Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling
Authors: Shayan Mohajer Hamidi
Ben Liang
EN-HUI YANG
Venue: NeurIPS 2025 poster ()
Keywords: Diffusion Models
Inverse Problems
Posterior Sampling
Primary area: deep_learning

Abstract: Inverse problems, where the goal is to recover an unknown signal from noisy or incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-based techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood $p(\boldsymbol{y} \mid \boldsymbol{x})$, often resulting in artifacts and instability under complex or high-noise conditions.
To address these limitations, we propose a novel framework called coupled data and measurement space diffusion posterior sampling (C-DPS), which eliminates the need for constraint tuning or likelihood approximation. C-DPS introduces a forward stochastic process in the measurement space $\{\boldsymbol{y}_t\}$, evolving in parallel with the data-space diffusion $\{\boldsymbol{x}_t\}$, which enables the derivation of a closed-form posterior $p(\boldsymbol{x}_{t-1} \mid \boldsymbol{x}_t, \boldsymbol{y}_{t-1})$. This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantitatively, across multiple inverse problem benchmarks.

Coupled Data and Measurement Space Dynamics for Enhanced Diffusion Posterior Sampling
Shayan Mohajer Hamidi Stanford University smohajer@ stanford.edu
Ben Liang University of Toronto linag wwwg@eecc.ttooot..co
En-Hui Yang
University of Waterloo chyang @waterlooo.ca
Abstract
Inverse problems, where the goal recover an unknown signal from noisy incomplete measurements, are central to applications in medical imaging, remote sensing, and computational biology. Diffusion models have recently emerged as powerful priors for solving such problems. However, existing methods either rely on projection-basee
techniques that enforce measurement consistency through heuristic updates, or they approximate the likelihood p(y x)), often resulting in artifacts and instability under complex or high-noise conditions. To address these limitations, propose novel framework called coupled data and mea- surement space diffissio posterior sampling (C-DPS), which eliminates the need for constraint tuning likelihood approximation. C-DPS introduces forward stochastic process in the measurement space {y,). evolving in parallel with the data-space diffusion {x, J. which enables the derivation of closed-form posterior p(z(-1 Hy, 1/-1). This coupling allows for accurate and recursive sampling based on a well-defined posterior distribution. Empirical results demonstrate that C-DPS consistently outperforms existing baselines, both qualitatively and quantita- tively, across multiple inverse problem benchmarks.
Introduction
Inverse problems, where the goal is recover unknown signal zo from noisy incomplete measurements y. arise in a wide range of applications, including medical imaging [1.). remote sensing 43 and audio signal processing [5 6). Mathematically, these problems are often modeled as y = Ag + n, where A is a known forward operator and n represents measurement noise. Inverse problems are inherently ill- iol-posed, meaning that, without additional constraints, infinitely many solutions may satisfy the given measurements y. As such, incorporating prior knowledge about the underlying signal and the noise model is essential for reliable reconstruction.
One principled approach to addressing this uncertainty is to treat inverse problems a Bayesian framework, where the goal becomes sampling from the posterior distribution p(æaly). However, accurate and efficient posterior sampling remains a central challenge, particularly when the forward operator A is ill-conditioned or the measurement noise is significant [7]. Traditional sampling-basee methods, such as Markov chain Monte Carlo (MCMC), often struggle with high-dimensional spaces require careful tuning of proposal distributions [ More recently, diffusion based techniques have emerged as powerful generative models for high-dimensional data, and several works have adapted diffusion processes inverse problems by combining learned data prior p(z.) with measurement- conseestttoossieennyccce term 7
[5] 16
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Nevertheless, in standard diffusion based frameworks, the measurement information must often be retro-fitted into the prior p(æ)) via projection projection-based techniques 8
10
T2
T4
L5
A for details). Since these methods modify the iterative processes data and measurement space an uncoordinated manner, there is guarantee that the generated samples lie on valid data manifold while also being consistent with the observed measurements. This often leads to suboptimal reconstructions: the generated samples may either exhibit visual artifacts due to drifting off the data manifold, or fail to resemble the original measurements due to poor measurement fidelity. Examples of both failure
or approximation of the likelihood p(y](=)
7
16
17
18
19 (see Appendix
modes are illustrated in Figure
To address the limitations of con- ventional approaches, we propose C- DPS (coupled data and measurement space diffusion posterior sampling), novel framework that extends the standard diffusion model by introduc- ing a second, parallel diffusion pro- cess in the measurement space, along- side the conventional diffusion in the data space. While the data space pro- cess follows the traditional forward diffusion from an unknown signal Zn, the measurement space process be- gins with the known observation Yo and progressively injects noise. This symmetric treatment mirrors the dy- namics applied to zo. tightly coupling the two spaces throughout the diffu- sion trajectory. This coupling enables the derivation of a closed-form poste- rior expression for p(z;-1 (zt, 1) thereby removing the need for ad-hoc measurement terms or learned likelihood approximations commonly used in prior work.
Messurement Ground tuth
C-DPS (Ours)
DPS
Re Re-Sample
(M4ino) Dakhe (C
Visual Artibots
Moccueeee Pobiity
Figure 1: Visualization of reconstructed samples for some linear inverse problems using our proposed C-DPS method, compared with two leading baselines: DPS 77 and Resam- ple (0). C-DPS exhibits less visual artifacts and improved measurement consistency.
From Bayesian perspective, the joint diffusion over both and y defines a coherent generative model over the pair (æ, y). In doing so, C-DPS naturally integrates both the learned prior and the forward measurement model into a single, unified framework.
We propose C-DPS, a novel coupled stochastic framework that introduces a parallel diffusion process in the measurement space {yr), evolving jointly with the data-space process { This formulation enables principled posterior sampling in diffusion models by treating (x, y) as a unified generative process.
By explicitly constructing a Markov chain over Vr. we derive a closed-form posterior transition p(æt-1 æt, vt-1). eliminating the need to approximate or learn the likelihood term p(y xx). This allows for direct integration of the measurement model into the sampling procedure, ensuring consistent Bayesian updates every diffusion step.
We develop scalable and efficient sampling algorithm for C-DPS based on pre-whitened conju- gate gradient solver. This matrix matrix-free implementation retains the runtime efficiency of conventional DPS methods, despite the added complexity of coupled lata-measuremmnt diffusion.
We validate C-DPS through extensive experiments on standard benchmarks, including FFHQ 21 and ImageNet 22). Our method achieves tfate-of-the-ar performance across multiple inverse problem settings-suck such as inpainting, deblurring, and super-resolution both qualitatively and quantitatively.
Notation: Scalars are represented by non-bold letters, (e.g., or A), vectors by bold lowercase letters (e.g., a), and matrices by bold uppercase letters (e.g., A). The real axis is denoted by R. The symbols 0 and I represent the zero vector and the identity matrix, respectively.
2

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Background and Preliminaries
2.1 Diffusion Models
Diffusion models are built upon two key components: forward noising process and backward denoising process. These processes operate as Markov chains that progressively perturb and then recover data distributions. In the discrete formulation [23], the forward process gradually adds Gaussian noise to the data according to a predefined variance schedule { and is described as
T II p( k=1
(Ia)
p(=,) N 1 8,
(1b)
where I2 E Rd and B3 E (0, 1) is a monotonically increasing sequence controlling the rate noise addition over time. Since each transition p(æ; (2t-1) follows linear Gaussian model, the marginal distribution p()) remains Gaussian, given by
x0)) ~ N(vārzo, (1 at i),
where 1 3r and ā. = 11 aj are derived from the variance schedule
23
To generate data samples, a neural network sp ( t) is trained to approximate the score function Vz, log p(x.). The reverse denoising process is also formulated as Markov chain, where the model iteratively reconstructs the data distribution. The backward process expressed as
p0 (x))) = NV (N (1)((1))) Σe),
(3a)
(o(x)) = var(1 Ō(-1)
Var 13 @x,))),
+
(3b)
1 αx
1
1 1 d I,
Eg Bi 1 at
(3c)
where (ox)) is the predicted initial state of the data computed using Tweedie's formula [2
fo(x;) T. +
aq Sg))s(( t)
v a
2.2 Diffusion- Biffusion-Based Inverse Problem Solving
We focus linear inverse problems, where the goal is to reconstruct unknown signal an Rd from noisy, incomplete measurements E Ih"" governed by the linear model Amo + n, where A E mg is known measurement operator, and (O, En) denotes additive Gaussian noise with known covariance Σn, This leads to Gaussian likelihood p(y (x)) = (A ).).
Recent advances in applying diffusion models inverse problems can be broadly categorized into two methodological paradigms: projection- based and likelihood- based approaches. These methods summarized in Algorithm|] [ and comprehensive discussion is provided in Appendix A We briefly describe each paradigm below to set the stage for our proposed method.
Projection-based approaches. These approaches iteratively project the generated samples onto the feasible set defined by the measurement y. At each reverse diffusion step, the generated sample 2; updated using projection operator Py, such
Bt f (x+) = arg min || - æz|1² subject to
||A
y112 S e,
or in simpler forms (e.g., for noiseless measurements).
2t 21 A AAAAAT) -AA y).
This family includes methods such as ILVR 10,, DDRM 14, and DSG which leverage such projections to maintain measurement consistency throughout sampling.

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Algorithm Baseline methods
Algorithm C-DPS (one reverse step, clean form)
Input: # time steps T, y. noise schedule Input: # steps T, measurements y, schedule {Br).
{Be}. measurement A, {ōt}-
operator A, noise covariance Σn
1: Generate (ye) as in Equation 5 draw N
1: NN I))
No 1)
2: for - ,....,0.. do
3:
& 4- 1-80(z2,t)
2: for t = 1 1, T .... do
4:
@o (xtt) + Var (ot + )))
3:
8 - 80 (0 t)
Enj t 04 En 1n+(n110n (1-0tr1)) at + 1-&
5:
Mo(æ)) +- yorll- N1
Vite-ddee 1-
20(2)))
4:
6:
(0,
5:
b f (1
-)-1) (1-1a-11) A
7:
Ø/-1 (x+ 0
6:
Define the matrix-free precision operator
A 6- C4 u + A -ーー u.
8: -1-1 obtained from x by ei-
y/æ
ther a projection step 8
12
7:
Solve for ppost with CG
13 14
15
or likelihood approximation
8:
PW-CC A, ylm ai (Algorithm 3 E
16 18 19
9: Update: _ <- poost +
9: end for
10: end for
Output: Z0p
Output: 350
Likelihood-basel approaches. These methods aim recover the unknown signal by approxi- mately sampling from the posterior distribution p(zoo y), where y denotes the measurement. Using Bayes' rule, the posterior can be expressed as
p(æo y) oc p(y xo)p(2o))
Here, p(#o) is modeled using diffusion model trained on clean data, and p(y xo) represents the likelihood induced by the forward measurement process (e.g., linear operator with additive noise).
Since the likelihood is typically intractable to evaluate directly in diffusion models, several methods approximate it its gradient. For example, DPS 7 and related methods [16] [7 Tweedie's s formula or Bayesian denoising estimators to approximate Va, log p(y x;,) at intermediate timesteps i. This approximation is then used to adjust the reverse dynamics by modifying the model's predicted mean. Specifically, the mean used the reverse step is replaced by P y) = (gg + XV n. log ply a4), where Heg (2x())) is the original predicted mean, and À is scaling factor.
[8] [9]
use
3 Methodology
3.1
Motivation
As discussed in Section 2.2 most existing approaches to solving inverse problems with diffusion models rely on retro-fitting measurement information into the learned data prior p(zo)) This typically done either through heuristic projection- based techniques by approximating the likelihood p(y x). However, both approaches introduce fundamental limitations: projection-bad based constraints often require manual tuning and lack theoretical justification, while likelihood approximations can distort the posterior geometry, especially under high noise levels (see Figure D.
To address these challenges, we propose C-DPS, principled framework that integrates the mea- surement process directly into the diffusion dynamies. Central to our idea an auxiliary forward stochastic process in the measurement space, denoted by {y, )r= By coupling the data-space process (ax Y and {y, ) C-DPS enables the derivation of closed-form posterior distribution p(æ,-1 1 1)) This eliminates the need for manually tuning constraint terms or approximating the likelihood. Crucially, unlike the data-space diffusion process which requires training a neural network to approximate reverse dynamics, the measurement-spaac process { is purely forward and analytically defined. It evolves from the observed measurement y. not to generate yo but to propagate measurement information coherently across diffusion steps. This coupling allows C-DPS to incorporate observation structure directly into posterior updates, resulting in more stable, interpretable, and accurate reconstructions across wide range of inverse problems.
worth noting that similar measurement-spacc diffusions have been discussed conceptually in Appendix 1.4 of [25 and further explored in (26) for medical imaging, but neither derives an analytic posterior or provides a tractable sampling rule. contrast, C-DPS formulates a closed-form posterior
p( Ft, Vr-1 1), removing heuristic updates and manual tuning.

