Eug [|s(w,, T t: t: 12) Ex=xxIXXXX X [ 12(())dr tr and T.
Note that the Lipschitz continuity assumption of the drift is a natural setup for the backward SDE to retain the uniqueness of (strong) solution, see Pidstrigach et al. (2024). It also follows from there that for almmssstt every y E mm the random variable W is distributed according to the posterior μ²,
5 Numerical experiments
We showcase our method in the context of inverse problems related to toy inpainting example, computerized tomography (CT) and deblurring problem. Details of the numerical implementation are described in Appendix D and enlarged depictions of posterior samples can be found in Appendix
To enable principled comparison, we examine the accuracy of posterior sampling using score approxima- tions trained with architectures that offer comparable runtimes. Methods that avoid forward evaluations during sampling, such as UCoS and conditional method, exhibit similar runtimes. However, we find that the performance of the conditional method is hindered by a more complex score approximation, especially when neural operators have very few parameters. In contrast, the computational cost of posterior sampling
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with unconditional methods increases due to the need for repeated forward evaluations during the sampling process. When comparing unconditional methods to UCoS, deviations from the true posterior are primarily attributed to the approximative nature of the conditioning on measurement information, rather than inac- curacies the score approximation. This is because the dimonsionality of the score approximation is the same for both UCoS and unconditional methods.
Following Baldassari al. 2024a), we utilize Fourier neural operator (FNO) Li et al. [202] to parameterise To and perform training outlined in Section 2.1 on Nvidia A100 GPU with 80 GiB. We compare UCoS to four different methods. First, consider
of the first conditional Jalal et al. [2021] Feng et al. (202)) and refer to it by SDE ALD. Additionally, we implement Diffusion Posterior Sampling (DPS) Chung et al. (2023a), projection based methods for general inverse problems (Proj) Dey et al. (2024) and the conditional score approach (Conditional) Baldassari et al. (2024a). Each score approximation given explicitly Appendix D The numerical implementation can be found
approximation considered in
https://github.com//aat http://ttabb.cmm/aaaaaaaaa////// hrtpaa/nte/..aataaaeeoooerrrrerr
Inpainting In this paragraph, the prior is modeled by Gaussian distribution with squared exponential covariance structure. This allows to compute all score functions in closed form without any learning, Uniquely to this example, we are able to sample from the (Gaussian) posterior. The forward mapping A R" + R" is diagonal and with diagonal entries one in the center of the image and zero elsewhere. The information on the center pixels is lost and in the inverse problem we seek to recover the full image. The image resolution is 64 64 pixels.
Figure 3 and Table illustrate that the posterior ensemble (N 1000) generated by UCoS (our method) and conditional method generate samples that are consistent with the true posterior distribution. Since score needs to be learned, UCoS and Conditional perform very similarly.
samples
bias
std
SDE ALD
truth
DPS
Proj
measurement
Conditional
UCoS
true posterior
Figure 3: Summary of posterior samples for inpainting problem. On the left column, the top image is ground truth and the bottom is measurement data. On the right column, posterior samples with true score corresponding to a Gaussian prior and their bias and standard deviation, with methods used from top bottom: SDE ALD, DPS, Proj, Conditional, UCoS and true posterior.
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method
bias
std
time
SDE ALD
0.1341
0.1419
202
DPS
0.1050
0.1242
467
Proj 0.2008
0.1772
254
Conditional
0.1011
0.1238
198
UCoS
0.1003
0.1226
197
True
0.1013
0.1304
<1
Table 1: Summary statistics of all methods for the inpainting problem. Bias and std are computed on 1000 generated posterior samples and averaged over the number of pixels. Computational time is given in seconds.
CT imaging Here, the forward mapping A models a sparse-view imaging setting with 45-degrce angle of view with 256 equiangular directions. The detector is assumed have 256 apertures spanning over the width of imaging area (i.e., 256 parallel line integrals per angle measured) and, consequently, the problem dimensions given by
256 Moreover, the measurement data is corrupted by additive Gaussian noise with signal-to sognal-to-noise noise ratio of 20 dB.
Our training data is the Lung Image Database Consortium image collection (LIDC-IRI) dataset Armato 3rd et al. 2011p, containing > 200.000 2D slices resolution 512 512, we rescale them to 256 x 256.
We analyze the influence of the complexity of the FNO approximation on the posterior samples. We employ architecture with 4 layers while varying the mumber of nodes uniformly across all layers. For the full 256 x 256 resolution, layer in Figure Zand and Table We see that UCoS and conditional method have significant edge in terms of computational time over the other methods as a consequence of avoiding repeated forward evaluations during sampling. With larger number of neurons per layer, conditional method produces relatively similar samples as UCoS in terms bias and standard deviation, but has worse statistics and visual sample quality when using only neurons. Due the complexity of the score approximation, the performance of conditional method at 32 nodes per layer differs the most from the performance using more nodes. Sample quality of UCoS seems comparable deviation and bias and a naive visual inspection. Noticeably, UCoS generates samples with low variance compared to the other methods.
analyze posterior samples for an architecture with 32, 64 and 128 neurons per
better to the unconditional methods (SDE ALD, DPS and Proj) in terms of standard
method
bias for nodes
std for k nodes
time for k nodes
k=32
k-64
k=128
k=32
k=64
k=128
k=32
k-
k= 128
SDE ALD
0.1014
0.1042
0.1044
0.0741
0.1064
0.0644
35
48
77
DPS
0.0605
0.0552
0.0788
0.068
0.0768
0.1723
69
99
176
Proj
0.0644
0.0610
0.0616
0.0667
0.078
0.0702
224
237
266
Conditional
0.2336
0.0621
0.0467
0.0923
0.0594
0.0516
15
27
57
UCoS
0.0489
0.0498
0.0541
0.0379
0.0316
0.0262
15
27
57
Table 2: Summary statistics of all methods for the CT imaging problem. Left table: bias, middle: std and right table: computational time in minutes. Bias and std computed on 1000 generated posterior samples and averaged over the number of pixels. In each table, the columns correspond to the mumber of neurons used per layer.
In addition to the previous experiment, for UCoS and conditional method, we train models for numerous number of neurons on a reduced 64 x 64 resolution. In Figure 5 we compare the number of nodes to the resulting posterior bias and standard deviation for the full 256 256 resolution (crosses) and the reduced 64 64 resolution (lines). We obscrve that, espocially terms of standard deviation, UCoS achieves saturation with lower model complexity. To provide context, halving number of nodes approximately halves the online sampling time.
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samples
bias
std
SDE ALD
truth
DPS
32
Proj
nodes
measurement
Conditional
UCoS
SDE ALD
truth
DPS
64 nodes
Proj
measurement
Conditional
UCoS
SDE ALD
truth
DPS
128
Proj
nodes
measurement
Conditional
UCoS
Figure 4: Summary of posterior samples for CT imaging problem. Top figure FNO with 32 nodes per layer, middle figure uses 64 nodes per layer and bottom row 128 nodes per layer. In each figure, the left column, the top image is ground truth and the bottom is measurement data (sinogram). Both are fixed for all FNO architectures. On the right column, posterior samples, their bias and standard deviation. Methods used from top to boltom: SDE ALD, DPS, Proj, Conditional, and UCoS.