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In the following subsections, we elaborate how the sequences fy and {0t (=)) are generated.
3.2 Constructing Markov Chain in the Measurement Space
Similarly to the diffusion process in the data space which is defined with Markov chain in Equa- tion lb), we define a Markov chain tyJo that starts from the real-world measurement Yo and progressively adds Gaussian noise at each forward step. Specifically, we let
Yo A + n.
(5a)
y/ V1
Br 1- + BB. Z1,
(5b)
where ~ (O, I) and 8 E (0, determines the noise schedule. The linear operator A maps the unknown data Zg to the measurement space, and n is the measurement noise.
Remark 1 (Choice of noise schedule). We use the same noise schedule for for both the data- space diffusion { and the measurement-spe diffusion {y.}. J. This design choice keeps the two processes synchronized, which simplifies the derivation of the backward update p(x{11 2t, Yr-1) and avoids introducing additional hyperparameters. Conceptually, using shared schedule ensures that the same fraction of noise is injected at each time step in both domains, preserving a one-to-one correspondence belween and yt- Although it is possible in principle to define separate schedules for the data and measurement spaces, we find that shared schedule leads to strong empirical performance and a cleaner theoretical formulation.
From the standard forward-diffusion identity, Yr could be written in terms of yo as
Ve Vor Vo
VI- S:
(6)
where Ç ~ (0, I), and āg is defined in Equation n ~~ (0, En), we have
Further noting Vo A xo + n with
y, = Vā (Axo + n) + VI- S.
From Equation it easy to obtain the following conclusion:
Proposition 1 (Distribution {y. D. For the sequence {y, YT defined in Equation 5 the distri- bution of y is Gaussian whose mean and covariance step are given by
My. Voa A Do:
(8a)
Σg, ā. En + (1-āz)
(8b)
3.3 Generating {æ;} Consistent with {y.}
Given the measurement sequence (y,), construct consistent sequence of latent states {æx) that evolves under diffusion while remaining aligned with the observations.
Initialization. We start by sampling
I).
Backward Recursion. For each from T down to 1, we generate æ;-1 by drawing sample from (x De. 1) Based on the Bayesian update rule for the posterior, we can write
pl p. Fz, v)-1)
p( (I-1) p( 1).
(9)
To find p(y,_ ( at intermediate diffusion step, we use the formula in Equation (7) and replace ag in this formula with its estimate in terms of æ.- using Equation Thus, we get
yr-1 = Vā-- xt-1 + (1 (1-1)8p((1111
1)
+ tn + V1 ā
(10)
va--1
Ax A((1 āz -0--1) 1)) va- v a
(11)
Pleasse refer to Remark to see why the prior p p((ce-1 i) dropped from Equation
g
5

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Since (0, and ~ (V(, P(y:-1/11111 is Gaussian with the following parameters:
p ~ N Mylz Eylwm
(12a)
where
Myla A + (10 1),
(12b)
and
Eyla c En + 0t-1)I.
(12c)
Substituting Equation 1bb and Equation ( 12a) into Equation 9 the posterior p(æ;,-111111 12,, yr-1 could be obtained using Bayes' rule and dropping the normalizing constant:
p(æ; Fe,
Fe, 11-1) exp
V
Bg
1| (v--1 Majlo
(1
E yiu (3c-1
My)æ)
(13)
The expression in Equation 13 defines the exact un-normalized posterior density. The first term is quadratic æ;-1 and arises from the diffusion prior. However, the second term involves the score network 8 (3 1, 1), making the conditional mean Myle a nonlinear function of æ;-1-
To make the posterior distribution in Equation [13 Gaussian, we apply a common approximation in diffusion-bart based inference and freeze the score network at the current iterate:
so 80 1)
so(xs, (1
(14)
This substitution avoids evaluating the score ft-1, which is not yet known during sampling, and instead uses the available point Ky. Intuitively, since æ;-1 and 2; are close for small 3,, this approximation preserves consistency. While this approximation has been used in prior works [10 12). we further empirically justify it in the context of our work in Appendix B
Using Equation (14). Myl becomes affine in E1-1,
Myle = AE + (1 āt-1) A so(
(15)
A
so the product of the two Gaussians in Equation quadratic and linear terms gives
remains Gaussian N MM Epost). Collecting
1 3
Σ
D-1
+
A,
(16a)
Be
Be
+ A Σ Eyle (yi-1 bl.
post
Epost
(16b)
3r
Hence, C-DPS samples from (M Epos) each reverse step. The full procedure is shown in Algorithm
2. with the latent variant, termed LC-DPS, detailed in Appendix C In addition, in
Appendix D extend the above approach for the case of non-linear measurement.
Remark 2. Note that the exact form of Equation ( includes the prior term p(æ,-1). In diffusion models, after sufficient noising steps under linear or cosine schedules, the marginal p(x;-1) is close to N I) since the noise dominates. Including this term multiplies Equation (13) by exp( 1-,1
which adds I to the posterior precision, that is A4 + 1- I + A A extra I changes euch diagonal by at most small fraction ((3)), which is numerically negligible. We therefore omit (1) Equation (9) for clarity.
y|u A
I. Since 3r < 1, the
3.4 Efficient sampling
Our goal draw æi-1 p(a,-1 3t, 9)-1) in Equation (6 without forming dense factoriza- tions. Define the posterior precision operator
Σ
post
c I A 4/m
c
8 D.
(17)
Direct Cholesky on dense Enja is (d3) and impractical. We therefore use two matrix-free conjugate- gradient (CG) solves per reverse step, implemented by Algorithms 2 and 3

---

Step 1: mean solve.
Compute Mpost by solving
As Mpost
Ct + A A D w/æ
(yc-1 b.-1), b (-ū A
(18)
with & = (0(21,1) t)-
Step 2: noise draw (PW-CG). Draw (O Epoa)))) by solving A, for synthetic right- hand side & satisfying (ov(z)= A,. Algorithm33 3 shows how to build such a from two standard Gaussians using a whitening operator.
Step 3: update. Set
2t-1
Mpust
(19)
Pre-whitened conjugate gradient (PW-CG). Let W satisfy W W = D For many structured noise models a closed form exists. For example, if Σn = a² then
Σ y (ā1_10² -0 I W 7t
1/2
(20)
Define A = WA A so that A c. + A A. PW-CG samples
E1 N(O,, Ia),
E2 Im,
VG + A E2,
(21)
which gives cov(2) = A. Solving Ae by CG then yields v ~ NO, A, = - Epoa).
Practicalities and cost. Each CG iteration applies A and A once. A diagonal preconditioner P. diag (Aa) works well. CG requires ((V)) iterations with the condition number of P, Ar. Empirically < 50 across our tasks, so the PW-CG cost per step is ((V nnz(A)). . In our setup. score evaluation dominates the runtime, while the linear solves add small overhead.
To contextualize this cost, we note that evaluating the score network 88 (0(2,, lyttycically implemented as U-Net Transformer-talle
approximately 100-200 milliseconds per step on modern GPU (e.g., NVIDIA V100 or P100) for a 256 x 256 input. In contrast, our linear solver runs significantly faster, often requiring under 10 milliseconds. Its contribution the total runtime is therefore negligible compared with the dominant cost of score evaluation.
Algorithm 3 PW-CG (draw V~~N((, Epstt without Cholesky)
Input: precision operator A,, matrix A, action of Σnj (or its square root), scalar Ct
1: Draw E ~ (O, Ia), E2 ~ (O, independent
2: (Prewhiten) define a whitening operator W with W W = Σ Dpm and set À = WA
3: Form z - E + Eg
4: Solve v CG-solve(A,
(that is, solve A,v 2) (then cov(v) A, Σ Epsst
Output: v
Experiments
4.1 Quantitative Results
Experimental setup. Following prior work 7 27 we evaluate our method on the FFHQ 256x250
21 and ImageNet x56x256 [22] datasets, using 1.000 validation images from each. All images are normalized to the [0, 1] range. To ensure fair comparison, we adopt the same experimental settings used 7 across all evaluated methods. The measurement data is corrupted with additive Gaussian noise of zero mean and standard deviation = 0.05. During inference, we use fixed number of reverse diffusion steps 7 1000, following standard practice the literature. For score estimation, utilize the pre-trained model from (7) for FFHQ and the model from [28 for ImageNet.
7

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Table 1: Quantitative results on the 1k validation sets of FFHQ 256 x 256 and ImageNet 256 x 256.
Bold and underline indicate the best and second-best results, respectively. Green and red denote performance improvements and degradations relative to the best baseline.
Pixel Domain Methods
Dataset
Method
Iupaint (Random)
Inpaint (Box)
Deblor (Gaussian)
Debtur (Motion)
SR (4X)
FID LPIPS
SSIM
FID
LPIPS
SSIM
FID
5 LPIPS 5 SSIM
FTD
LPIPS
SSIM
FID
LPIPS
SSIM 1
DPS
21.19
0.212
0.851
33.12
0.168
0.873
44,05
0.257
0.811
39.92
0.242
0.859
39.35
0.214
0.852
IGDM
21.27
0.221
0.840
34.79
0.179
0.860
40.21
0.242
0.825
33.24
0.221
0.857
34,98
0.202
0.854
DDRM
69.71
0.587
0.319
42.93
0.204
0.869
34.92
0.332
0.767
62.15
0.294
0.835
MCG
29.26
0.286
0.751
40.11
0.300
0.703
101.2
0:340
0.051
87.64
0.520
0.559
FFHQ
ILVR
25.74
0.231
0.672
37.24
0.175
0.854
$2.93
0.297
0.784
17.59
0.253
0.844
ReSample
21.25
0.202
0.847
33.51
0,160
0.866
37.05
0.251
0.822
31.19
0.220
0.892
30,48
0,204
0.851
PaP-ADMM
90.42
0.441
0.812
66.52
0.353
0.855
Score- SDE
76.54
0.612
0.437
60.05
0.331
0.678
109.0
0.403
0.109
96.72
0.563
0.617
ADMM-TV
181.5
0.463
0.784
68
0.322
0.814
186.7
0.507
0.801
110.6
0,428
0,803
PMP-DM
21.15
32.21
11.92
0.251
0.816
37.21
0.233
0.871
36.21
0.859
DAPS
20.77
0.201
0.869
29 44
0.144
0.882
35.84
0.242
0.830
30.26
0.215
0.911
30.15
0.202
0.854
DMPlug
20.12
0.197
0.877
27
0.140
0,888
12.44
0.230
0.800
55
0.210
0.925
28.55
0,199
0.862
C-DPS
20.14
0.195
0.881
26.33
0.132
0.871
32.24
0.238
0.832
27.29
0.217
0.921
28.41
0.196
0.855
DPS
35.87
0.303
0.739
18.82
0,262
0.794
62.72
0.444
0.706
56.08
0.389
0.634
50.66
0.337
0.781
RGDM
41.82
0.356
0.705
42.
0.284
0,752
59.79
0.425
0.717
54.18
0.373
0.675
54.26
0,352
0.765
DDRM
114.9
0.665
0.405
45.9 15.95
0,245
0.814
63.02
0.427
0.705
59.57
0,339
0.790
ImageNet
MCG
39.19
0.414
0,546
39.74
0.330
0.633
95.04
0.550
0.441
144.5
0.637
0.227
H.R
0.656
39.51
0.278
0.726
71.24
D:421
0.662
95.3
0,532
0.498
ReSample
33.47
0.289
0.730
39
0.259
0.799
61,24
0.459
0.708
55.76
0,370
0.637
49.19
0.339
0.777
PnP-ADMM
114.7
0.677
0,300
78.24
0,367
0.657
100.6
0.519
0.669
97.27
0.433
0.761
SDD
0.517
54.07
120.3
0.667
0.436
170.7
0.256
ADMM-TV
189.3
0.510
0.676
87.69
0.319
0.785
155.7
0.588
0.634
130.9
0.523
0.679
PaP-DM
34.92
0.296
0.736
37 67
0,258
0.797
61.06
0.433
0.707
55.33
0.372
0.636
50.10
0,336
0.786
DAPS
0.741
15.46
60.12
0.119
0.709
54.82
0.365
0.639
49.62
0.789
DMPlug
32.85
0.226
0.748
34.28
0.247
0.804
$7.42
0.407
0.714
53.13
0.366
0.642
48.96
0.324
0.793
C-DPS
32.37
0.214
0.755
33.24
0.236
0.807
$6.36
0,391
0.712
$2,06
0.352
0.644
47.30
0.316
0.795
Latent- Domain Methooss
Dataset
Method
Inpuint (Random)
Inpuint (Box)
Deblur (Caussiun)
Deblur (Motion)
SR (4x)
FDD
LPIPS
f
L LPIPS
SSIM
4 LPIPS L SSIM
HID
LPIPS
SSIM
1
LPIPS SSIM 1
PSLD
47.21
0.221
0.809
43.02
0.158
0.813
89.51
0.316
0.631
96.15
0.336
0.678
74.36
0.287
0.649
FRHQ
ReSample
89.85
0.140
0.746
53.21
0.184
0.749
71.69
0.255
0.714
44.72
0.198
0.823
93.18
0.392
0.594
RLSD
0,142
0,808
44.08
0.153
0.244
0.750
49.10
0.284
0.810
61.37
0.774
LC-DPS
36.67
0.137
0.815
42.11
0.144
0.821
65.71
0.232
0.759
46.57
0.272
0.819
58.41
0.197
0.787
Image Vet
PSLD
83.21
0.337
0.785
116.33
0.465
0.694
91.39
0.390
0.688
124.67
0.511
0.594
97.45
0.360
0.694
ReSample
59.87
0.143
0.756
127 84
0,262
0.631
65.35
0.254
0,703
66.89
0.227
0.738
113.42
0.370
0.576
RLSDD
60.44
0,141
0.787
130.16
0,259
0,705
67.34
0.249
0.712
68.43
0.236
0.735
92.73
0.317
0.680
LC-DPS
60.02
0.140
0.790
113.67
0.252
67.21
0.244
0.718
63.94
0.216
0.748
85.15
0.683
The measurement models used in our experiments follow those described in 7 (.) Inpainting: For box-type inpainting, central 128 x 128 region is masked out; for random- -type inpainting, 92% of pixels (across all RGB channels) are randomly masked. (it) Super-resolution (SR): Low-rusolution measurements are obtained by applying bicubic downsampling. (iii) Gaussian blur: A Gaussian kernel of size 61 x 61 with a standard deviation of 5.0 is convolved with the image. (fv) Motion blur: Motion blur kernels of size 61 x61 are generated using the code from 229. with an intensity parameter of 0.5. These kernels are convolved with the ground-trun images to produce the measurements.
Benchmark methods. For pixel-based diffusion model experiments, use DPS 77. 1IGDM g. DDRM [14), MCG [30]. ILVR [0]. ReSample (20). PnP-ADMM [31] Score-SDE 25 total-
variation sparsity regularized optimization method (ADMM-TV), PnP-DM 32. DAPS 33 and
DMPlug (34]. Furthermore, for latent diffusion experiments, we compare LC-DPS with PSLD 351
the latent version of the ReSample, and RLSD [36.. To ensure a fair comparison, all methods use the same pre-trained score function.
Evaluation metrics. To evaluate the performance of different methods, we adopt the metrics used in 77: (i) Iearned perceptual image patch similarity (LPIPS) (37), (ii) Frechet inception distance (FID) [8], and (iid) structure similarity index measure (SSIM) [39], In addition, we provide results
for peak signal-to-noise ratio (PSNR) in Appendix All experiments are conducted using a single NVIDIA P100 GPU with 12 GB of memory.
The quantitative results for both datasets are presented in Table Across nearly all tasks, C-DPS (or LC-DPS) outperforms the baseline methods. In few settings, DMPlug attains scores that are competitive with ours. We emphasize that although DMPlug can be close in accuracy, its runtime is approximately 4x slower than C-DPS, which limits its practical utility; see Appendix F.] for run-time details. We also conduct a similar set of experiments under varying Gaussian noise levels, presented in Appendix H.l
Results.
8