Deblurring In this example, the forward operator corresponds a two-dimensional convolution with a Gaussian kernel over the image domain. We observe the output function on the same grid with an additive Gaussian noise vector.
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0.225
X
ours conditional
0.18
outs
0.16
0.200
oura
cooditione))
conditioonll
0.175
0.12
0.150
0 0.10
0.125
0.00
0.100
0.04
0.02
20
20
60
80
200
120
40
100
120
nodes per layer
modes por layer
Figure 5: Comparison of the error dependence of the conditional method and UCoS on the parametrization of the score approximation. X-axis: number of nodes per layer in the FNO, Y-axis: L2 norm of the bias (left) std (right), averaged over the number of pixels. Dashed lines correspond to 64 x 64 resolution and crosses correspond to the problem with 256 x 256 resolution.
The training data is given by the Large-scale CelebFaces Attributes (CelebA) Dataset Liu et al. (015 which contains > 200.000 face images of celebrities. We scale the images to square resolution 175 x 175.
method SDE ALD
bias
std
time
0.0878
0.0915
300
DPS
0.0872
0.0801
239
Proj 0.0785
0.0622
15822
Conditional
0.2178
0.0950
85
UCoS
0.0884
0.0345
85
Table 3: Summary statisties of all methods for the Deblurring problem. Bias and std are computed on 1000 generated posterior samples and averaged over the number of pixels. Computational time is given in minutes.
We demonstrate in Figure 0 and Table that UCoS and conditional achieve the fastest sampling speed. For the chosen FNO parameterization, the conditional method produces samples with higher bias and standard deviation as well as low visual quality. Visually, the posterior ensemble (N = 1000) produced by UCoS is consistent with the ground truth, while other methods introduce artifacts that are inconsistent with human facial features. In terms bias, UCoS at a roughly similar or slightly better level than unconditional methods, while the pixel-wise standard deviation is noticeably lower than other methods.
6 Conclusion, limitations and future work
This paper addresses the challenge of balancing the computational cost between offline training and online posterior sampling in large-scale inverse problems. In many such problems, evaluating the forward operator during sampling constitutes a major computational bottleneck. We introduce a novel, theoretically grounded method UCoS that entirely removes this burden during sampling by shifting it to the offline training phase, without introducing any approximation error. Our approach combines the computational benefits of the conditional method introduced in Baldassari et al. (2024a) with more scalable score-learning formulation. Furthermore, demonstrate that this method is rigorous in infuite-diimensional setting and, therefore, independent of discretization.
We validate the correctness of our approach with numerical experiments related to CT imaging and De- blurring. UCoS generates samples with approximately comparable or better quality than unconditional approaches while offering tangible computational benefits during sampling. However, dependent, requiring re-training when the forward problem is modified. The quality of posterior samples is assessed using simple statistics such as bias, wixel-wise standard deviation, and visual inspection. Hence, the
remains task-
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samples
bias
std
truth
SDE ALD
DPS
Proj
measurement
Conditional
UCoS
Figure 6: Summary of posterior samples for the Deblurring problem. On the left column, the top image is ground truth and the bottom is measurement data. On the right column, posterior samples and their bias and standard deviation, with methods used from top to bottom: SDE ALD, DPS, Proj, Conditional, and UCoS.
results should be viewed as proof of concept. Further experiments are needed for more comprehensive parison of sample quality. While UCoS does not accelerate posterior sampling compared to the conditional method when using score approximation with similar complexity, we show that conditional methods, unlike UCoS, involve more complex function approximations. This complexity can hinder performance, particularly for neural operators with a limited number of parameters.
The general principle of transforming task-dependent unconditional score into a conditional score, devel- oped in this work, extends beyond the specific OU diffusion process considered here. This opens up intriguing questions about selecting the appropriate process to achieve more ellicient balance of computational elfort.
That said, the theoretical foundation of UCoS relies critically on the interplay between linear forward model and a Gaussian likelihood, which together enable the diffused posterior to be reinterpreted as a diffused prior under a new, task-specific process. This structure allows us to derive an exact identity between the conditional score and a transformed unconditional score, as captured in Theorem 3.7 However, extending this exact correspondence beyond settings where one can 'complete the square' as done in Equation appears nontrivial. Investigating whether similar structures can be identified in nonlinear non- non-Gaussian problems remains an important avenue for future work.
Acknowledgements
This work has been supported by the Research Council of Finland (RCoF) through the Flagship of advanced mathematics for sensing, imaging and modelling (FAME), decision number 358944. Moreover, TH was supported through RCoF decision numbers 353094 and 348504. M.L. partially supported by PDE- Inverse project of the European Research Council of the European Union, the FAME and Finnish Quantum flagships and the grant 336786 of the RCoF. Views and opinions expressed are those of the authors only and do not necessarily reflect those of the European Union or the other funding organizations. Neither the European Union nor the other funding organizations can be held responsible for them. MVdH gratefully acknowledges the support of the Department of Energy BES, under grant DE-SC0020345, Oxy, the corporate members of the Ato-Mattenatical Imaging Group at Rice University and the Simons Foundation under the MATH + X Program. The authors acknowledge the National Cancer Institute and the Foundation for the
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National Institutes of Health, and their critical role in the creation of the free publicly available LIDC/IDRI Database used in this study.
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Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
Yang Song, Liyue Shen, Lei Xing, and Stefano Ermon. Solving inverse problems in medical imaging with score-based generative models. In International Conference on Learning Representations, 2022.
Andrew M Stuart. Inverse problems: a bayesian perspective. Acta namerica, 19:451-559, 2010.
Yu Sun, Zihui Wu, Yifan Chen, Berthy T Feng, and Katherine Bouman. Provable probabilistic imaging using score-based generative priors. arXiu preprint arXiv:2310.10833 2023.
Pascal Vincent. A connection between score matching and denoising autoencoders. Neural Computation, 23:1661 1674, 2011.
Luhuan Wu, Brian Trippe, Christian Nacsseth, David Blei, and John P Cunningham. Practical and asymp- totically exact conditional sampling in diffusion models. Advances in Newral Information Processing Sys- tems, 36, 2024a.
Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, and Katherine Bouman. Principled proba- bilistic imaging using diffusion models as plug-and-play priors. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024b.
Jiachen Yao, Abbas Mammadov, Julius Berner, Gavin Kerrigan, Jong Chul Ye, Kamyar Azizzadeneshelli and Anima Anandkumar. Guided diffusion sampling function spaces with applications to pdes, 2025. URL https://arxii .rw. htrg/bbs/5505. 17004
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A Probability measures on Hilbert spaces
A.1 Gaussian random processes in an infinite-dimensional Hilbert space
This section introduces notations and outlines some basic properties of probability measures on Hilbert spaces. For more comprehensive introduction, we refer to Da Prato & Zabezyk 2014 Hairer 20223
Gaussian measures on Hilbert space Tet HH, ()) be a separable Hilbert space with norm || In V (.. A bounded linear operator C:: H + H called self-adjoint if (a, Cy) H (Cr, y) for all æ, y E and posilive definite if (C 2 0 for all æ E H. We say that a self-adjoint and positive definite operator C is of trace class if
(CH = Σ (Cens en) < cc.
where } is orthogonal basic of H. We denote by (H) the space of all self-adjoint, positive definite and trace class operators H. A random variable laking values in H is called Gaussion if the law of (h, X)n is Gaussian for each h E Н. Gaussian random variables are determined by their mean m = E[X] E H and their covariance operator defined as
(Cg, = [o, X m) X m)]
In this case, we denote X ~ (mm, C). m===i = O, is called centred. It can be shown that if X is a Gaussian random variable on H then C'E Li (H), moreover, [|I| trm(C).
The Cameron-Martin space We define the Cameron Cameron-Martin space associated with a Gaussian measure # = (0(0, C)) on H to be the intersection of all linear spaces of full measure under H,, and denote it by Hμ or le. It can be shown that H₁ = C1/2 and Hy compactly embedded and dense in H. In infinite dimensions, is necessarily the case that p(H,,) = Morcover, Hμ can be endowed with a Hilbert space structure with an inner product
(g, (g, = (g, = C-12g))))))))
Exomple 1. Let H Rd and μ = N/0, C) be a Gaussian measure on H with a positive definite covariance matrix C Rdd Then since C1/2 H, the Cameron-Marte space is the whole space Rd
Cameron-Marting theorem The Martron space H2 plays a special role in that it characterises procisely those directions in which one can translate the Gaussian measure μ without changing its null sets, thus 14 (h, C) and μ = (0, C)) are equivalent if and only if h Hn. Moreover, the Radon-Nikodyne derivative of μo with respect to μ is given by
dph
(x) exp - J.a.s., H. du
1
(21)
Note that since H, C1/2H H is dense in H, the random variable (h,x) 1/2 C-122 n, H, can be defined properly using limiting process, Remark 2.24 in Da Prato & Zabezyk 2014
A.2 Wiener processes B, in an infinite-dimensione Hilbert space
In this subscction, we recall the definition of the Wiener processes B, in an infinite-dimensional Hilbert space. As H is an infiniteddimensional Hilbert space, the process B, can not be defined as random variable taking values in the Hilbert space H, but we need to start our considerations with a larger Hilbert space having the additional propertics that are described below. At cach time E [o, 7] the process B, is defined be a Gaussian random variable taking values in the Hilbert space Y that the expectation of B, is zero and the covariance operator (Cy, where Cy: Y Y is a symmetric and injective trace-class operator in
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Y. That is, for o, v Y and 0 <ttsss T we have
= 0, B[, 0)r (Be, ,))r) = 1(0, Cr
( Be: 0)y (Bi+s - Be, 1/)r) s(6, Cy
We require that Y and Cy are such that
= Ran(C C/
Under the above assumptions, H is the Cartron-Martin of the Gaussian random variable B, for 0. Then, we can define for b e Y and ø E (Y) extension of H-inner product (u, o) by setting
0) = (u, Cy 6)x.
Using these definitions, holds that for all 0, E Cr c and 0 stttt+ we have
(((B,,0)H)
E((B,, (Be, v) h) U(b H)
EE([r++ B,, o) H (Br+s
= s(b,
Motivated by these formulas, we call B,, 0 s s T. a Wiener process in H having the (generalized) covariance operator tI.
We also note that in formula Equation 1 we use the symmetric trace-class operator C: + and the increments CB/ These increments can be interpreted the differences of the random process + Cl/2B,. At any time < sttT T, the random variable C1/2 B, takes values in the space H, and its covariance operator in H is C: H - H.
B Proofs for Section 3
B.1 Task mask-depedent score
Recall that C. I'+ (e' - 1)ACA* И + H and E, (e' - -1CC (e* - 1), AC::
AC + H.
Proof of Lemma 3.3 It is clear that M, is self-adjoint. For positive definiteness, we write
Σ, C1/2 ((e' -1] (e' (18 AC1/2) C1/2
C1/2 1 I 1-T A AC!/2)
C1/2
et
where we applied Lemma B.1 (a) for the last identity. Since 1 AC also positive definite, immedi- ately follows that
0< tr (E)) = (e' ()) (e' (CA' AC s (e' ()t < oa
and, consequently, Σ, is trace-class.
We replicate the proof of Lemma in Pidstrigach et al. (2024) for the procoss X!. The density Pe( (o) of X conditioned X 2o Gaussian centered at 2o with covariance Ex-
Proof of Lemma 3.6
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For what follows, let (o())) be the density of X and pe(x) the density of X We apply Leibniz's rule to obtain
1 Vx [((o(r))(Fo(x)))) p((x)
V. log
Ex
zo) (o dpo(ro)
- Σq
(x Σy
pe(x))
| (x 2o (x)())))))))
E(XIX* =
where we utilized Bayes" formula for the second last identity. This proves the claim. Lemma B.1. For any t > O, we have that
(a) the linear operator = - (cd--)1- (e' (6'-1))01/2000 AC -
is bijective and
1
E,¹ e It 1 AT 1 /C
(b) the linear operator E/ = 1 - (e' C) A is bijective and
(E)) I - (e' 1)CAT=1A and
(c) it holds that
(e' - EAT-
Proof. a) Invertibility from the right can be derived by straightforward computation
1 C1/2 /// 1
=I (c' / AC - -1) - (1AAA AC
Similarly, invertibility from the left follows from an analogous computation. The invertibility in (b) can be established using the same arguments. For the identity in (c), we have
E
(e'- 1)CA" (e' C) ACA" + - -I--))CAT--AACA
-(e' - ASSCA'OTAAAATTTT ACA"
(6' - 1)CA" +('' -1"CA) ACA"
-(e' - -1)CAAC ((e' 1)ACA + T) ACA"
ACA
(e' 1)CA"
The desired identity follows by inverting Cz.
B.2 Proof of Theorem 3.7
We follow the structure in Section 3.1 and first prove the following equivalent of Equation 10
Theorem 3.7
follows directly from substituting Equation T6 into Theorem B.2 while utilizing Lemma B.8
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Theorem B.2. Let Hc be the Cumeron-Martin space of C and assume that the prior satisfies ((Hc) 1. For > O, let En be given by Equation 12 and define
m2 y) = e x + (e' - 1)CA"C (y e² Ax). Az))
(22)
Then the conditional score function s(x, ) related to it holds that
s(r,ti = X(t) 8 (me .tp) + me(a, y) - e2
(23)
for (x. TRm a.e. ( Y) and >.0. 0.
Theorem B.2 direct consequence of the following proposition.