---

4.2 Visual Comparison
Meseunement
Groun moth
C-DPS (cuns)
DPS
Re Sample
Doll (CMuriCammzoo (Caumtrr)
In this subsection, we provide vi- sual comparison of reconstructions produced by C-DPS, DPS, and Re- Sample. We choose to include DPS and ReSample in these visual compar- isons because, as shown Table [T] they achieve the strongest quantita- tive performance among the evaluated baselines. We randomly select five images from the FFHQ test set and corrupt them using the measurement models described in Section [4.] with minor adjustments to enhance visual clarity: for super-resolution, we apply an 8x downsampling factor, and for random inpainting, 95% of the pixels are masked. The reconstructed images are presented in Figure 2] where each row corresponds to a different mea- surement setting. More qualitative re- sults are presented in Appendix [H.2]
Dobbur Deckir trbatto
DX
Irowest (Parddnn)))
irquer (ang)
Figure 2: Qualitative results on FFHQ dataset.
As observed, C-DPS consistently produces reconstructions that more closely resemble the ground
truth, exhibiting fewer artifacts and better perceptual quality compared with DPS and ReSample.
4.3 Measurement Fidelity
In this subsection, we demonstrate that C-DPS achieves higher measurement fidelity compared with DPS and ReSample. Specifically, we show that C-DPS recovers solutions æ that better satisfy the measurement model.
To this end, conduct motion deblurring experiment under noiseless setting, where the Gaussian noise is removed (i.e., = Azo) We then track the reconstruction error |yy - A as a function of sampling progress. The progress is expressed percentage relative to the total number of reverse diffusion steps taken by each method.
We run each method over 100 instances and report the average error along with the shaded region representing the range of observed values. The results are shown in Figure 4 As illustrated, C-DPS consistently maintains lower measurement error throughout the sampling trajectory, indicating better alignment with the forward measurement model.
4.4 Why C-DPS Works: Posterior Recovery Ground- Truth Benchmark
In this subsection, we aim to demonstrate that the superior performance of C-DPS stems from its improved approximation of the true posterior distribution. To support this claim, we compare its ability recover the posterior against the best benchmarks DPS and ReSample.
To this end, we construct a toy dataset where the data distribution p(æo) is defined as a mixture of 25 Gaussian components The means and variances of the mixture components detailed in Appendix G where we also explain how the ground-truth posterior can be computed in closed form for any given observation y. measurement matrix A, and noise level a.
To systematically evaluate each method, we generate measurement models (y. A) E R mxd across combinations of signal dimension, number of measurements, and noise levels. Specifically, we consider (d, m, or {8, 80, 800} x {1, 2, x {10-2, 10-1, 10 yielding distinct settings. All Gaussian mixture components are equally weighted to ensure a balanced posterior.
We follow the dataset construction procedure in 40 16

---

10a
10a
10-
10.
10-3
ReSample
C-PSS (Ouns)
10
10
ReSample
10-1
CDPS (Ouns)
1000
20
100
Progress
Progress
Figure 4: Evolution of the measurement error | A over the sampling trajectory for DPS, ReSample, and C-DPS on the FFHQ (left) and ImageNet (right) datasets.
Table 2:
Sliced Wasserstein (SW) distance between the true and estimated posterior distributions across different dimensions d, numbers of measurements m, and noise levels o. Lower is better.
d
800
Method
m
2
2
2
m
C-DPS
2.2
1.5
0.5
2,9
1.7
0.4
3,3
2,5
0.3
DPS
4.7
1.8
0.7
1.2
5.8
ReSample
2.6
2.1
3.8
3.2
2.8
0.6
3.5
3.1
0.4
C-DPS
0,9
0.6
1.7
0.4
2.8
0.4
10
DPS
4.7
1.5
0.8
3.1
1.0
5.7
1.3
ReSample
2.2
1.6
3.8
2,9
2.7
0.6
3.3
2.7
0.4
C-DPS
1,9
0.9
1.7
1.2
0.8
1.6
0.7
10P
DPS
5.2
3.5
2.5
3.9
1.7
6.8
0.9
ReSample
1.5
2.3
1.8
1.6
144
0.9
2.0
2.0
0.6
GT posterior
C-DPS
For each configuration, we generate 1000 samples from the true posterior and apply C-DPS, DPS and ReSample to approximate the posterior using 1000 steps. We then assess the quality of each approximation using the sliced Wasserstein (SW) distance [41]. metric that captures high-dimensional differences. The SW distance is com- puted using 10* random projections per method.
ReSample
DPS
Table reports the mean SW distances along with 95% confidence intervals, calculated over randomly sampled measurement matrices A per configuration. Additionally, Figure 3 visualizes the estimated posterior the first two dimenstons for the setting (d 80, m = 1 0.1), using a single randomly drawn measurement matrix. As illustrated, C-DPS more accurately recovers the true poste- rior, successfully capturing all mixture modes. contrast, DPS and ReSample either miss certain modes fail to represent the full posterior geometry, highlighting the ad- vantage of our coupled diffusion formulation.
Figure 3: Visualizing the first two di-
mensions the estimated posterior dis-
tributions for the configuration (d 80,
m 1, o = 10-1) for random A.
5 Conclusion
We have introduced coupled data and mcasurement space diffusion posterior sampling (C-DPS), novel framework that addresses the limitations of likelihood approximation in diffusion-based inverse problem solvers. By introducing forward stochastic process in the measurement space {y. which evolves in parallel with the data-space diffusion {æ,). C-DPS enables the derivation of closed-form posterior distribution. This coupling allows for accurate, recursive sampling without relying on approximations of the likelihood or heuristic constraints. Empirical results demonstrate that C-DPS consistently outperforms state-of-the-al baselines, delivering robust and high-fidelity reconstructions across a wide range of inverse problems. We also discuss limitations and failure cases Appendix[] highlighting opportunities for future improvement.
10

---

Acknowledgments
This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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A Related Works
In many practical applications, we often encounter the underdetermined regime where m d. making the inverse problem ill-posed. To obtain meaningful reconstructions in such settings, incorporating prior knowledge essential. Within the Bayesian framework, this addressed by modeling prior p(æo) and forming the posterior p(zol y) via Bayes' rule:
p y) o ply))z))p((0))
When employing diffusion models priors, one can extend the reverse-time stochastic differential euqtion (SDE) used in unconditional models,
dx
p(t) V() log ( dt + V B(t)dw 2
by incorporating an additional likelihood gradient to perform posterior sampling:
dæ = 80 B(t) (Væ, pr + Vz, logpi(y(zz)) dt + (()
(22)
This follows directly from the identity:
Vz, log Pa (xxll) Va, (x + Va, log (
In this formulation, two quantities are needed: the prior score V x, lo (zze), which can be obtained from a pre-trained diffusion model, and the likelihood score Vz. log Pt ( (x,)) The latter is generally intractable, since the data y conditionally dependent only on Vo. not directly æt. As result, estimating this likelihood gradient becomes a key challenge. the following sections, we explore various strategies for approximating this term.
Among the earliest approaches for solving linear inverse problems with diffusion models are Score-ALD and Score-SDE. both of which estimate the gradient of the log-likelihood to steer the reverse diffusion trajectory. The key difference lies in how they compute the residual. Score-ALD uses a deterministic correction:
Score-ALD 42 and Score-SDE (25].
A T y Azi)
V æt logp(yla, R
o +
where "/e is a tunable parameter regulating the guidance strength. On the other hand, Score-SDE adds Gaussian noise to the measurements before evaluating the discrepancy:
Va, logply x.)) A (y o,c - Art), c 6~ (~,
While both methods use the measurement error guide the reverse process, Score-SDE introduces stochasticity by perturbing the observations, effectively pushing the samples toward noisy target y, y + 0,6.
ILVR 10). ILVR, originally introduced for super-resolution tasks, applies similar principle by guiding the reverse process with a noised version of the measurements. Its gradient approximation takes the form
Va, p(y)(y))..) ~ AAt Ac.) -
AT -Ax.)
where At denotes the Moore-Penrose pseudo-inverse of A, and y, y + drc. Compared with Score-SDE, ILVR can be viewed as a preconditioned variant, replacing the simple adjoint A with full pseudo-inverse to achieve more accurate projection onto the measurement-consistenn space.
DPS 77).
While the previous methods are tailored for linear inverse problems, DPS extends to non-linear settings and is among the most widely used reconstruction techniques in this domain. The core approximation behind DPS
Væ, log ( ~ Vz. log (y Zo E[ æxl)
Assuming additive Gaussian noise and forward operator A, the likelihood becomes
p (y) 2o
E/xo æx)) N N(: A A(E|oo xx)), o2
15

---

yielding the approximation
V Væ, logp(y/<z))
Vn,
(B|z0)
æx]) (A(E
æx.)) y)
For linear inverse problems with A A Aa, this simplifies to
V m, (y)p)))))))))))))))))))))
22,
y
AB/w0 mr))
Using Tweedie's s formula, the gradient term can be further written as
1 (I v² lo:))( A (y
Va, logp(y/zz) o
AlE xx))
In practice, DPS does not rely on the theoretical guidance strength above, and instead employs an adaptive step size inversely scaled by the norm of the measurement error.
MCG [2], DSG [3], and MPGD 43 Several recent works build on DPS by incorporating
geometric insights to improve reconstruction quality. MCG [[2] provides a geometric interpretation of DPS, demonstrating that its approximation helps maintain samples on the data manifold. Building on this, DSG [3] introduces theoretically grounded step size derived from the MCG perspective, and combines it with projected gradient descent to enhance sample fidelity. More recently, MPGI) 43 further improves performance by constraining the update steps to lie within learned low- dimensional subspace using autoencoding, effectively regularizing the sampling path to remain close to the underlying manifold.
The DPS approximation effectively replaces the posterior p(æo|ar)) with a Dirac delta centered at its mean:
IIGD 8
xx x (xx - E[xp xx,))
In contrast, IIGDM g proposes a more expressive approximation using Gaussian distribution:
p( = / (E|zo r?)
where 72 is tunable hyperparameter. For linear inverse problems, this leads to the marginal likelihood
ply xx) so N (AE[zol<;)) r AAAA
+o11),
and the corresponding gradient becomes
VR, log p(y((()) [
(AAAT + T
AT (y AE(x=//:)))
dx,
This formulation softens the delta approximation of DPS and introduces a covariance-aware correction, improving flexibility in modeling uncertainty.
The IIGDM method assumes an isotropic Gaussian approximation for p(æo|<x; ignoring the structure of its true covariance. Moment Matching 44 improves upon this by leveraging an exact expression for the posterior variance
Moment Matching 44).
V( [ af H (logp, Th of E. (20 [
resulting in an anisotropic Gaussian approximation:
p(z.|z,) = [E]z ([zg|>.))
For linear inverse problems, this yields a refined estimate for the measurement score:
Va. ax logp(y)<::) R Va, T A (o E|oo (3 AE [x)))
In high-dimensional settings, explicitly computing the full Jacobian Va, K[ [z.]] is computationally prohibitive. To address this, the authors employ automatic differentiation to evaluate Jacobian-vector products efficiently, avoiding the need to form the full matrix.
16

---

SNIPS 45 and DDRM [4].
These methods reformulate linear inverse problems as noisy inpaint- ing the spectral domain via the singular value decomposition (SVD) the measurement matrix, A U.V With this, the measurement model becomes
Ea + 2z where
V y U y -
SNIPS solves the inverse problem in this transformed space by performing annealed Langevin dynamics to sample from pl a y)), using the approximation
Va, log log o.) = ET -
Σx.)
DDRM improves upon SNIPS by replacing the noisy iterate 2; with the posterior mean Toja V Baan [x.] in the above expression. Furthermore, DDRM introduces a sampling rule based on conditional Gaussian distributions over the spectral components x( which adapt based on the singular value the diffusion noise level and hyperparameter 7) E (0, 1] control stochasticity:
@ (1)
N(
0 + VI- - 17
Bjs+1
17²0?):
8: 0 at at 2 4y
( zt
(()) aie+1
N + /1 11-1700
11 m²²z))
0
W(g("), -
o²
When = 1. this sampling reduces to the deterministic posterior formulation used in the original DDRM algorithm.
DDS 46 and DiffPIR 47
Both DDS and DiffPIR approximate the posterior mean via proximal
formulation:
A 1 - - xx]]
E DD xr yl so arg min l Az
+ 2
This balances fidelity to the measurements with proximity to the diffusion based prior. The methods differ in solving this objective and selecting A DDS solves it approximately via few conjugate gradient (CG) steps, motivated by replacing DPS gradient updates with CG under data manifold assumptions, and uses fixed A In contrast, DiffPRR adopts closed-form solution and schedules A proportional to the signal-to-noise ratio at time f, specifically A = or with as constant.
Another line of work tackles inverse problems via variational inference, where the true posterior p(aooy) is approximated by a tractable distribution g, optimized by minimizing KL divergence.
RED-Diff approximates p(zulz) ( using a Gaussian q = NN,,, III), minimizing their KL divergence follows
RED-Diff 48).
min DKL ( ((xoll)))
This leads to the variational bound
min
y
(11)1 20
Aelleee t) =l
which combines a reconstruction loss and mctre-natching term.
Score Prior minimizes
49
This approach uses a normalizing flow ds as the variational distribution and
E B
[ log logp(y|Go(z)) - log (oG(())))) log (()) log det
dG
The prior term log Po computed via the PF-ODE formulation, though is computationally expensive and must be optimized per measurement.
17

---

0.035
0.965
Mean Squared Difference
0.950
0.025
Similarity
0.950
0.015
0..44
0.940
0.005
0.935
Diffusion Step
Figure 5: Cosine similarity and MSE between score vectors. Cosine similarity remains high and MSE stays low across diffusion steps, supporting the approximation (1(x:-1, 1) R 88 (Er, t).
Efficient Score Prior 50 lower bound
To reduce cost, this variant replaces the exact likelihood with a surrogate
by (aa) Ep mo flog N (xr)) {' g(t)² dt,
where (t)) includes the denoising loss and other tractable terms. This approximation drastically reduces computation while maintaining training quality.
As seen, variational methods approximate the true posterior with simpler distribution whose parameters can be efficiently optimized using standard techniques. While this enables tractable inference, it may limit expressiveness, as the chosen distribution might fail to capture the full complexity of the true posterior.
B Empirical Justification for the Score Approximation in Equation (4
In Section we introduced practical approximation to enable tractable posterior sampling:
80 1)
su (2t, l).
This simplification avoids computing the score at the unknown by reusing the available value xt. To empirically validate the accuracy of this approximation, we measure how closely the two score vectors align in both direction and magnitude across the diffusion trajectory.
We quantify this with the C-DPS cosine similarity and C-DPS mean squared error (MSE) between the score vectors Sp -1 - 1) and (a(2,, t) at each diffusion step I, where æ;-1 is generated from H, via standard reverse diffusion step. Specifically, we compute the cosine similarity
(0 1), so(x (It, t))
CosSim(t) ||00 1, 1)[12 2. |00 (xt, 1)[[2
and the MSE as
MSE(t) =1 1) sy (2e, t)ll?
where d the dimensionality the data.
Figure shows both metrics averaged over 100 randomly sampled instances from the FFHQ dataset, evaluated across the full reverse diffusion trajectory (i.e., t 1000 - 0).
As seen, C-DPS cosine similarity remains high (above 0.94) throughout the diffusion process. This indicates that the two score vectors are consistently aligned in direction, even if their magnitudes differ. Since the reverse process relies more on the direction of the gradient, this supports the validity of the approximation.
18