Proposition B.3. Let the assumptions of Theorem hold. Then
E(X (1, v)) B(X y, = =x).
(24)
for (I.,)) H T.m a.e. in (X,; Y) and > 0.
Throughout this section, we denote the transition kernel densities
(JD. Σx) dZ (e' - 1)() (e)22), (x(00.)) = dV(0,, Σ,) (z), (e x) = d0 -e' 1)C)
whenever the Radon Nykodym derivatives make sense.
A plan of proof for Proposition B.3 is as follows. We first develop auxiliary results: Lemma B.4 shows that the laws of (x,",, Y) and X²" coincide when conditioned on Xo 0. Lemma B.5 is used to express each expectation in Equation 24 in terms of transition kernels n, and ńn of the corresponding forward SDEs, which in turn can written as Radon Nykodym derivatives certain measures. After that, we use Lemma B.6 to show that the measures (0, (e' 1)C) and (0, Ex) are equivalent, in particular their Cameron-Mart -Mrttin spaces equal, which concludes the proof. Finally, put together the argument at the end of the section.
Lemma B.4. Let Z1 (1, (1 ())),, Zz ((O,T) and Za =N be mutually independent Gaussian random variables on H. Moreover, let 2o E H be arbitrary. It holds that
L(mi(e (0t2(e-1/220- 11/220
+ Z1, Azo + Z2)) = L(xo + Zz).
for any 0.
Proof. We have that
m/(e-t1220 + Z1, Axo + 72)
To + 1/ 71 + (e'
3o 12 E,--1,, EATT-1
is a Gaussian random variable centered at zo with covariance
(e 1/2x0 + Zı, Azg + Zz)) = EAT'-IA
e - ExC
(1TT-1,
1
E, (TT-1A 1 + C-1 Ex Ex.
This proves the claim.
Lemma B.5. The following holds
(1) Por (x,9) v) E HHH. m. a.e. in (X,,,, Y),
Snr zane cranalo, (dzo)
B(XPLY = X!" = x))
Je nu (o x 2))e00 (daro)
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(8) For E H [(x)' a.c.,
B(X = =) JH 1004 ()) In Me( x) (dzo)
Proof. (1): We first observe that, X Nxoo C) then ax ~ (oroo S2 for any >> 0, and as a direct consequence
Man. Naar,, oC)
(25)
for any a > 0 and B(H. By using this property for a e/2/ it holds that
dN(zo, (e' - 1)C)
(x(00 =
dN(0, (e' - 1)C) (c
dN(e W/(e-1/2000 (1 e (1))) (x)
dN (1 - ())
= B(X = X = x),
for (x E H E R a.e. in [(x Y), where the last equality follows from the proof of Theorem 2 in Pidstrigach et al. (2024) and that V(e-t/2a (1 - e) is the transition kernel of the forward SDE in Equation 4
(ii): We repeat the aforementioned argument from Pidstrigaah et al. (0024) adapted to our case. The joint X
distribution of , is given by ñ(vo, (O, (dx) & p(dx)). Indeed, for any A E B (X)),
n(xo, (0, (d) J| AXB
dN Σe) JJ
(d)d))udd))))
AXB dN E,) /| N(xo, E (drr)
P(X E X. B).
where it is used that N(ro,, 2x) is the forward transition kernel of the process XP, We show that
f(x) J (2u, x p(do)) f ni (x x 2) (dzo)
is a version of the conditional expectation E(X x). Let P, be the law of X7, that
(A(A) = F(X E A), A
The function f(x) af mmesuuuall by Fubini's theorem and for any A E (X()),
SA Sn zoño (x x)) dIP (x:
Exx,((p)) (LAf(x))
JH in (No, x))(drr))
I1 JH zone (No, x))((do))
(i x) x)) (p(dro) i (To. (0, (d)d)))((á))
/ xoñ p) (dro) JH n(0o, (dd0d)
NO, (dx)
Jn i x ())
// Coñū(Zn x) (d ((,
2n Na (1
The above properties define the conditional expectation and we can conclude the proof.
Lemma B.6. We have that
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( (H) C(H); moreover,
E E((H)
(ATT-AAA A+ (e* -1)- "leurr
(26)
(8) H(el-1)c= Hx..
(ti) The measures N(O, (e' 1)c) and (N(,,,)) are equivalent.
(tt) For To E H1e--1)e, we have
( ) Hs, (Eo, A AT)-1A-))) + (Fo, Hit--e
(27)
in NNO, Ea.-a.e
Proof. For the purpose of this proof, we abbreviate
C. = (e' 1)C.
(28)
(i): We first show that (H)) (H). Note that we can write Ex as
Σ, E, E. (I- (
where the last factor on the right-hand side is invertible (see Lemma B.1 (b)) proving that ranges of E, and Ce coincide. To prove Equation
26 let
E 2g (H) = C(H), then for some x H,
== En = E5 - C,A
We have
(AT-)
AT AĆ2a A 1CCx
ACx x ACC
x x AT ACxx - AT (r+ AC,A*) (0T AC²x = I,
since I' I'+ AČ,A Cr. This shows that
--12=-= (AT--I++
(ii): Notice first that we can write
11 H₂. cl c. (C,(H)) Нz, = -l (Ex(H))
where alll.lm, (H2) denotes the closure of H2 c Hi w.r.t I1-, Let prove that the norms I1 and Il are equivalent on HH) = (H) as, together with (i), this will prove the statement.
To this end, let x E EHH = (.( and apply (i) and Cauchy Schwarz inequality to obtain
|| || -ー < ||
(x, AT Ax) + ||0 A~
s 21/2 A - AC//21
((H,H) | + |1rllii
1+ c 1
((H,H) Iee
where the operator norm
21/2 21/2 11 ACC/22
is finite.
((H.H)
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(iii): From (ti), the covariance operators of μ and v have the same Cameron-Marthn space, by the Feldman- is Hilbert-Schmidt. To this end, note that
Hajek theorem we need to show that C, 1/1122, ,,, 1/2
C c- 1/2 - -- A Ac B.
The operator 8 is of trace class, since
(B= trx (2,22-00 A
A c(m.m) tri(Cc)) < co.
proving the claim.
(iv): Let xo e (.)... By (i), we have
(Do, )Hs, = -Ex Iu. 1)
= (ATT AAo, )) + (C, xz.
(29)
(ATT Aro, 1) + (Bos)) NO -aee
Now since EH) = C(H) dense Hy. H. the identity above can be uniquely extended to Hy, by the white noise mapping, see (Da Prato 2006 p. 23). This completes the proof.
7
Corollary B.7. We have
2 X2\ + (ro, AT) (vo, mx(X Y) Hr.
(30)
in distribution.
Proof. Combining Lemmas B.6 (wv) and B.4 it follows that
(zo, m2 (X#, Y)) nr. (ro, mi (X,, y) (5o, 1 Axi Am V)
in distribution. Therefore, we obtain
(Eo, (X Y) us.