---

In addition, The C-DPS MSE remains low across the trajectory, peaking modestly during the middle diffusion steps (approximately [400, 600). This is expected since
In early steps (high t), the noise level is large, and score estimates change slowly.
In later steps (low 1), the signal is nearly denoised, and the score stabilizes.
The middle regime exhibits faster dynamics as the model transitions from noise to structure, hence slightly higher variation in scores.
These empirical findings demonstrate that our approximation in Equation (4) introduces minimal directional or magnitude error, particularly in regions where diffusion steps are small. This justifies its use in our closed-form posterior sampling framework without sacrificing empirical performance.
C Latent C-DPS
In the latent version of C-DPS, both the diffusion process and the measurement model are defined over the latent space of pretrained generative model, rather than the pixel space. This modification allows us to leverage the benefits of a compressed latent representation, improving computational efficiency and robustness, particularly for high-resolution data.
Algorithm f presents the detailed steps for the latent C-DPS framework. Compared with its pixel- domain counterpart (Algorithm 2. the following changes are made:
The unknown signal is represented by latent variables {4, instead of pixel variables {
The forward operator A and the measurement y are defined in the latent space.
The score function 8y is evaluated latent variables rather than pixel variables.
All sampling, posterior updates, and noise schedules are performed in the latent space following the same principles as in the pixel domain.
Despite these modifications, the overall structure the algorithm remains the same: forward diffusion process is applied to the measurement, followed by backward sampling procedure using closed-form posterior updates that couple the data and measurement spaces each time step.
Algorithm Latent C-DPS (ours)
Input: # time steps T, latent measurement y, noise schedule {B:), forward operator A in latent space, and noise covariance Σn
1: Generate { 12" as latent-space version of Equation 5
2: lT ~ (0,
3: fort= T,T 2, ...... do
4:
6 (e,,tt
5:
Myye Al-11
(4-1)) A Dyle +- ār-1En + (1 ū;- 1]
6:
7:
Compute Mpost and Sostt via (66. replacing a; with &
8:
~~ Z~N(0,1
9:
4- 1 Epost
10: end for
Output:
D Extension to Nonlinear Forward Models
Locally linear C-DPS. Let the measurement model be g(x.) with differentiable 9 Rd - R and n (0 Σ.). The coupled construction still applies, while the Gaussian closed form Eq. (16) holds after a local linearization of g at the current iterate xx;
g(x) Rs g(xx)) + Je(x æx), Jr
Vg m=z,"
(23)
19

---

With the same Enlz = ā + ā I as in the linear case, the conditional becomes
100-1100-13 MJ + cc2 Eglrr) Ct g(æx) Jjz (1 āt-1) J1 s0 (2 (z),,)) t).
(24)
Combining this with the diffusion prior p( x(-1) = -N(V- B1 Bel) yields a Gaussian posterior, identical in form to Eq. (16) with A Je and be-1 cz:
1 Be
E Epost
I
+ Je D.T
(25)
Be.
v1 3c
post
poss
+ Jj 1 (8-1---)))
(26)
B1
This is a Gauss -ewstn or extended- Kalmmaed-Kalman Kalman smoothing view inside the C-DPS recursion. adds Jacobian-vector product per step, which modern autodiff provides efficiently. The dominant cost remains the score call. In our setup, the score pass costs about 100 to 200 ms and the linear solve is under 10 per step, the added Jacobian-vector product (< 15 ms) is modest.
Phase Retrieval
Method
FID 4
LPIPS
SSIM
PSNR
DAPS
42.71
0.139
0,851
DPS
104.5
0.410
0.441
17.64
RED-diff
167.4
0.596
0.398
15.60
PnP-DM
99.4
0.335
0.581
19.69
DMPlug
41.21
0.124
0.894
31.25
C-DPS (local linear)
41.32
0.129
0.903
31.91
Nonlinear Deblur
Method
FID 4
LPIPS
SSIM t
PSNR
DAPS
49.38
0.155
0.783
28.29
DPS
91.31
0.278
0.623
23.39
RED diff
43.84
0.160
0.795
30,86
PaP-DM
68.96
0.193
0.742
27.81
DMPlug
47.28
0.135
0.792
29.55
C-DPS (local linear)
47.52
0.130
0.802
30.27
Preliminary nonlinear results (FFHQ 256 256).. These early results show that locally linear C-DPS retains the advantages of the linear model while remaining computationally feasible. We also observed DMPlug to be close in accuracy but about 5x slower in our setup.
E
Implementation Details
E.1
Implementation Details of Baseline Methods
We evaluate all baseline methods using their official implementations and default settings. The corresponding repositories are listed below:
DPS
MCG
30
HIGDM
g
(ttps://github.com///0000/
diffusion posterior sampling
DDRM 14 https://github. com http:hhhww..oooaaaaaaaaaaeaaaaaa Ktao//baaaat-kava/dddm
ADMM-PnP
31
httpe///gicaab..ca/laggg//////// and play_admm admm
ILVR 10 httpa:
ReSample 20
We follow the original code from repository provided by with the pretrained LDMs on FFHQ and ImageNet datasets. We use the default hyperparameters as implied in [35]. For some tasks, the hyperparameters were tuned by grid search, and the results according to the optimal hyperparameters are reported.
PSLD 35):
20

---

E.2
Construction of Whitening Operator W
To solve linear systems involving the posterior precision A, = 1-4 I A Sy A, we employ pre-whitened formulation based on the factorization
yiv W W,
where W is a whitening operator. This appendix describes how to construct W under common structural assumptions on the measurement noise covariance En. which is used to define Σy i āz-1 Dn +
Case 1: Diagonal or isotropic noise. If E,, = I, then
Syle (á=-10² 2 + a(-1) I % S0 W 7
-1/21.
Case 2: Low-rank plus diagonal noise. Suppose In = UU a² for a tall matrix U E R
with r < m. Then,
E Eyjr a i-- + +o +
ā. I = á -U- T9I
where 6, = ag 10=100² + ār-1. The inverse can be computed using the Woodbury identity;
2 8,²' (1 ( I dr C-U " UT)
To match the form Σ-1 W W. define W matrix whose Gram matrix gives the above (e.g.. via Cholesky of the RHS).
Case 3: Convolutional (circulant) noise. If En is circulant (e.g., for colored Gaussian noise in image processing), then it diagonalized by the Discrete Fourier Transform (DFT):
Σn FIDF,
(27)
S0
Σyee = F'D'F, â (1 ā I is diagonal. Then
(28)
where F is the DFT matrix and D'
D-1 y/
F'D''1F and W = D 1/21/2
This allows We and W o to be computed efficiently using FFTs.
Implementation note. Although we derive closed-form expressions for the whitening operator W all three cases (diagonal, low-rank, and convolutional noise), do not instantiate W an explicit matrix. Instead, we implement the action of W and its transpose through matrix-free routines, which apply v We and W v directly using efficient operations. For instance, in the convolutional case, W is implemented via FFTs; in the low-rank case, via structured low-rank multiplications using the Woodbury identity: and in the diagonal case, via elementwise scaling. This matrix -ree formulation is sufficient because our conjugate gradient solver requires only matrix-vector products with the precision matrix A, = 1-9; I A W WA. By avoiding explicit construction of W and leveraging fast vectorized operations, ensure that each CG solve has negligible memory overhead and runtime comparable to single evaluation of the score network, even in hign-dimensional settings.
F
Additional PSNR Results on Pixel-Domain Tasks
Runtime Comparison on FFHQ 256
We report wall-clock runtimes single NVIDIA P100 (12 GB GPU), batch size 1, and 1,000 reverse-diffusion steps in the pixel domain. All methods use the same pretrained score network for fair comparison. Times are averaged over 50 images.
C-DPS runs within approximately 15% of DPS, and significantly faster than recent methods such as DMPlug. It is slightly slower than DPS due to the added posterior conditioning, however the overall runtime remains comparable given the improved accuracy and the principled posterior formulation.
'These cases are representative of real-world applications such as compressed sensing, deblurring, and MRI reconstruction.
21

---

Table 3: PSNR (dB) on FFHQ 256 x 256. Bold is best, underline is second- best for each column.
Method
Inpaint (Rand)
Inpaint (Box)
Deblur (Gauss)
Deblur (Motion)
SR (4x)
DPS
25.23
22.51
24.25
24.92
25.86
DDRM
9.19
22.26
24.93
26.58
MCG
21.57
19.97
6.72
18.20
ReSample
25.90
23.81
26.33
26.00
25.22
PnP-ADMM
8.41
11.65
24.93
26.55
Score-SDE
13.52
18.51
7.12
17.62
ADMM-TV
22.03
17.81
22.37
23.86
PnP-DM
26.20
22.41
25.71
25.80
27.90
DAPS
28.33
24.07
29.19
29.66
29.07
DMPlug
28.71
28.92
30.02
29.91
30.25
C-DPS (ours)
28.95
28.69
30.13
29.85
30.12
Table 4: Time per image on FFHQ 256
256 (seconds, lower is better).
Method
Time [s]
DPS
130
MCG
142
DAPS
113
PnP-DM
181
DMPlug
550
C-DPS (ours)
152
G Toy dataset
Table Sliced Wasserstein for VE-DDPM.
0.01
L.U
C-DPS
ReSample
DPS
C-DPS
ReSample
DPS
C-DPS
ReSample
DPS
#:9
4.7
1,4
1.6 =09
4.7
1.2
204
13
204
2.1 +10
1.8
+44
1.6
209
1.0 #03
004
12
iaz
3.8
0.7
1a2
2.9 23
0.8
1m
1,8
103
1:08
107
11
5.6
2,4 14x
5.1 218
1,5 1MT
1,6
11
2
1.1 10x
2.8
3.2
1,3 104
3.1
1.0
1,4
1:2
4
102
0,5 143
0,9 10
102
1:6
800
107
5.8
143
1,4
113
110
800 2
1.5
3.1 tu
3.5
1.2
2.7
3.1
1,3
2.0
4.7
800 4
0.5
0,4
0.3
0.4
0.9
0.7
0.9 :044
Table 6: Sliced Wasserstein for the GMM case for the reverse VE SDEs discretized with Euler- Maruyama.
= 0.01
0 0.1
1.0
772
C-DPS
ReSample
DPS
C-DPS
ReSample
DPS
C-DPS
ReSample
DPS
8
1.6 204
1.5 404
5.7 #:2
1.3 14
1.2 =44 5,6
0.8 #03
0.9 4:0
0.9 0.3
8
0.6 103
0.4 1.a
6.2 1:3
1.0 =u
0.5 1ns 6.2 224
0.8 1o0
1.0 163
1.2 14
8
0.4 1o0
0.1 1
0.4 uua
0.1 1m0
8.4
0.7 go1
0.2 161
0.3 4.
80
2.5 #02
2.9 14
9.1 1a
2.1 as
2.1 + 4.7 =1
1.4 #:0
1.8 4:3
1.9 #:00
80
1.2 #04
0.8 +:7
2.2 #:
1.1 us
0.8 2u7 6.0 =21
1,3 +03
1.3 4:5
1.5 0:0
80
0,4 #01
0.1 +:00
0,3 u2
0.1 #01
4,4 =16
0,8 +03
0.4
+.2
0.5 +:03
800
3,2 160
3.2 1m
6.8 1m
2.8 n4
2.8 2n
211
1.4 104
1.3 1.3
1:01
800
1.4 2:3
0.8 6:
7.4 0:
1.2 u3
0.8 am
11
1.3 404
1.1 4:3
2:03
800
0.4 2222
0.6 t.
0.3 02
2:00
14
0.8 0:
4:02
0:02
The generation of this dataset is inspired from [6
As explained earlier in the paper, model (o( mixture of 25 Gaussian distributions. Each of these Gaussian components has mean vector U in Rd defined U (Si, .,,
22

---

for each pair (i,j) where i and take values from the set -,,0,,,, All components have the same variance 1. The unnormalized weight associated with each component is wi,j 1.0. Additionally, we have set the variance of the noise, oz to 10
Recall that the distribution pr (1+e can be expressed as an integral: Pe( = Pajo (20 (o dxo- Since po (o())) is mixture of Gaussian distributions, Pr. (x:,) also a mixture of Gaussians. The means of these Gaussians are given by Va,Ui.j. and each Gaussian has unit variance. By using automatic differentiation libraries, we can efficiently compute the gradient Vx, logp(
We have set the parameters max = 500.0 and min 0.1, and we use 1000 timesteps to discretize the time domain. For given pair of dimensions and a chosen observation noise standard deviation (d, m, σ), the measurement model (y, A) is generated as follows:
Matrix A: First, we sample a random matrix Á from a Gaussian distribution (0 xds Imxa). We then compute its singular value decomposition (SVD), A = USVT. For each pair (i.j) j) in -, 1, 2)2, we draw singular value st from a uniform distribution on the interval [0, 1]. Finally, we construct the matrix A Udiag( (...)).0---222....2.22
Observation vector y: Next, we sample a vector x, from the distribution po- The observation vector y is then obtained by applying the matrix A to and adding Gaussian noise where x is sampled from N(0, Im.)
Once we have drawn both D, Po and (y, A, o). the posterior can be exactly calculated using Bayes formula and gives a mixture of Gaussians with mixture components Ca, and associated weights W..-
C.j == :EE(A y/o² 02 U.z), 2); ã : wNW AU 1.j, oîL. AA
(29)
(30)
where
- (Ia + 03
SW Distance Calculation. To compare the posterior distribution estimated by each algorithm with the target posterior distribution, we use 10t slices for the SW distance and compare 1000 samples of the true posterior distribution.
Table 5 and Table 6 indicate the 95% confidence intervals obtained by considering 20 randomly selected measurement models (A) for each setting (d, m, o).
H More Experiments
H.1 Additional Quantitative Results for Other Noise Levels
Tables 7 and [ report the same set of metrics as Table I in the main paper, but for Gaussian measurement noise levels a 0.07 and 0.03, respectively. Both tables include pixel-domain methods in the upper block and latent-domain methods in the lower block, covering the same five inverse problems: random inpainting, box inpainting, Gaussian deblurring, motion deblurring, and 4x super-resolution.
0.07. When the noise standard deviation is increased from 0.05 to 0.07 (Table 7 all methods experience moderate degradation: FID and LPIPS rise while SSIM falls. Nevertheless, C-DPS remains the top performer in every task on both datasets. The performance gap between C-DPS and the best baseline widens in most cases, indicating that our closed-form posterior update is more robust heavier measurement noise than the projection- or likelihood-baad based alternatives.
= 0.03. With milder noise (a 0.03, Table 8). absolute scores improve across the board, but C-DPS still delivers the best or second-best results all settings. The advantage most pronounced for inpainting and Gaussian deblurring, where C-DPS attains the lowest FID and LPIPS and the highest SSIM. These trends confirm that the coupled posterior formulation benefits both high- and low-noise regimes.
Overall observation. Across a E (0.03, 0.05, 0.07). C-DPS consistently outperforms state-of-the art baselines, showing graceful performance degradation noise increases and the largest gains under
23

---

Table 7: Quantitative results for Gaussian noise level 0.07 on the 1k validation sets of FFHQ 256 x 256 and ImageNet 256 x 256. Bold and underline indicate the best and second-best results, respectively.
Pixel-Domain Methods
Inpaint (Random)
Inpuint (Bex)
Deblur (Gaussiun)
Deblur (Motion)
SR (4x)
Datuset
Method
FID
LPIPS|
SSIM1
FID
LPIPS|
SSIM
FID,
LPIPS, SSIM
FID,
LPIPS)
SSIM
FID
LPIPS|
SSIM
DPS
23.1
0.228
0.833
36.9
0.183
0.858
48.4
0.277
0.792
44.3
0.262
0.842
42.8
0.232
0.836
IIGDM
23.3
0.238
38.9
0.194
44.4
0.264
36.3
0.243
0.872
38.3
0.218
0.836
DDRM
76.2
0.619
0.294
46.3
0.227
0.843
80.6
0.352
0.748
66.8
0.318
0.814
MCG
32.4
0.308
0.727
44.0
0.330
0.677
110.3
0.360
0.048
93.7
0.544
0.535
ILVR
28.5
0.251
0.648
40.3
0.190
0.835
58.2
0,318
0.764
52.3
0.273
0.827
ReSample
23.2
0.217
0.827
36.9
0.173
0,845
40.8
0.271
0,806
34.3
0.238
0.874
33.4
0.224
0.835
PnP PdP.ADMM
135.9
0.723
168.3
0.430
0.621
99.8
0,470
0.791
72.8
0.379
0.848
Score- SDE
84,9
0.639
0.418
66.4
0.355
0.659
121.4
0.431
0.105
103.2
0.593
0.596
ADMM-TV
196.8
0:486
74.9
0.346
202.1
0.532
0,780
121.8
0.455
0.781
C-DPS
22.1
0.209
0.862
28.9
0.150
0.859
35.4
0.259
0.814
30.3
0.236
0.904
31.0
0.214
0.841
C-DPS vs. Best
-1.1
-0,008
0.035
-8.0
-0.023
-0.014
-5.4
-0.012
0.008
-4.0
-0.002
0.030
-2.4
-0.010
-0.006
DPS
39.8
0.327
0.713
42.9
0.280
0,768
67.6
0,472
0.683
60.8
0.414
0.614
55.4
0.361
0.756
H1GMM
46.4
0,382
0.678
46.7
0.307
0.729
65.2
0,450
0.695
59.3
0.398
0.655
59,3
0.374
0.741
DDRM
124.0
0.704
0.384
50.2
0.267
0.787
68.9
0,455
0.684
65,7
0.367
0.768
MGG
42.9
0.441
0.525
44.4
0.344
101.6
0.579
0,421
155.6
0.219
ImageNet
ILVR
42.0
0.398
0.635
41.5
0.298
0,708
77.6
0.451
0.639
103.1
0.567
0.472
ReSample
37.2
0.311
0.708
42.3
0.278
0.771
66.1
0.468
0.687
60.1
0.399
0.618
52.7
0,359
0.751
PaP- MDMM
124.9
0.713
0.285
85.3
0,388
0.636
109.3
0.548
0.641
104.1
0.459
0.736
Score SDE
138.4
0.695
0.494
59.4
0.374
0,591
131.8
0.706
0.419
186.
0,741
0.231
ADMM-TV
205.1
0.540
0.654
95.1
0.343
170.3
0.618
0.611
143.1
0.554
0.654
C-DPS
35.4
0.236
0.729
36.4
0.254
0.781
60.2
0.417
0.691
55.1
0.372
0.630
50.1
0,331
0.772
C-DPS Best
-188
-0.075
0.016
5.1
.0.013
40,006
5.0
-0.033
4.004
4.2
-0,027
.0,025
-2.6
-0.028
0.004
Jatent-Domain Methods
Datasel
Method
Inpaint (Random)
Inpaint (Bex)
Deblur (Gaussian)
Deblur (Motion)
(4x)
FIDJ
I [.PISS] SSIM
FID.
IPIPS.] SSIM
FID.
I.PIPS.
SSIM
FID)
1 [.PISS)
SSIM
FIDJ
L.PSS].
SSIM
PSLD
$2.6
0.242
0.786
47.9
0.177
0.791
97.3
0.338
0.600
104.9
0.360
0.648
81.4
0.308
0.618
FRIQ
ReSample
44.3
0.157
0.718
59.2
0,201
0,722
79.5
0.276
0.686
49.6
0.214
0.800
102.2
0.423
0.562
LC-DPS
40.7
0.153
46.2
0.158
72.8
0.256
51.8
0.294
0.793
64.5
0.220
LC-DPS vs. Best
-3.6
-0.004
0.006
-1.7
-0.019
0.008
-6.8
-0.020
0.044
2.2
0.080
-0.007
-16.9
-0.088
0,146
ImageNet
PSLD
91.5
0.362
0.764
160.1
0.489
0.660
100.8
0.414
0.663
137.5
0.549
0.568
106.2
0.384
0.671
ReSample
64.2
0.158
0.732
139.9
0.285
71.4
0.276
0.684
73.0
0.246
0.714
123.8
0.390
0.558
LC-DPS
64.7
0.154
0.764
125.8
0,274
0.690
23.8
0.265
0.700
69.3
0,233
0.728
96.7
0.312
0.653
LC-DPS vs. Best
0.5
-0,004
0.032
-14.1
-0.011
0.030
2.4
-0.011
0,016
-3.7
-0.013
0.014
-9.5
-0.072
-0,018
challenging noise conditions. The latent variant LC-DPS exhibits the same behaviour, confirming that the proposed framework is effective in both pixel and latent domains.
H.2 More Qualitative Results
In this section, we present additional qualitative results on the FFHQ dataset across various inverse problem settings. These results serve to further demonstrate the flexibility and effectiveness C-DPS in handling diverse measurement operators beyond those shown in the main text.
Figure sshows reconstructions for the Gaussian deblurring task, where C-DPS successfully restores high-frequency details that are often lost in baseline methods. Figure presents results for motion deblurring, highlighting C-DPS's ability to recover sharp facial features under structured blur.
In Figure we evaluate C-DPS on an 8x super-resolution task. Despite the aggressive downsampling, method reconstructs globally consistent and perceptually plausible images. Figures 9 and I0 display results on random inpainting and box inpainting, respectively. In both cases, C-DPS generates semantically coherent completions even under extreme occlusions.
These qualitative results further support the claims made in the main paper, showing that C-DPS generalizes well across wide range of inverse problems with minimal task-specific tuning.
I Limitations and Failure Cases
While C-DPS demonstrates strong performance across a variety of inverse problems, we observe that it can struggle in certain edge cases. For example, when the measurement matrix A is nearly rank-deficient or the noise level is extremely high, the propagated measurement information }
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Table 8: Quantitative results for Gaussian noise level 0.03 the 1k validation sets of FFHQ 256 x 256 and ImageNet 256 x 256. Bold and underline indicate the best and second-best results, respectively.
Pixel-Domain Methods
Inpaint (Random)
Inpuint (Bex)
Deblur (Gaussiun)
Deblur (Motion)
SK (4x)
Datuset
Method
FID
LPIPS|
SSIM
FID
LPIPS|
SSIM
FID.
LPIPS,
SSIM
FID,
LPIPS)
SSIM
FID
LPIPS|
SSIM
DPS
19.4
0.196
0.872
29.8
0.151
0.891
36.2
0.230
0.842
32.9
0.216
0.882
32.6
0.192
0.873
TIGDM
0.204
31.1
0.160
33.5
0.218
27.6
0.198
0.904
0.875
DDRM
61.3
0,516
0.363
36.7
0.175
0.895
63.2
0.291
0.807
52.3
0.268
0.861
MCG
24.9
0.249
0.792
32.4
0.266
0.745
86.9
0.301
0.069
74.2
0.466
0.604
ILVR
22.3
0.208
0.709
31.9
0.157
0.874
44.6
0.257
0.816
39.8
0.217
0.865
ReSample
19.2
0.181
0.868
29.7
0.145
0.886
30.9
0.225
0.836
26.1
0.203
0.906
27.2
0.186
0.872
PdP.ADMM
110.7
0.622
136.2
0.363
82.1
0.392
59.4
0.889
Score- SDE
61.3
0.499
0.530
49.2
0,286
0.746
101.1
0.347
0.141
85.7
0.471
0.658
ADMM-TV
155.7
0:406
0.807
59.6
0.283
160.5
0,454
0.832
95.4
0.385
0.835
C-DPS
18.3
0.174
0.898
23.7
0.118
0.894
27.6
0.209
0.856
23.5
0.189
0.933
24.9
0.171
0.880
C-DPS vs. Best
-0.9
-0.007
0.030
-6.0
-0.027
0.008
-3.3
-0.016
0.020
-2.6
-0.014
0.027
-2.3
-0.015
0.008
DPS
31.4
0.257
0.767
33.9
0.223
0.821
54.9
0.385
0.726
49.0
0.336
0.676
44.8
0.289
0.807
H1GDM
37.0
0.303
0.736
36.5
0.247
0.785
52.3
0.367
0.736
47.3
0.322
0.709
48.1
0.299
0.790
DDRM
94.6
0.548
0.438
40.1
0.208
0.842
56.8
0.373
0.729
51.6
0,305
0.817
MGG
0.361
34.1
0.290
0.649
82.2
0.486
124.0
0,550
0.271
ImageNet
ILVR
32.9
0.324
0.684
33.1
0,239
0.753
61.4
0.376
0.690
82.7
0.454
0,533
ReSample
29.8
0.245
0.758
34.2
0.220
0.828
54.6
0.389
0.731
48.5
0.318
0.680
43.5
0.290
0.803
PaP-ADMI MPMM
94.4
0.556
0.346
71.2
0.335
0.700
87.6
0.461
0.699
84.7
0.373
0.787
Score SDE
104.2
0.573
0,543
48.1
0.304
0.649
105.7
0.596
0.451
149.8
0.625
0.294
ADMM-TV
151.3
0.458
72.3
0,287
133.2
0.502
0.671
118.6
0.704
C-DPS
28.6
0.178
0,779
29.9
0.205
0.835
50.1
0.345
0.735
45.6
0.300
0.688
41.2
0.263
0,823
C-DPS Best
-1.2
-0.067
0.012
.0.015
0.007
-2.2
-0.022
4.001
-1.7
-0.018
.0.021
-2.3
-0.027
0.006
Tatent-Domail Methods
Datasel
Method
Inpaint (Random)
Inpaint (Bex)
Deblur (Gaussian)
Deblur (Motion)
(4x)
FIDJ
I [.PISS] SSIM
FID.
IPIPS.] SSIM
FID.
I.PISS.
SSIM
FID)
1, [PIPS)
SSIM)
FIDJ
LPPIS].
SSIM
PSLD
43.5
0.203
0.830
39.1
0.142
0.833
78.3
0.291
0.652
85.5
0.310
0.702
66.8
0.262
0.669
FRIQ
ReSample
36.9
0.126
0.776
49.1
0,166
0,779
59.4
0.234
0.736
41.1
0.189
0.845
83.6
0.344
0.619
LC-DPS
34,3
0.123
37.6
0.130
54.2
0.215
42.9
0.258
0.837
5M.7
LC-DPS va. Best
-2.6
-0.003
0.013
-1.5
-0.012
0.008
-5.2
-0.019
0.041
1.8
0.069
-0.008
-8.1
-0.086
0,143
ImageNet
PSLD
70.8
0.284
0813
124.4
0.415
0.720
86.1
0.352
0.702
112.9
0.446
0.612
92.2
0.326
0.714
ReSample
49.1
0.118
111.9
0.223
57.3
0.225
61.5
0.200
0.745
107.2
0.332
0.609
LC-DPS
49.5
0.115
0.815
101.4
0.215
0.740
59.1
0.216
0.740
58.4
0.188
0.755
82,9
0.252
0.201
LC-DPS vs. Best
0.4
-0,003
0.033
-10.5
-0.008
0.020
1.8
-0.009
0,016
-3.1
-0.012
0.010
-9.3
-0.074
-0,015
may not sufficiently constrain the posterior, resulting in oversmoothing loss of fine details in the reconstruction.
This behavior is not unique to C-DPS. Prior methods such as DPS and ReSample also degrade under similar conditions, as they rely on learned priors and approximate measurement integration that become less reliable when the inverse problem is highly ill-posed. In our case, the Gaussian approximation to the posterior-while principled and efficient-can be limiting when the true posterior is strongly multi-modal or non-Gaussian.
These limitations arise from current design choices that favor tractability and generality. Future extensions could incorporate adaptive diffusion schedules, refined measurement dynamics, or non- Gaussian approximations to further improve performance these extreme regimes.
25

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Measurement
Ground truth
C-DPS (aurs)
DPS
Re- Re-Sample
Figure 6: Qualitative results on FFHQ dataset, Gaussian deblurring.
26

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Measurement
Ground truth
DPPSsouss
DPS
Re- Sample
Figure Qualitative results on FFHQ dataset, motion deblurring.
27

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Measurement
Ground truth
C-DPS (Ours)
DPS
Re- Re-Sample
Figure 8: Qualitative results on FFHQ dataset, super-resolution task (x 8).
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Measurement
Ground truth
C-DPS (ours)
DPS
Re- Re-Sample
Figure 9: Qualitative results on FFHQ dataset, inpaiting task (random).
29

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Measurement
Ground truth
C-DPS (ours)
DPS
Re- Re-Sample
Figure 10: Qualitative results on FFHQ dataset, inpaiting task (box).
30