= (wo, ma 1) + (20, AmT Y))
= X c. + (2o,
- + 2/2 AIT A
+ (20, (6' ACA' e
-002 X!1) c2
H- + 222 (zo A + (2o. A' (X -2
(6' () AT ACA CY e222x
(00 X!) 1 + e 1/2 ( A AX')
A + (2o, A x222x()))
= = X(") Xt") H + (zo. AT--lY)
in distribution, where we have used that
" AC. (e'-1) ACA' 1 AAT (T (e' 1)ACA") - A
=C.
This completes the proof.
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Proof of Proposition B.3. Let 20 E Hr, H where C. (e' -)CC. By virtue of Lemma B.5
will prove
that
In wonelas x)1 JH (roñi(ao, mz( y) (u(d2)) Sn ni (Fo,
(31)
Sn (no, mz( y) p(doo))
for (x,y) E H E m a.c. in (xX, Y). Notice that it suffices show that
du (mo) = 7(y) (I me v)),
ne( x)
(32)
du
for To E pla.c. and for (a, y) H Rm a.c. in (X," Y), where Z(y) does not depend on zo, hence will be canceled out in Equation 31 By Bayes' theorem Equation
du
(+) = Z(y)
cxp (-
1 ||A - u1l in (1)),
dp
Let us write
1
1
1 2 1lyl
|1Az yllr-
(zo, ATT
3)
2 (20. AT (xo))
H
2
and set Z(v) Z(y) exp
(1lllli): For xo E 11z,
1
ne X exp
(wo, (T (ro, A AST)
18/1
2(V) cxp
(2e (ro: X!")
||00| |100| H- Ce
+
(00, ATT-1
(wo. 1-T - AAz)
2
(33)
1
1
Z exp
222 Xi')
(50, A- -
|| H~
(ro, A-D-1 A-r-1Aro)
2
exp
(To, Me (X, Y) Mzz
| (
(
2
in distribution, where we have used Corollary B.7 and the identity
| #= + (200 A Ano |
(34)
which follows from Lemma B.6 (ir))
Now by Lemma B.6 (ii), zo E Hy, and thus the Cameron Martin theorem gives
exp (mo me (Xt Y), - ||:0 | Hy,
- metto, Y))
in distribution, which together with Equation 33 lead to Equation which completes the proof.
32
Therefore, Equation 31 holds true,
B.3 Gaussian example
Proof for Lemma 8.8 Let Č be covariance operator such that n (T))((H) 1. As / is the
intersection of all linear subspaces of full measure under a (see Prop. 4.45 in Hairer 2023 it holds (HH ( (H)
We now find another covariance operator C such that the score function corresponding C bounded linear, Let be such that C2/2(XX c s for any linear subset X of H. This implies
(H) = C(H) E.(H).
(35)
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We identify the law of X by using the relation
x Mr 2%),
where 2o realisation of the prior μ = (N, So). Hence
X²" ~ (O Σ, + So).
By the reasoning of Lemmn 4.4 in Hairer et al. (0066) it holds that
X6 ~N(mmc)
with some covariance operator C" and
ml
(2, + 5o) Xx.
Note that in Lemma 4.4 of Hairer et al. extension of the bounded map
(0066), the operator Ex + So) is defined measurable
(E, + (H) + H,
THE So)
to the whole space H per Theorem II.3.3 in Daleckij [991]. In case, it is possible to give explicit
formula for possible extension using the inclusion Equation 35 We define
Ex + So) = ((), + Sn) -x,)
((T + -,)
This map coincides with A (E, E (0) and we will now show, it is defined even the whole space
H. The operator Σ, H + (HH ( is bounded and invertible. Hence also Σ (H) + H is bounded. By Equation 35 the map 2, is well defined and bounded. We can now identify the score function by using the previous equality for the conditional expectation
-(z (=-B((XIIX) =))
8(2,6:)) (1
So + Sa) Sa) -12
[So(E, So) 1 2.
-E -0)-
This yields the claim.
B.4
Conditional score matching
Recall that mz(x,)) = e/22x + (e' I)CA e/2 and (t :- CAT + X(1)x for x E H and E RT. Moreover, R₁ = (e' - 1)1 + (e' - C, A: H + H.
Lemma B.8. The operator Rs H + H is bijective and the identity
mi(x,y) ReEr(x,)).
holds for all a E H and y E RM
Proof. First note that bijectivity is implied by Lemma B.1 (b). By direct computation,
Rs (CATT x(t)=))
(e' - 1)CAT- (c' -
ACATT-1
tel/2 - Ar
1/2 - (/2 - -I)CA'T Ar
+(6' - 1C) I)ACA" + (e'- 100,-111110 - ACA")I
=01
e/2 + (6' -1CCA (v- (v-c!/aAz)
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This completes the proof.
Proof of Lemma 3.10
We replicate the arguments of Baldassari et al. (2024a] adapted to setting. First,
observe that
(R, R, t: pi) s(2,, t.) #ellir
(R, Te, 1: + (18( 2.1l. 2(re "Tc, (1), 8( t; 3z)
It holds by definition Equation 14
E Rn (x))) (To Ti
Ir, 1; u), 8(0r, ti Iz))
=-B (ro "2, t: p)) E 20 20 xx (3r 30) $1 H
so 21 S (ro (Re 2,, 1; n), T0)
Hence holds
T( 3, ((pp) ||ro(Re 15- 1;11) (1) -Bell
- 2(t) 2n 2 (x)) E 2 (0,
-oo 1
with
('()) = 1( (x 15(0z, t, t.) - - -(t)) (Xx)
|| Err)2E)n~_2((3
(36)
To conclude, we add expectation with respect to ~ [6, Note that by Assumption 3.9 the first term on the rhs of Equation 36 uniformly bounded in & and by clementary calculations
E
20~2 (X:) N(t) |0oll S A(8) |
such that
; = E.-.T V'tt
sup V'(t) < cc.
8-8 tE[6,7]
This concludes the proof.
C Proofs for Section 4
Proof of Theorem Below, we the notation f S 9, if f(z) s Cg(x)) for all x with some universal constant C Recall that true solution w."" of the time-reversed denoising process and corresponding approximative solution process v.e satisfy
dw!" = (-2 W." -((W"" T -:t)))))) dt CL/2dB
av!" Vu [d X(T (() [ro (V("
v), v), T- 11:b)) ((F-14)/2 di dB/2 [e]
In what follows, we abbreviate the expectation Ey E (wr s)) _VAMe) as E, unless otherwise speciffee Decomposition of the error. Let us first consider the difference
E [ - 12l12 [w We - (vg ( + (wo - vo)l
Ww 100 (vz-vo)lie + EINIT,
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where EINIT = E |wo v-lla- Now decompose the difference into three terms follows:
Up) L//
W/ we (or (00)-
where we have
1
[Tr -2 (w- (((r) + s(w,, -=; - s(w[r]; - [T];
I2(r) = s(Wir] T |T|: - X(T - [-]] [ro(Er-1r](0))
[T]:4]] - c00-111)/2011) and ro
Ia(T) = AATT- - [T]) (r0()r--1)((11))))) 8). [T]:
To (ET (0- [T]:nA}
-(X(T (T]T--1])e(t (T-)//21 1) ( (/11).
Bound for EINIT- Recall that to (1 TZ and wh (1- (1-6-T)2 + c -o where Z ~ NO,,C) and Xo ph. It directly follows that
CINIT S S Exwny | = E - Eanp ||0||
where we applied marginalization of the joint distribution. Contribution from I
We observe that
T-
ENUM ==
E
1 (T) dr
T-5
(w, w
- -s(1]=), -[Tl: dr
1
14 Wir wr 11 dr + / T-6 |s(,,, T;11²) - s(w]r) T T 1=): dr.
S E
By Lemma 2 in Pidstrigach et al. (2024). it holds that
s(XI" it; p²) = c (1--))/2 (s(XH", ,=)"))="))) 0 O
for ay.a. y E R Therefore, we deduce by Lemma in Chen (223a that for the time-reversod process it holds that
( T - - T;) s(w) T (T])
< 4E
((wr,
-T:10))) T
((e(T--F1)/2) W|], 7; T::))) 11
+ 2 (1 1=) 2
E Bl|s(u+,7 - )1
< 4T. - -)) -T)B [c- W[
2 I + 2 -- T -=;;1²))
where we used the Lipschitz continuity of We note that
(e- walw: ~ N (e (t- s)2 ue, (1 (t-s9)/2
(1-e-(t-)))))
(37)
for T > t 0, see Itō 1984
It immediately follows that
wr ( LT11/2
1 = - (r-()---))
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and, consequently,
2 + (1 (1--e (1-e(t-L-D//)@B 1 11
E
Wtrrllln 1 SB wr (-11/2
[[]
H
2 (1 - ---
+ +(1 (1- elt- Ey Eenpy || + ((1 (1----0-(zD) tzx(C))
(1- - ttrr + (1- - c,//2)2 Expp || + trn(C))
Combining the arguments yields
ENUM S (T- (11- (()r--)))n((()))) (1--))))) (Baw || + trn(C)))
(T - - (1- = T; 11
Since the last expectation and Ezzp ||=|| are bounded by assumption, and note that 1- At O(At), (1
e O(At), we obtain the required upper bound.
Contribution from Iz: Applying Theorem
7 we obtain
I2(r) X(T - (00, [T]:m) + -
-=
= X(T |
[T|; + [=-
-re -ro -
[r]:n)}
and now it follows by reversing time and applying Lemma B.4 that
T-5
E
[[2(() dr ELOSS:
where éLOSs is given by Equation
20
Contribution from Iz: By triangle inequality and the assumption uniform Lipschitzness of we have
| и s X(T ( (r) - (r-
+ >(T
+ || -111|
$ k. (T (r]) -|
where we abbreviate (T')') = Lo(T))((')) + A( + for convenience. Note that by assumption kx()) (,2 T).
Combining the estimates, we obtain
T-6
- E Blwr-s - or SENUM + ELOSS + EINNT +E (T - 1r1) [ J dr.
Applying Grönwall's inequality, it follows
(f---aaa-- [T])2dT -L((T-L-)(dr))
B| -T- z (ENUM ELoss + +INIT)
Ra(TT
We may factor in the truncation by utilizing Equation
37
BllTT WT o11
E Ey E, - (1 6/2 )/22x-1 + v 5
(1-e 6/2 - Erop |
(1 (c))
0()).
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Published in Transactions on Machine Learning Research (11/2025)
Hence, conchude
VeT - -r- (eNUM ELOSS + +ENITT)epp exp 2 1-'6
)
([]))
+
which yields the result.
Details of numerical implementation
Neural network architecture In Figure 7 we highlight the FNO architecture that we use to approximate the score function in our approach. Here, we implement 4 hidden layers and denotes the number of pixels in both horizontal and vertical directions, while à represents the number of hidden nodes. The Conditional method employs the same architecture but omits the transform f(t) and includes an additional input node y. This modification affects only the dimensionality of the first layer, changing it to R*xxx3 while all subsequent layers remain unchanged. Consequently, the total number of parameters remains in the same order. The unconditional score approximation also follows the same architecture as UCoS but without the transform (()).
f(t)
K. b1)
(WW2 + K2 by)
aWW's + K2 by)
W, KG t-
ro(æ, (;
(
B
Raxaxh
Raxaxh
Raxaxh
Rexaxh
Rexaxh
Figure 7: Our FNO architecture. The scalar multiplication is given by f(t)x 1+std(a)) Activation function given by Relu. The lifting block R is given by linear layer and Q by two linear layers and activation function. The Fourier layers are followed by batch-noommliizatoon layer before the activation function. Below each layer is the dimensionality of tensor after passing through each layer
We use the FNO architecture from Baldassari et al. 20244) and choose å = 10-3 and 1. Similar the works of Pidstriggah (2024): Baldassari (2024a), we run the forward SDE with a non-constant
speed function leading to the SDE
1 dx, 3(1) VB(t)aWW,
with ((t) = 0.05 + (110-.00:5
We train the neural network for 10 epochs for the 32-nodes architecture, 30 epochs for the 64-nodes archi- tecture and 65 epoch for the 128-nodes architecture, where training is done Nvidia A100 GPU with 80 GB of memory. Training takes from 14-46 minutes for the unconditional method (depending on the architecture), 17-47 minutes for the conditional method and 24-51 minutes for UCoS.
During the training process we use the Adam stochastic optimizer with linearly decaying learning rate ranging from 0.002 0.0005. Samples are generated on the same machine. A uniform Marcy-Maruyama approxi- mation with 1000 steps is employed. Additional information can be found in the numerical implementation taepl--ttak--pppapaa http:////t/ttakkdddddeeeettteeee
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Published in Transactions on Machine Learning Research (11/2025)
Comparison methods For clarity, let give the precise score approximation of the following methods that we compare our method with. Note that some of these methods are used in combination with some other sampling method in the reference but we will always utilise backwards in time Euler Maruyama approximations.
SDE ALD uses the score approximation of Jalal (2021), which given by
s0(2,1;p1)) = 0(2, (1) + (T + 1 Ax)
(38)
for the hyper-paramette Tr under the assumption T 2 for some 0² 0. In line with Feng et al. 2023), we tune " such that the additive term has equal norm to the score function.
DPS Chung et al. 2023a employs similar idea to SDE ALD by removing the hyper-parameter and changing the mean of the Gaussian likelihood to be an estimate of zn:
sp (x s(r,4,)) - vv |y A(fo(x))
for some which is chosen such that p p= E/ lly - ( for some constant &. We use a grid-search algorithm to find the optimal ç. Above 8o (z) is an estimate EXxp X₂ = x) using the definition of the score function Equation 2.1
Proj Dey et al. (0024): This projection- based approach adds a data consistency step before every reverse time Euler Maruyama slep:
Et = -AATA+(1-X))) ((1 - vt)
We tune the hyper-paramcter by grid search algorithm. Since the operator (ΛΛ A (1- (1-A)) very large in terms of GPU-memory, employ an iterative scheme with 10 iterations each time step to solve the inverse (CT imaging problem) very small batch-size (Deblurring problem). This significantly increases runtime compared to applying precomputed operator.