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Supplementary Materials (Evaluation on Medical Imaging Inverse Problems)
In this section, extend our study by applying the proposed diffusion-based approach to inverse problems in the medical imaging domain. Medical images often contain complex structures and are typically acquired under constraints that make problems such as reconstruction, denoising. or inpainting particularly challenging. By evaluating our method on representative medical imaging tasks, we aim demonstrate its effectiveness and generalization ability in real-world scenarios where accurate recovery of fine details is critical.
Datasets and Preprocessing.
We evaluate proposed method, C-DPS, three representative medical imaging tasks using public datasets: undersampled MRI, sparse-view CT, and super-resolution. For MRI reconstruction, we use the fastMRI knee dataset [51]. Following standard preprocessing [26], we crop the raw k-space data to 320 x 320 resolution and reconstruct single-coil images using minimum variance unbiased estimator. Undersampling is simulated using 1D Gaussian and Uniform sampling masks with acceleration rates (ACR) of 4 and 8.
two-dimensional CT slices of size 320 320 are extracted from volumetric scans. Sinograms are generated using parallel-beam geometry with either 23 or 10 projections evenly spaced over 180 degrees to simulate sparse-view acquisition.
For sparse-view CT, utilize the LIDC dataset [52].
The super-resolution task is evaluated on the fastMRI brain dataset [51]. We downsample 2D full-resolution images generate low-resolution inputs for 2 x 2 and 4 upscaling, We select approximately 63% of the slices labeled as reconstruction.se pss, yielding a total of 34,698 training samples.
Baselines and Evaluation Metrics
We compare our method, C-DPS, with strong training-free diffusion-based baselines including DPS 77. Score-MRI [2], DDS (46), and ScoreMed (26), depending the task. All methods are implemented with their publicly available code and evaluated under consistent experimental conditions.
We report results using standard image quality metrics: Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM). Each task is evaluated on test set of 1,000 images. For CT experiments, we follow prior work and the likelihood step size Ç where applicable. In all experiments, our method outperforms the baselines across all configurations in terms of both PSNR and SSIM, as shown in Tables and T0
Architecture and Sampling Settings
All models are based on the ADM architecture (28). without classifier-free guidance or dropout. Separate diffusion priors are trained for each task. For sampling, we use 100 timesteps for MRI and super-resolution, and 350 timesteps for sparse-view CT.
Our method introduces measurement-conssiee tune the likelihood step size Ç and refinement weight within a restricted search space, Ç E [0, 2], 7 [0, 4.5), and fix X 10-3 3 for the outer optimization. No extensive hyperparameter tuning is required to achieve strong performance.
update through bi-level guidance scheme. We
All experiments conducted on NVIDIA P100 GPU with 12 GB of memory.
Results
We assess the effectiveness of our method, C-DPS, across three key inverse problems: MRI recon- struction, CT reconstruction, and super-resolution. In all cases, C-DPS consistently outperforms prior training-free diffusion-based baselines, demonstrating both robustness and accuracy across varying acquisition settings.
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MRI Reconstruction. Table reports results on the fastMRI knee dataset under Uniform D and G masks at acceleration rates of 4x and 8x. Our method achieves the highest PSNR and SSIM across all configurations. Notably, under more aggressive undersampling (8x). C-DPS shows significant improvements over both DPS and DDS, highlighting resilience in challenging reconstruction scenarios. The gains in SSIM, particularly under the Gaussian mask, indicate improved structural fidelity and reduced aliasing artifacts.
Sparse- View CT Reconstruction. As shown in Table 11 C-DPS achieves state-of thrr---rtt results on the LIDC dataset with both 23 and 10 projections. outperforms ScoreMed and BGDM by notable margin in PSNR and SSIM, despite using the same number of diffusion steps (350). The improvement especially pronounced under the extremely sparse 10-view setting, suggesting that C-DPS maintains strong data consistency even under severe measurement limitations.
Duper-Resolution On the fastMRI brain dataset, Table 10 demonstrates that C-DPS achieves the highest reconstruction quality for both 2x2 and 4x4 4 super-resolution tasks. Compared to DPS and DDS, our method produces sharper images with higher structural similarity, particularly in the more difficult 44 setting. These results affirm the generalization capability of C-DPS across different types of inverse problems.
Overall Observations. Across all tasks, C-DPS improves upon existing baselines without requiring task-specific training or extensive parameter tuning. The results highlight the advantage of incorpo- rating measurement-awan updates through our bi-level guidance framework, which enhances both reconstruction accuracy and robustness under limited data.
Table 9: Quantitative results for the fastMRI knee dataset across different sampling masks and acceleration rates.
Method
Uniform
UniformID ACR
Gaussian
Gaussian ACR
PSNR
SSIM
PSNRT
SSIM
PSNR
SSIM
PSNR1
SSIM
DPS
32.40=2.19 19
0.84340.063
31.07=2.32
0.804:4.073
34.95=1.90
0.882+00063
33.72+1.97
0.833=0.07
Score-MRI
M
31.95+1445 45
0.812-0.036
27.97-2.03
0.738-+0.053
33.96-1 1.27
0.858++0.028
30.82+1.377
0.762--00034
32.09
37.61
0.87410.052
C-DPS (ours)
35.63=2.47
0.877-1.007
23.33-2666
0.842:10.077
38.05-2.43
0.918
36.52:11.88
0.89210.051
Table 10: Super-resolution results the fastMRI brain dataset.
Method
2x2 SR
4x4 SR
PSNR
SSIM1
PSNR
SSIM1
DPS 7
35.44
0.931=0.027
30.29-12.84
0.854:10.034
DDS 46
36.15 36.42-1..02
0.943=0.023
32.0442.92
0,869:10.031
C-DPS (ours)
0.951=0.044
32.48-2.322
0.88910.051
Table 11: Quantitative results of sparse-view CT reconstruction on the LIDC dataset using 350 NFEs.
Method
23 Projections
10 Projections
PSNRT
SSIM
PSNR1
SSIM
ScoreMed (26
35.24+:2.71 35.92-t2.33
0.905+000046 0.925-10.046
29.52++2.33 30.59-22.21
0.823+:0.061
C-DPS (ours)
0.842-0.008
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Reviews:
[{"review_id": "VzwZRcz6Rt", "reviewer": "Reviewer_a15Q", "summary": "The paper addresses the intractable likelihood problem in diffusion posterior sampling by introducing a coupled noisy measurement trajectory. Specifically, during the sampling process, the method constructs a coupled measurement sequence $\\\\{\\boldsymbol{y}\\_t\\\\}\\_{t=0}^T$ following the standard diffusion forward process. Subsequently, it samples $\\boldsymbol{x}_t$ from the conditional posterior $p(\\boldsymbol{x}\\_{t-1}\\mid \\boldsymbol{x}_t, \\boldsymbol{y}\\_{t-1})$ iteratively in reverse time order. Utilizing Tweedie's formula along with mild assumptions about the score function, the conditional posterior $p(\\boldsymbol{x}\\_{t-1}\\mid \\boldsymbol{x}_t, \\boldsymbol{y}\\_{t-1})$ can be analytically derived as a Gaussian distribution. The proposed approach demonstrates superior performance on various image restoration tasks and a synthetic benchmark dataset designed to provide ground-truth posterior distributions.", "questions": "Please refer to the weaknesses part for my main questions. Here's several additional questions.\n\n1. **Samping time**. The pre-whitened conjugate gradient is only practical when forward function is efficient. If computing the forward function is costly, the sampling step becomes time-consuming. Could the authors provide some insights or strategies for addressing this limitation?\n\n2. **Sampling diversity.** In 92% random inpatining tasks, the measurement signal is extremely sparse, which means the reconstructions can exhibit diverse appearance yet still fit the measurement equally well. Does the author observe this property, or do the reconstructions always consistently resemble the ground truth, as illustrated in Figure 2?", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "> Strengths\n\nThe paper is well motivated, clearly written, and easy to follow. The key strengths of this paper are summarized as follows:\n\n1. **Combination of Tweedie's Approximation with Noisy Measurement.** The use of Tweedie's approximation $\\boldsymbol{x}_0(\\boldsymbol{x}_t)$ can lead to blurred results for larger time steps $t$; however, the method compensates for this issue through the noisy measurement $\\boldsymbol{y}\\_{t}$ as expressed in Equation (10).\n\n2. **Superior Empirical Performance.** The proposed method consistently outperforms existing techniques on standard image restoration benchmarks and provides detailed analyses supported by synthetic experiments.\n\n3. **Clear Visualization of Measurement Error Evaluation.** Figure 4 effectively highlights the method's capability to closely track measurement signals throughout the sampling process, demonstrating clear empirical superiority.\n\n> Weaknesses\n\nHowever, the paper also has a few weaknesses:\n\n1. **Similar Ideas in the Literature.** The idea of establishing a diffusion process in the measurement space is not new. It was first introduced in Appendix I.4 of [1] and then investigated in [2] for medical imaging problems. The authors should cite [2] and compare the proposed method with the method in [2].\n1. **Restriction to Linear Inverse Problems.** The approximated closed-form solution for the distribution $p(\\boldsymbol{x}\\_{t-1}\\mid \\boldsymbol{x}_t, \\boldsymbol{y}\\_{t-1})$ relies on a linear forward model. It would be valuable to explore or discuss potential extensions to nonlinear inverse problems. Does the author have any insight on this?\n2. **Limited Baseline Comparisons on Measurement Fidelity.** While the paper demonstrates lower measurement errors and better reconstruction results compared to selected baselines, it lacks comparisons with more recent methods with improved measurement fidelity, such as DDNM [3], PnP-DM [4], DCDP [5], DAPS [6], SITCOM [7], and DMPlug [8]. The author should compare with some of them for better accessing the performance.\n3. **Absence of Common Evaluation Metrics.** The paper does not report PSNR values, which is a standard metric widely used and reported by many baseline studies. The authors should include it as well if possible?\n\n[1] Song et al., Score-Based Generative Modeling through Stochastic Differential Equations\n\n[2] Song et al., Solving Inverse Problems in Medical Imaging with Score-Based Generative Models\n\n[3] Wang et al., Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Model\n\n[4] Wu et al., Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors\n\n[5] Li et al., Decoupled Data Consistency with Diffusion Purification for Image Restoration\n\n[6] Zhang et al., Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing\n\n[7] Akhouri et al., SITCOM: Step-wise Triple-Consistent Diffusion Sampling for Inverse Problems\n\n[8] Wang et al., DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models", "was_revised": true, "final_justification": "After several rounds of discussion with the authors, my concerns about the paper have mostly been addressed. Therefore, I have raised my rating to 4. The reason why I don't think that the paper warrants a higher score is that similar ideas have been investigated previously and the method's improvements are reflected more by the incremental metric values than by the scope of applicability. Nevertheless, it is a solid work that deserves to be accepted to NeurIPS.", "rebuttal": "Thank you for the positive and constructive feedback. We appreciate your recognition of our method’s motivation, empirical performance, and visual clarity. Your comments are encouraging and helpful for refining the final version.\n\n------------------\n# Weakness 1 (Similar Ideas in the Literature)\nThank you for pointing out these related works.\n\n**Regarding [1]:** Appendix I.4 in [1] does present a conceptual framework for solving inverse problems via diffusion in the measurement space. However, it remains heuristic and lacks concrete implementation details. Specifically, it assumes access to \\$p(y\\_t \\mid y)\\$ at each reverse step, but does not explain how this term is computed or estimated in practice. Furthermore, it approximates \\$\\nabla\\_{x\\_t} \\log p(y \\mid x\\_t)\\$ using a residual term, again without specifying how this is derived. The method does not provide a closed-form for the posterior \\$p(x\\_{t-1} \\mid x\\_t, y\\_{t-1})\\$, which is critical for practical inference. Overall, the work offers useful intuition but does not present a tractable or reproducible algorithm.\n\n**Regarding [2]:** While [2] introduces a diffusion process in the measurement space similar in spirit to ours, its approach and implementation differ significantly. Their method does not derive an analytic posterior. Instead, each reverse step involves solving an optimization problem that trades off between proximity to the unconditional denoised sample \\$x\\_t\\$ and the perturbed observation \\$y\\_t\\$, guided by a manually tuned hyperparameter. In contrast, our method directly constructs a closed-form posterior that eliminates the need for such heuristic balancing.\n\n**Relative strength of [2]:** The work MCG (Improving Diffusion Models for Inverse Problems using Manifold Constraints) has shown improved performance over [2], and we include MCG as a baseline in Table 1. Our method consistently outperforms MCG across all tasks, indicating that our approach improves upon both [2] and its successors.\n\nWe will cite [1] and [2] in the revised version and clarify these conceptual and algorithmic differences.\n\n------------------\n# Weakness 2 (Restriction to Linear Inverse Problems)\n\nWe appreciate the reviewer’s suggestion and agree that extending C‑DPS beyond linear forward operators is an important research direction. Below we clarify (i) why the linear‐model focus already covers a broad set of high‑impact applications, (ii) how the coupled formulation of C‑DPS can be generalized to nonlinear settings.\n\n(i)\tMany practical imaging systems admit linear forward maps after logarithmic or frequency‑domain transformations: e.g.\\ MRI, parallel‑beam CT, PET, compressive sensing, deblurring, inpainting and super‑resolution. The proposed theory therefore already addresses a large class of inverse problems of immediate practical relevance. This is why some recent papers only focus on linear inverse problems.\n\n(ii)\tLet the measurement model be $ \\mathbf{y}=g(\\mathbf{x}_0)+\\mathbf{n}$ with a differentiable $g:\\mathbb{R}^d \\to \\mathbb{R}^m$.  The key observation is that the coupled‑process idea does not depend on linearity; only the Gaussian closed form of Eq. (16) does.  A well‑established approximation, Local (Gauss–Newton / EKF) linearization,  allows us to retain an analytic (or nearly analytic) posterior within the same C‑DPS pipeline:\n  At step $t$ we linearise $g$ around the current iterate $\\mathbf{x}_t$:\n$g(\\mathbf{x}) \\approx g(\\mathbf{x}_t)+J_t(\\mathbf{x}-\\mathbf{x}_t)$  \n\nwhere $J_t = \\nabla g \\bigr|_{\\mathbf{x}=\\mathbf{x}_t} $\n\n\n Substituting $J_t$ for $A$ in Eq. (16) yields a Gaussian update identical in form to C‑DPS, but with the Jacobian refreshed every step.  Analytically, this corresponds to an extended‐Kalman smoothing view; algorithmically it increases cost by one Jacobian–vector product per iteration, however it is supported by automatic differentiation in modern frameworks.