Conditional Baldassari et al 2024a This approach approximates the conditional score function
8(x, (2)) = e e --- = =x,Y=v)
directly.
E
Additional figures
More samples and in higher resolution for all methods can be found below in Figure g for the inpainting problem, in Figures 10 and 11 for the CT imaging problem and Figure 12 for the Deblurring problem.
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Published in Transactions on Machine Learning Research (11/2025)
Figure 8: Posterior samples for Inpainting problem. Methods from top to bottom: SDE ALD, DPS, Proj.
Conditional, UCoS, true posterior
Figure 9: Posterior samples for CT imaging problem with a FNO architecture that uses 32 nodes per layer. Methods from top to bottom: SDE ALD, DPS, Proj, Conditional, UCoS.
35

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Published in Transactions on Machine Learning Research (11/2025)
Figure 10: Postcrior samples for CT imaging problem with a FNO architccture that uses 64 nodes per layer. Methods from top to bottom: SDE ALD, DPS, Proj, Conditional, UCoS.
Figure 11: Posterior samples for CT imaging problem with FNO architecture that uses 128 nodes per layer. Methods from top to bottom: SDE ALD, DPS, Proj, Conditional, UCoS.
36

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Published in Transactions on Machine Learning Research (11/2025)
Figure 12: Posterior samples for Deblurring problem. Methods from top to bottom: SDE ALD, DPS, Proj.
Conditional, UCoS.
Below in Figure
13
reprint Figure in larger size.
0.18
0.225
ours
conditional
conditional
0.16
0.200
conditional
0.14
conditional
0.175
0.12
0.150
Bis
0.10
0.125
0.08
0.100
0.06
0.075
0.04
0.050
0.02
20
40
60
80
100
120
20
60
100
120
nodes per layer
nodes per layer
Figure 13: Comparison of the error dependence of the conditional method and UCoS on the parametrization of the score approximation. X-axis: nunber of nodes per layer in the FNO, Y-axis: L2 norm of the bias (left) or std (right), averaged over the number pixels. Dashed lines correspond to 64 x 64 resolution and crosses correspond the problem with 256 x 256 resolution.
37


Reviews:
[{"review_id": "k1N1Zk4q8H", "reviewer": "Reviewer_SMGZ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths.**  \nThe central idea of the paper is interesting and mathematically well grounded. The authors provide a precise formulation in infinite-dimensional Hilbert spaces, supported by a clear theoretical identity and convergence guarantees. This is a strong point of the work.\n\n---\n\n**Weaknesses.** \nThere are concerns regarding the practicality of the proposed framework, particularly in terms of the computational complexity of the final model and its implementation. The paper does not provide an order-of-complexity analysis of inference, which makes it difficult to assess how the theoretical advantages translate into practice. Moreover, the experimental results, while demonstrating that UCoS achieves comparable posterior quality to existing baselines, do not convincingly support the central claim of improved runtime efficiency. In below, I will explain my concerns in detail.\n\n**Major comment on the clarification of the runtime efficiency claims:**  \n\nThe authors claim that UCoS achieves more efficient inference by avoiding applications of the forward operator $A$ during sampling. However, the empirical runtimes reported in Tables 1 and 2 are identical to those of the Conditional baseline, which does apply $A$ at inference. This suggests that the claimed computational savings are not realized in practice. One possible reason is that the UCoS framework still requires evaluating a neural network $r_\\theta(\\xi(x, y), t)$ at every sampling step. Therefore, although $A$ is not directly applied to $x$ at inference, the network $r_\\theta$ still processes a full $x$-dependent input and performs a costly forward pass, particularly given the use of a Fourier Neural Operator (FNO) as the backbone.  \n\nTherefore, to support the central hypothesis that eliminating applications of $A$ leads to more efficient inference, the authors should provide a clear theoretical comparison of the per-step computational complexity between UCoS and Conditional. In particular, they should quantify the cost of applying $A x$ versus evaluating $r_\\theta(\\xi(x, y), t)$. Furthermore, an ablation study is necessary to assess the trade-off between network complexity, denoising performance, and runtime. For instance, it is unclear whether simpler architectures could retain similar posterior quality while improving inference time, or whether the cost of evaluating $r_\\theta$ outweighs the benefit of removing applications of $A$. None of these analyses are presented in the current version, and as a result, the paper does not convincingly support its core claim regarding the efficiency of the proposed formulation, neither theoretically nor experimentally.  \n\nIn fact, although the authors acknowledge the similar complexity of the FNO backbone used in both methods, this undermines the central motivation for the UCoS reparameterization. If the neural network architecture ultimately dominates inference time, then moving operator applications offline does not yield tangible benefits in practice. This suggests that the complexity of posterior sampling, especially in high-dimensional or functional inverse problems, is not solely driven by the cost of operator evaluations, but also by the cost of score network evaluations and data flow. The authors are encouraged to clarify this point and moderate claims regarding efficiency gains, and provide additional experiments to demonstrate that decoupling from $A$ leads to clear speedups in a different regime (e.g., with a lighter network or a more complex forward model).\n\n**Some minor comments:**\n\n- The flow of the paper could be improved. At present, detailed derivations are somewhat merged with the high-level description of the approach. For clarity, I suggest moving parts of the theoretical development (e.g., Section 3.1 on theoretical foundations) to the supplementary material.  \n\n- Figures would benefit from clearer annotation. It would be preferable to indicate directly within the figure (not only in the caption) which model each row corresponds to and what each column represents. In addition, including a brief statement of the main conclusions drawn from the figures and tables in the captions would be helpful.  \n\n- There are minor typographical issues, for instance the word *equation* is occasionally repeated (e.g., “equation equation”).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses the problem of posterior sampling in Bayesian linear inverse problems with additive Gaussian noise, where the posterior $p(x \\mid y)$ is typically intractable. Score-based diffusion models (SDMs) have recently been proposed to approximate such posteriors, but existing methods often require repeated evaluation of the forward operator $A$ and its adjoint $A^*$ during inference, which becomes computationally expensive in high-dimensional or infinite-dimensional settings.  \n\nTo address this, the authors propose **UCoS**, an unconditional representation of the conditional score function. The central idea is to shift the computational cost to an offline training phase, where a task-dependent but measurement-independent denoising network is trained. At inference, the conditional score is recovered by applying an affine transformation to the latent variable and measurement, without requiring further applications of $A$ or $A^*$.\n\nThe method is formulated in infinite-dimensional Hilbert spaces to ensure discretization-invariance and is supported by a theoretical identity connecting the conditional score to the trained unconditional score, as well as a convergence result bounding the sampling error.\n\nThe authors validate their approach on two imaging inverse problems: *computed tomography (CT) reconstruction* using the LIDC-IDRI dataset and *image deblurring* using the CelebA dataset. They compare UCoS with different baselines, including conditional score-based diffusion, DPS, Proj, and SDE ALD. The reported results show that UCoS achieves posterior samples of comparable or superior quality to these baselines, while maintaining similar runtime despite not applying $A$ during inference.\n\nThe authors conclude that UCoS provides a theoretically principled and practically effective framework for posterior sampling in high-dimensional inverse problems, with the key advantage of decoupling inference from forward operator evaluations.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Provide a theoretical comparison of the per-step computational complexity of UCoS versus the Conditional baseline, explicitly comparing the cost of applying $A x$ versus evaluating $r_\\theta(\\xi(x, y), t)$.\n\n- Add an ablation study to analyze the trade-off between network complexity, posterior sample quality, and runtime. In particular, test whether a lighter network architecture could offer improved inference time without degrading performance.\n\n- Clarify and moderate the paper’s claims regarding inference efficiency, especially in light of the empirical runtimes being similar to baselines that do apply $A$.\n\n- Include additional experiments (e.g., with a more complex forward model or simpler network) that clearly demonstrate the benefit of decoupling from operator applications during inference.\n\n- Improve the structure of the paper by separating high-level explanations from detailed derivations. Consider moving some theoretical sections (e.g., Section 3.1) to the supplementary material.\n\n- Revise figure annotations to include model and axis labels directly within the figures. Additionally, update figure captions to briefly summarize the key takeaway from each figure or table.\n\n- Correct minor typographical issues, such as repeated words (e.g., “equation equation”).", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "a4v3KpSFAb", "reviewer": "Reviewer_9hUC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "# Strengths\n- The main identity (Eq. 15) appears novel and of independent interest: it cleanly reduces a conditional score to an affine transform of a task-dependent unconditional score.\n- It outperforms the “Conditional” baseline (a model trained directly on (x, y) to approximate the conditional score) in several benchmarks.\n- The exposition is clear and the paper is well structured.\n\n# Weaknesses\n- The approach is not training-free: a new task-specific unconditional model must be trained for each linear inverse problem. In terms of wall-clock, the reported training and evaluation times are broadly comparable to directly training a conditional model for that task.\n- While improvements over the conditional baseline are present, the gains are within the reported standard deviations, so some margins appear modest.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proves that, in linear–Gaussian inverse problems, the posterior (conditional) score can be written as an affine transform of an unconditional score. Building on this, the authors propose training a task-dependent but unconditional diffusion model whose dynamics already encode the observation operator. With that model in hand, conditional sampling for the given inverse problem becomes a cheap affine post-processing step that avoids per-step applications of the forward operator.\n\n# Questions\n- “Matrix-free” meaning: Matrix-free seems to denote the fact that you do not need to evaluate or invert $A$ during sampling. This naming convention is a bit unusual to me, but maybe I missunderstood.\n\n# Summary recommendation\nA solid, well-argued paper with a useful and (to my knowledge) novel identity at its core. Recommend accept to TMLR, with the above additions likely to further improve clarity and empirical persuasiveness.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "# Changes (to strengthen the paper)\n1) Running toy examples. Include worked-through cases (e.g., C = I, Gamma = epsilon * I, A a masking/projection operator) to give immediate intuition for the covariance shaping, the R_t / Sigma_t constructions, and the resulting per-coordinate noise levels.\n2) Direct use of the conditional-score identity. For a few problems where A is cheap (e.g., inpainting/masking) or moderately sized, demonstrate literal use of the affine conditional-score formula (even if slower) to show the identity’s practical fidelity and to triangulate performance against the proposed task-dependent training. Here, comparison with DPS would also be of interest to the community, even if the results are negative.", "broader_impact_concerns": "The same as for any work in generative modelling.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "7qd2k9uotZ", "reviewer": "Reviewer_pG4G", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Regrettably, I am not familiar enough with Bayesian inference in infinite dimensions or using diffusion models to adequately assess the soundness of the theoretical part of this submission, which is a pity, as it is the main contribution. Therefore, this review is with low confidence.  \nThat being said, as far as I can tell, a major strength of this work is its technical rigor. The proposed method is grounded in theoretical arguments and insights and comes with mathematical guarantees.  \nHowever (mostly unrelated to the acceptance criteria), with my level of expertise I found the submission quite unaccessible and I would have appreciated more discussions and interpretations of the results, e.g. in form of a short paragraph in the vincinity of the theorems and throughout.  \n\nThe empirical evaluation (which is clearly not the focus of this work) is rather limited and from my perspective serves only as a proof of concept. Only two tasks (CT-Imaging and Deblurring) are studied and each only on the basis of a single dataset.   \nMore importantly, **the experiments do not verify the central claim of the submission**, i.e., that \"[the authors] address the prohibitive sampling cost for large-scale inverse problems by demonstrating that this computational overhead can be offloaded to the training phase, thereby accelerating posterior sampling.\" Looking at the reported runtime, the method is exactly as fast as its most related baseline \"Conditional\" (Baldassari et al., 2024a).   Therefore, I currently answered no wrt claims and evidence.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This work covers Bayesian inference in infinite dimensions using score-based diffusion models. Specifically, an \"unconditional representation of the conditional score (UCoS)\" is introduced which lowers computational costs as forward model evaluations are avoided during sampling. The main contribution of this work is theoretical, i.e., in the proofs of (1) that the conditional score is connected to a task-dependent unconditional score which can be learned in an offline phase, and (2) the convergence results that quantify how far the UCoS samples are from the true posterior.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I have some questions  that are crucial for my understanding of this work and I want the authors to reply to here and in a revision.\n\n1. Why is it beneficial that *\"the computational effort can be shifted to the offline task of training the unconditional score of a specific diffusion-like random process*\". Relatedly, is it really a shift of computational effort, or does its amount change (decrease or increase)?\n2. What is the practical benefit of the infinite dimensional setting (if there is any)? Does it allow to handle new types of tasks or data; is it more natural/appropriate than the finite dimensional setting?\n3. Your submission largely builds upon the work by Baldassari et al., (2024a). In which terms is is similar, and where precisely does it differ. I have seen the comment *\"This is in stark contrast to the conditional\napproach Baldassari et al. (2024a), where the conditional score $(x, y, t) \\mapsto s(x, t; \\mu^y ) : R^n \\times R^m \\times R → R^n$ in Def. 2.3 is approximated\"*, but it was too brief and superficial to answer my questions.\n\n4. Minor. You state that *\"This method extends rigorously to infinite-dimensional diffusion models, making it attractive for large-scale inverse problems, as it eliminates the need to evaluate the forward map during sampling.\"* How is the causality here? Do you mean that the method extends to infinite dimensions because there is no need to evaluate the forward map during sampling. If so, why?", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)