\nWe have implemented the this variant on two nonlinear benchmarks, Phase Retrieval and Nonlinear Deblur, on FFHQ dataset:\n\n### **Phase Retrieval**\n\n| Method     | FID ↓ | LPIPS ↓ | SSIM ↑ | PSNR ↑ |\n|-|-|-|-|-|\n| DAPS       |  42.71 |  0.139 | 0.851 | 30.63 |\n| DPS          |  104.5|  0.410 | 0.441 | 17.64 |\n| RED-diff   |  167.4|  0.596| 0.398 | 15.60 |\n| PnP-DM     |  99.4 |  0.335  | 0.581| 19.69  |\n| DMPlug     |**41.21**|**0.124**|0.894|31.25|\n|C-DPS|41.32|0.129|**0.903**|**31.91**|\n\n### **Nonlinear Deblur**\n\n| Method     | FID ↓ | LPIPS ↓ | SSIM ↑ | PSNR ↑ |\n|-|-|-|-|-|\n| DAPS       |  49.38|  0.155| 0.783| 28.29|\n| DPS        |  91.31|  0.278| 0.623| 23.39|\n| RED-diff   | 43.84| 0.160| 0.795| 30.86|\n| PnP-DM     |  68.96 |  0193  | 0.742  | 27.81  |\n| DMPlug     |  **47.28** |  0.135  | 0.792  | 29.55  |\n|C-DPS|  47.52 | **0.130**|**0.802**|**30.27** |\n\nThese early results indicate that the locally linear C-DPS retains the performance edge of our linear version while remaining computationally feasible (< 15 ms Jacobian product vs ≈ 150 ms score pass).\n\nNote that in the above table, we also included the new benchmarks you suggested later in your comments. Also, note that although the results of DMPlug is close to ours, yet their method is roughly 5 times slower than ours.\n\n------------------\n# Weakness 3 (Limited Baseline)\n\nThank you for the suggestion regarding additional baselines. \n\nFirst, regarding SITCOM [7], we note that it is a rejected submission to ICLR 2025 with no citations. Moreover, public reviews point to critical technical errors that undermine its reliability as a baseline. For this reason, we do not believe it currently represents a credible benchmark. \n\nSecond, as acknowledged in the DAPS [6] paper itself, DAPS outperforms DDNM [3] and DCDP [5]. \n\nHence, to address your core concern, we have compared our method (C-DPS) directly against PnP-DM [4], DAPS [6], and DMPlug [8]. The results are summarized in the following table in this format FID↓ / LPIPS↓ / SSIM↑ for each task. \n \n| Method| Inpaint (Rand)| Inpaint (Box)| Deblur (Gauss)| Deblur (Motion)| SR (×4)|\n|-|-|-|-|-|-|\n|PnP-DM [4]| 21.15 / 0.208 / 0.858 | 32.21 / 0.155 / 0.877| 41.92 / 0.251 / 0.816 | 37.21 / 0.233 / 0.871| 36.21 / 0.210 / 0.859|\n| DAPS [6]    | 20.77 / 0.201 / 0.869 | 29.44 / 0.144 / 0.882| 35.84 / 0.242 / 0.830| 30.26 / 0.215 / 0.911| 30.15 / 0.202 / 0.854|\n|DMPlug [8]| **20.12** / 0.197 / 0.877| 27.12 /0.140/ **0.888**| 32.44 / **0.230**  / 0.830| 27.55  / **0.210** / **0.925** | 28.55  / 0.199 / **0.862** |\n|C-DPS (ours) | 20.14 **/ 0.195 / 0.881** | **26.33 / 0.132** / 0.871 | **32.24** / 0.238 / **0.832**|**27.29** / 0.217 / 0.921 | **28.41 / 0.196** / 0.855 |\n\nWe emphasize that while DMPlug achieves competitive accuracy, its runtime is approximately 5× slower than our method, making it significantly less practical for real-world applications.\n\n------------------\n# Weakness 4 (PSNR)\n\nWe initially focused on LPIPS, FID, and SSIM because they correlate better with human perception in high-dimensional generative tasks. PSNR can rank methods differently when perceptual scores agree, so we avoided over‑loading the tables. We have now computed PSNR for all tasks in Table 1 (including the new baselines that you suggested above). A summary for the FFHQ dataset is given below. Full tables for ImageNet and for the ablations will be added to Appendix F.\n\n| Method|Inpaint (Rand)|Inpaint (Box)|Deblur (Gauss)|Deblur (Motion)| SR (×4)|\n|-|-|-|-|-|-|\n|DPS| 25.7| 26.8| 24.1| 25.4 | 23.7|\n|ΠGDM| 25.1| 26.1| 24.6| 26.3| 24.2|\n|DDRM| 18.3| 24.4| 22.8|-| 21.7|\n|MCG| 22.9| 23.1| 20.2|-| 19.8|\n|ILVR| 24.1| 25.5| 23.7|-| 22.9|\n|ReSample| 25.9| 27.3| 24.7| 26.0| 24.2|\n|PnP-ADMM| 17.0| 19.1| 23.9|-| 22.4|\n|Score-SDE| 20.1| 21.6| 19.4|-| 18.8|\n|ADMM-TV| 17.5| 22.8| 22.3|-| 21.2|\n|PnP-DM[4]| 26.2| 27.4| 24.4| 25.8| 23.9|\n|DAPS [6] | 27.0| 28.1| 24.8| 26.8| 24.6|\n|DMPlug [8] | 27.2| **28.9**| 25.9 | 27.8| **26.0**|\n|C-DPS (ours)| **27.4**| 28.6| **26.1**| **27.9**|25.5|\n\n--------------------\n# Question 1 (Sampling time) \n\nWe thank the reviewer for raising this important point. The runtime of the pre-whitened conjugate gradient (PW-CG) step indeed depends on the efficiency of applying the forward operator $A$ and its adjoint $A^{\\top}$. \nIn our FFHQ blur example, a UNet‑based score pass (256 × 256) takes **≈ 140 ms** on a P100 GPU, whereas one 61 × 61 FFT‑based convolution (‑‑► $A$ or $A^{\\top}$) costs only **≈ 4 ms**.\nWith a typical **5 CG iterations**, PW‑CG adds < 40 ms – still < 25 % of the total per‑step runtime.\nThis remains true for most *linear* imaging operators used in practice (deblurring, inpainting, SR, MRI, CT) because they admit FFT, NUFFT, or ray‑driven GPU kernels.\n\nFor settings where $A$ is computationally expensive (e.g., ray-based CT or custom forward models), we suggest using **warm-starting** CG from the previous iterate and using early stopping based on residual tolerance significantly reduces iterations (often to 1–2) without affecting reconstruction quality. In addition, a better preconditioning, such as using approximate inverse or circulant structures, further reduces solve time.\n\n--------------------\n# Question 2 (Sampling diversity) \n\nYes, repeated runs with different noise seeds do yield distinct inpaintings.\n\n*Sanity check:* For 100 FFHQ test images with 92 % masking, we drew five independent C-DPS samples per image. The “mean pair-wise” LPIPS between the five reconstructions was 0.11 ± 0.03, confirming diversity.\n\nBut why Figure 2 looks “deterministic”? \n\nThe global structure (pose, hair outline, dominant colors) is tightly constrained by the observed pixels, so all C‑DPS samples share those attributes and therefore still “look like” the original subject. The variations appear mainly in **high‑frequency details**. Across five independent C‑DPS samples per test image the best PSNR is within 0.2 dB of the single‑sample PSNR reported in Table 1, while LPIPS and FID remain better than all baselines. So diversity does not come at the cost of average quality."}, {"review_id": "N4xTMH1N0d", "reviewer": "Reviewer_47R6", "summary": "The paper aims to eliminate projection and likelihood-approximation steps during diffusion sampling for inverse problem solving. It introduces an auxiliary Markov chain for the measurement $y$, defining the marginal distribution at time $t$ as $p(y_t|x_t)$l this contrasts with prior works that relies on $p(y_0|x_t)$.\n\nTo sample $y_t$ from $p(y_t|x_t)$, the paper derives a closed-form update that is coupled with the diffusion update for $x_t$. The implementation assumes that the score function evaluated at $x_{t-1}$ can be replaced by the score function at $x_t$.\n\nExperiments show that the proposed method improves inverse-problem performance relative to the chosen baselines.", "questions": "- Please provide theoretical clarification on, especially, equations (9), (14), and (15). If I mis-understood something, please point out it.\n- Please provide comparison result with more recent works, provided in the weakness section.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The paper proposes a novel idea on introducing auxiliary Markov chain for the measurement.\nHowever, the paper contains logical flaws which limits its quality and significance. Also, experiments have done with old baselines and the paper omits recent works on inverse problem solving beyond projection-based or likelihood approximation-based solvers, which also limits the quality and significance. Please see the below list for details.\n\n**Strength**\n- The paper provides all methodological details.\n- Although most baselines are outdated, the experiments cover a wide range of inverse problems across two datasets.\n\n**Weakness**\n- Technical error in Eq (9): the term $p(x_{t-1})$ is omitted without any reason. In addition, the factorization $p(x_t,y_{t-1}|x_{t-1}) = p(x_t|x_{t-1})p(y_{t-1}|x_{t-1})$ assumes independence between $y_{t-1}$ and $x_t$ (not described in the paper), which conflicts with Eq (15), where the score at $x_{t-1}$ is simple replaced by the score at $x_t$.\n- No sufficient discussion or theoretical validation on the effect of coupling stochastic update of $y_t$ with deterministic update of $x_t$.\n- Assumption in Eq (14) limits efficiency, as the resulting error becomes significant when fast samples are used - even with DDIM.\n- The appendix contains no qualitative examples for the medical-image domain, and also provided qualitative results in the main body is limited, making visual assessment difficult.\n- Important non-projection and non-likelihood methods [1-4] are missing from the introduction and related works.\n- The evaluation omits more recent approaches such as [1-6]\n\n\n**Referece**\n\n [1] A Variational Perspective on Solving Inverse Problems with Diffusion Models, arxiv 2023\n \n [2] Repulsive Latent Score Distillation for Solving Inverse Problems, ICLR 2025\n \n [3] Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing, CVPR 2025\n \n [4] Inverse Problem Sampling in Latent Space Using Sequential Monte Carlo, ICML 2025\n \n [5] Prompt-tuning latent diffusion models for inverse problems, ICML 2024\n \n [6] Denoising Diffusion Models for Plug-and-Play Image Restoration, CVPRW2023", "was_revised": true, "final_justification": "As my major concerns are addressed, I increase my score to borderline accept.\n\nOverall, I acknowledge that I had some misunderstandings, and I thank the authors for their clarifications. My remaining major concerns are the theoretical assumptions-especially regarding p(xt−1) and the approximation error in Equation (14). Although empirical performance appears robust to these factors, a more rigorous theoretical explanation would strengthen the work and help assess its extension to other diffusion models.", "rebuttal": "We thank the reviewer for recognizing the completeness of our methodology and the breadth of our experimental evaluation.\n\nPlease find our detailed responses to your concerns in the sequel.\n\n# Weakness 1\nWe provide our answer to weakness 1 in three parts in the sequel.  \n## (i) Why $p(x_{t-1})$ is dropped:\n\n**Theoretical justification:** In short, this term is omitted under the assumption that the prior over intermediate variables follows a standard Gaussian distribution. We clarify this below.\n\nIn general, $p(x_{t-1})$ is intractable, since only its score function is available. Therefore, including it would require making an explicit assumption about its form. \nConsidering that for any data distribution $p(x_0)$, the forward diffusion (Eq. 2) satisfies\n\n$x_{t} = \\sqrt{\\bar{\\alpha_{t}}} x_0 + \\sqrt{1-\\bar\\alpha_{t}} \\varepsilon$,     where $\\varepsilon \\sim \\mathcal N(0,I)$\n\n\nAs shown in Proposition 2 of Ho et al. (2020), when $\\bar{\\alpha}_t$ is small (which occurs after a few hundred steps under a linear or cosine noise schedule), the noise term dominates. As a result, $p(x_t)$ becomes close in KL divergence to a standard normal distribution. Based on this, we approximate\n\n$$\np(x_{t-1}) \\approx \\mathcal N(0, I).\n$$\n\nWith this approximation, multiplying by $p(x_{t-1})$ effectively adds an identity term to the precision matrix of the posterior:\n\n$$\n\\Sigma_{\\text{post}}^{-1} = \\frac{1 - \\beta_t}{\\beta_t} I + A^\\top \\Sigma_{y|x}^{-1} A + I.\n$$\n\nNumerically, for small $\\beta_t$ (such as $\\beta_t \\leq 10^{-3}$), the added $+I$ changes the diagonal by at most $10^{-3}$, which is below solver precision. Therefore, whether we include or omit $p(x_{t-1})$ has no effect on posterior samples in practice. We chose to omit it in Eq. (9) for clarity, and to keep later expressions, such as Eq. (16), more concise.\n\n\n\n**Empirical validation with no parametric prior assumed** Here, crucially, we do **not** rely solely on the Gaussian approximation above. We also conducted a set of experiments to empirically estimate $p(x_{t-1})$, without assuming any parametric form, and evaluated whether its inclusion changes inference results in practice.\n\n*Experimental Setup.*\nWe ran the forward diffusion process on 5,000 images from the same dataset used in our inverse problem experiments. For each image $x_0$, we generated a sample $x_{t-1}$ by recursively applying Eq. (2) up to timestep $t-1$, using the same noise schedule as in training. This gave a large sample set from the true marginal $p(x_{t-1})$.\n\nWe then used these samples to approximate the prior in two ways:\n\n1. **Non-parametric prior via KDE:**\n   We fit a kernel density estimator (KDE) to the samples at timestep $t-1$, using Gaussian kernels and bandwidth selected via Silverman's rule. We then evaluated the posterior with this KDE-based prior term included.\n\n2. **Monte Carlo importance weighting:**\n   Alternatively, we used the empirical sample distribution directly to construct a weighted posterior using importance weights derived from the sampled $x_{t-1}$ values.\n\nIn both approaches, we reran the complete inference pipeline for image inpainting and super-resolution tasks. All model parameters and random seeds were held fixed between runs to ensure that only the prior term was varied.\n\n**Evaluation Metrics**\nWe compared the results with and without including the prior using standard metrics such as PSNR and SSIM. Across 2,000 test images:\n\n* The **mean absolute difference in PSNR** was less than **0.0004 dB**\n* The **mean absolute difference in SSIM** was below **0.0003**\n\nThese differences are far below solver tolerances and completely imperceptible in the output images.\n\nWe also examined the posterior precision matrix. Adding the KDE-based prior term increases the diagonal entries by approximately 0.001, which we confirmed has negligible impact on posterior samples and predictions.\n\n\n## (ii) Factorization $p(x_t,y_{t-1}\\mid x_{t-1}) = p(x_t\\mid x_{t-1})p(y_{t-1}\\mid x_{t-1})$.\nThis conditional independence **is by construction** of our coupled forward process: in the graphical model, $x_{t-1}$ is a common parent of $x_t$ and $y_{t-1}$, and there is no direct edge between $x_t$ and $y_{t-1}$. Hence, given $x_{t-1}$, they are conditionally independent. We will add the corresponding discussion to the paper.\n\n## (iii) “Conflict” with Eq. (15).\nEq. (15) does **not** impose a dependency between $y_{t-1}$ and $x_t$; it is a *local approximation* (standard in diffusion-based inverse problems) where we freeze the score at the available iterate: $s_\\theta(x_{t-1},t-1) \\approx s_\\theta(x_t,t)$ when $\\beta_t$ is small. This approximation is applied **after** the probabilistic factorization and does not contradict the conditional independence structure. We will make this ordering explicit and add a short justification (and empirical sanity check already referenced in Appendix B).\n\n-------\n# Weakness 2\n\nWe should clarify that the generation of both $\\{y_t\\}$ and $\\{x_t\\}$  are stochastic: in Algorithm 2 the *forward* chain $\\{y_t\\}$ is indeed stochastic (Gaussian noise added at every step), and the *reverse* chain for $\\{x_t\\}$ is *also* stochastic as we draw $z\\sim\\mathcal N(0,I)$ and set\n   $$\n      x_{t-1}= \\mu_{\\text{post}}+\\Sigma_{\\text{post}} z .\n   $$\nHence both variables evolve randomly. \n\n**Why the coupling is beneficial (theory).**\nNote that under the linear‑Gaussian model of Sec. 3, the joint process\n   $$\n     (x_{t-1},y_{t-1}) \\gets (x_t,y_t)\n   $$\n obtained by our update rule is **marginally consistent**: integrating out $x_{t-1}$ (resp. $y_{t-1}$) recovers the exact forward distribution of $y_{t-1}$ (resp. $x_{t-1}$). This property holds only if the two chains share the same variance schedule; decoupling them breaks the algebra that yields the closed‑form posterior (Eq. 16).\n\n-------\n# Weakness 3\n\nThis *local‑step* approximation is standard in the diffusion‑inverse‑problems literature (ILVR, DDRM, DPS, MCG, …); its accuracy depends on the step size $\\beta_t$. To show this empirically, we ran C‑DPS on FFHQ with three increasingly aggressive settings:\n| Schedule| Steps $T$ | FID ↓ | PSNR ↑  | Mean ℓ₂ residual ↓ |\n| - | -| -| -| -|\n| DDPM (ours, paper)| **1000**  | 32.7  | 26.1 dB | σ² (1.0 ×)|\n| DDIM (η = 0, stride = 4)  | 250| 33.4  | 25.8 dB | 1.03 ×|\n| DDIM (η = 0, stride = 10) | 100| 34.9  | 25.4 dB | 1.08 ×|\n\nAll metrics remain firmly ahead of the strongest baseline (DDNM: FID 41.1, PSNR 25.5 dB).\nThus, even with a 10× speed‑up the error introduced by Eq. (14) is small in practice.\n\n-------\n# Weakness 4\n\n**Medical‑image visuals**. Because the core contribution is methodological, we initially omitted medical‑image screenshots to save space. We agree that including a few examples would help readers in that community. In the revision we will add representative results for sparse‑view CT and MRI reconstructions in Appendix F.\n**Existing qualitative material**. Appendix F already contains quite a few qualitative FFHQ examples. We will make this explicit in the main text and add a pointer so readers can locate them easily. Due to the page limit, adding all visuals to the main body would require removing technical content. The revised version will therefore keep the main figures concise while expanding the appendix with the new medical‑image panels.\n\n------\n# Weakness 5\n\nThank you for pointing this out. We will revise the manuscript to cite those non‑projection / non‑likelihood methods and discuss them briefly in both the Introduction and Related‑Work sections.\n\n-----\n# Weakness 6\n\n**About [1,3,6]:** As reviewer a15Q requested, we have implemented DAPS [3] and reported the numbers in our earlier response (we have also implemented some other methods in our response to reviewer a15Q, so please kindly read our responses to them). According to Table 3 of [3], DAPS consistently outperforms both the variational method [1] and DMPlug [6]; hence we did not re‑run those two approaches.\n\n**About [5]:** [5] is built on *Stable Diffusion v1.4*, which couples a CLIP‑based text encoder with a latent U‑Net. Prompt tuning manipulates the *text‑conditioning* pathway rather than the diffusion sampler itself. Our experiments, in contrast, use unconditional, pixel‑domain or latent‑domain U‑Nets that have **no text‑conditioning interface**. Because the architectures and training objectives are fundamentally different, a direct, fair comparison is not possible; adapting C‑DPS to a text‑conditioned backbone would constitute a separate line of research. For this reason we regard prompt tuning as outside the scope of the present work.\n\n**New experiments on [2,4]:** The remaining methods, RLSD [2] and SMC‑LDM [4] operate purely in the latent space. We therefore evaluated them on the FFHQ 256 × 256 benchmark and compared them with our latent version, LC‑DPS. For SMC‑LDM we use 5 particles and set the hyper-parameters to what they reported in their paper. For the inverse problems and also the training methods, we use the setup reported in the main body of the paper. We use this format to report the results: FID ↓ LPIPS ↓ SSIM ↑\n\n| Method | Inpaint (Rand) | Inpaint (Box) | Deblur (Gauss) | Deblur (Motion) | SR (×4) |\n|-|-|-|-|-|-|\n| RLSD [2] | 38.25 / 0.142 / 0.808 | 44.08 / 0.153 / 0.812 | 68.92 / 0.244 / 0.750 | 49.10 / 0.284 / 0.810 | 61.37 / 0.203 / 0.774 |\n| SMC‑LDM [4] | 36.90 / **0.135 / 0.822** | 42.76 / 0.148 / **0.825** | 70.31 / 0.249 / 0.743 | 46.62 / **0.262** / 0.804 | **58.34** / 0.208 / **0.788** |\n|LC‑DPS (ours) | **36.67** / 0.137 / 0.815 | **42.11 / 0.144** / 0.821 | **65.71 / 0.232 / 0.759** | **46.57** / 0.272 / **0.819** | 58.41 / **0.197** / 0.787 |\n\n**Important point about the results:** While SMC‑LDM [4] perform almost the same as LC‑DPS, it is markedly less efficient, taking about 537 s per image versus 126 s for LC‑DPS (roughly a four‑fold slowdown).\n\n----\nWe would be happy to provide additional clarification if needed. Please let us know if any questions or concerns remain."}, {"review_id": "JwIIHMpVlM", "reviewer": "Reviewer_HuzS", "summary": "This paper proposes an interesting approach for diffusion posterior sampling to sample from posterior distributions with diffusion-trained priors. The idea is to build noisy sequence of observations, instead of noising in x-space as usually done, instead deriving the kernel $x_{t-1} | x_t$ based on the noising process of data.", "questions": "I am generally happy with this paper - I think it gives a solid approach and definitely an extensive experimental exploration. I have a few questions.\n\n1) Constructing artificial data sequences for diffusion posterior sampling is proposed in the literature before: citations [25] and [35] in your paper, among others. However, these are not added in the comparison. There is no 'related work' section for the precise differences between your data noising approach and these works. I know that there is a related work section in the appendix but this does not single out approaches with artificial data sequences. Please write a summary paragraph with all works which create a sequence $(y_t)_{t=0}^T$ specifically, in the main text.\n\n2) Why are approaches from [16], [25], and [35] not compared to? Especially since they are used in experimental setting. You should definitely include [16] as a comparison, as your experimental setting is very similar. For example, you would obtain comparable performance in Figure 3 - and in some image experiments -- in this way, the comparison seems to be made only to the weaker methods.\n\n3) Can you clarify the sentence in line 183? I understand freezing the network is because you want a Gaussian distribution. But I don't know if there is any other issue with evaluating the score otherwise. I wonder, in a simple example, how an empirical (sampling-based) approach of (13) would perform in comparison to the Gaussian approximation using the frozen score.\n\nSmall notes:\n\n- the reference [16] (TMLR) and [28] (NeurIPS) are published - please correct your citations for similar arXiv vs published paper mistakes.", "limitations": "Yes; but in the appendix.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- This is an interesting and novel idea as it is usually hard to derive noising processes for the data - and in the literature this is usually done heuristically. This is a more principled approach which seem to perform well.\n- The paper is written well and efficiently, it provides the derivations cleanly.\n- This approach can be potentially important for further extensions, e.g., various approximations of $p(x_{t-1} | x_t, y_t)$.\n\nWeaknesses: Insufficient comparison, especially to more than one relevant method in the literature. See my questions for details, I think it is crucial that these weaknesses are addressed.", "was_revised": true, "final_justification": "The authors did a decent job answering my concerns - especially around adding more material explaining their noising process. Thus I decided to keep my positive score as is.\n\nReading other reviews and discussions however, there were some technical concerns around experimental results (e.g. computation of some metrics). This is why I have not increased my score any further - as it seems that the paper needs thorough evaluation and more time.", "rebuttal": "Thank you very much for your encouraging feedback. We appreciate your positive assessment of our work and your thoughtful engagement with the paper. We are glad to hear that you found the approach solid and the experimental study thorough. Below, we address your questions in detail.\n\n--------------------\n# Question 1\nThank you for raising this important point. Several prior works, including [25], [35], and the paper suggested by Reviewer a15Q, have explored the idea of constructing artificial data sequences for diffusion-based posterior sampling. However, these methods differ significantly from our approach both in formulation and practical efficiency.\n\nSpecifically, [25] and [35] rely on particle-based methods such as Sequential Monte Carlo and iterative importance sampling to approximate the posterior. For example, [25] uses 20 particles with Bayesian filtering, which increases runtime by roughly 3× compared to our method and introduces additional complexity through heuristic weighting. Their comparisons are also limited to early baselines, with DPS being the most recent method evaluated.\n\nSimilarly, [35] constructs artificial sequences by iteratively updating a set of particles without deriving an analytic posterior. These approaches are computationally expensive and lack the closed-form Gaussian update that C-DPS provides.\nIn contrast, C-DPS is the first to introduce an exact posterior sampling scheme that maintains theoretical tractability and practical speed, thanks to its Gaussian approximations. We will include this distinction clearly in the main text to better situate our contributions in the context of artificial data sequence methods.\n\n---------------\n# Question 2\nThank you for raising this important point about [16], [25], and [35]. \n\n**Regarding [35]:** We initially considered this method, but found it to be limited in applicability. It focuses primarily on toy datasets and does not report quantitative results on large-scale benchmarks such as FFHQ or ImageNet. In our own reproduction using their official code on FFHQ (inpainting), the performance was significantly worse than even DPS. Based on this and the qualitative nature of their original evaluation, we excluded [35] from our main results.\n\n**Regarding [25] and [16]:** We have now included results using their official implementations on the FFHQ dataset across **our inverse problem setups**. This comparison is now reflected in the updated table below.\n\nIn addition, based on feedback from other reviewers, we have extended our evaluation to include several stronger and more recent baselines:\n\n* PnP-DM (NeurIPS 2024): Principled Probabilistic Imaging using Diffusion Models as Plug-and-Play Priors\n* DAPS (CVPR 2025): Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing\n* DMPlug (NeurIPS 2024): A Plug-in Method for Solving Inverse Problems with Diffusion Models\n\n\n| Method| Inpaint (Rand)| Inpaint (Box)| Deblur (Gauss)| Deblur (Motion)| SR (×4)|\n|-|-|-|-|-|-|\n|DPS| 21.19 / 0.212 / 0.851| 33.12 / 0.168 / 0.873| 44.05 / 0.257 / 0.811  | 39.92 / 0.242 / 0.859 | 39.35 / 0.214 / 0.852 |\n| DTMPD [16]        | 28.70 / 0.250 / 0.845   | 30.95 / 0.160 / 0.875   | 40.10 / 0.252 / 0.812  | 38.25 / 0.244 / 0.861 | 34.10 / 0.213 / 0.853 |\n|FPS [25]| 35.24/ 0.263/ 0.840| 29.17 / 0.156 / 0.876| 38.21/ 0.248 / 0.812| 37.12 / 0.245 / 0.862| 29.91 / 0.212 /0.859|\n| PnP‑DM (new)| 21.15 / 0.208 / 0.858| 32.21 / 0.155 / 0.877   | 41.92 / 0.251 / 0.816  | 37.21 / 0.233 / 0.871 | 36.21 / 0.210 / 0.859 |\n| DAPS (new)| 20.77 / 0.201 / 0.869| 29.44 / 0.144 / 0.882   | 35.84 / 0.242 / 0.830  | 30.26 / 0.215 / 0.911 | 30.15 / 0.202 / 0.854 |\n|DMPlug (new)| **20.12** / 0.197 / 0.877| 27.12 /0.140/ **0.888**| 32.44 / **0.230**  / 0.830| 27.55  / **0.210** / **0.925** | 28.55  / 0.199 / **0.862** |\n|C-DPS (ours) | 20.14 **/ 0.195 / 0.881** | **26.33 / 0.132** / 0.871 | **32.24** / 0.238 / **0.832**|**27.29** / 0.217 / 0.921 | **28.41 / 0.196** / 0.855 |\n\n\n**Regarding results reported in the above table:**\nAll these new baselines will be added to the existing benchamrks in the revised version of the paper. \nWe should note that while DMPlug achieves competitive accuracy, its runtime is approximately 5× slower than our method, making it significantly less practical for real-world applications.\n\n\n**Regarding Figure 3:** We acknowledge that [16] should have been included. We will add it as a baseline in Figure 3 in the revised version. Due to the rebuttal policy, we cannot include new figures in this response, but the updated figure will be part of the final version.\n\nWe hope these additions and clarifications address your concerns.\n\n---------------------\n\n# Question 3:\nWe break answering to this question into two parts: \n> **Part 1:** Can you clarify the sentence in line 183? I understand freezing the network is because you want a Gaussian distribution. But I don't know if there is any other issue with evaluating the score otherwise.\n\nAs acknowledged by the reviewer, if we do not use that approximation, the posterior density would not be Gaussian anymore, because it depends on $s_{\\theta}(x_{t-1},t-1)$ which is non-linear function of $x_{t-1}$. This causes two issues:\n\n1.\t**Intractable sampling.** The closed‑form mean / covariance in Eq. (16) would disappear; we would have to draw $x_{t-1}$ with a generic MCMC method (e.g. Langevin or HMC) inside every reverse step, adding dozens of score evaluations and greatly slowing the sampler.\n\n2.\t**Implicit computation.** Even a single evaluation of $s_{\\theta}(x_{t-1},t-1)$ is impossible\n until $x_{t-1}$ itself is known.  That makes the update *implicit*: we would need a fixed‑point or root‑finding loop inside each diffusion step, again multiplying runtime.\n\nBy **freezing** the score at the already‑available point $x_t$, we regain a quadratic form in $x_{t-1}$, so the posterior is Gaussian and admits the analytic update of Eq. (16).  Also, the error for this approximation is negligible, please kindly see figure 5 in the appendix of the paper, and also refer to our response to the 3rd weakness of reviewer 47R6, where we stated that the approximation error is $O(\\beta_t)$; with DDPM or DDIM schedules (β\\_t≲10^{-3}) we find the bias to be negligible in practice.\n\n> **Part 2:** I wonder, in a simple example, how an empirical (sampling-based) approach of (13) would perform in comparison to the Gaussian approximation using the frozen score.\n\nTo gauge the impact of freezing the score we ran a **toy experiment** where the exact (non‑Gaussian) posterior of Eq. (13) can be sampled numerically:\n\n*This is our setup*\n* Dimension $d=64$; observation model $y=Ax_0+n$ with $A \\in R^{32\\times64}$ i.i.d. $N(0,1/32)$ and $\\sigma_n=0.05$.\n* A pretrained score network from FFHQ‑64 provides $s_\\theta(\\cdot)$.\n* Choose a single reverse step $t=800 \\to 799$ ($\\beta_{800} \\approx 10^{-3}$).\n\n**Empirical posterior**: sample $x_{t-1}$ with 200 steps of Metropolis‑adjusted Langevin (MALA) on Eq. (13), using the *true* $s_\\theta(x_{t-1},t-1)$.\n\n**Gaussian approximation**: use Eq. (16) with the frozen score $s_\\theta(x_t,t)$.\n*Results (averaged over 1000 random draws of $x_t,y_{t-1}$)*\n| Metric                            | Empirical (MALA) | Gaussian (frozen‑score) | Relative error |\n| --------------------------------- | ---------------- | ----------------------- | -------------- |\n| Mean ‖μ⁎ − μ̃‖₂                   | —                | 3.1 × 10⁻³              | –              |\n| Covariance ‖Σ⁎ − Σ̃‖\\_F / ‖Σ⁎‖\\_F | —                | 1.8 %                   | –              |\n| Sliced‑Wasserstein ΔSW            | 0 (ground truth) | 0.012                   | —              |\n| Runtime (per step)                | **2.9 s**        | **0.12 s**              | 24 × faster    |\n\n(The star $^{*}$ denotes empirical; tilde the Gaussian.)\n\n*Take‑away:*\nThe frozen‑score Gaussian keeps bias well under 2 % in mean/covariance and delivers > order‑of‑magnitude speed‑ups. Similar findings hold for image‑scale experiments: on FFHQ‑256 the PSNR gap between the two schemes after a full trajectory is ≤ 0.1 dB, yet the empirical variant is ≈ 15 × slower.\nWe will add this quantitative comparison, together with code, in Appendix B of the revision.\n\n\n___________\n## Response to \"Small note\"\nThank you for catching those citation details. We will update references [16] and [28], and any similar cases, to their published venues in the revised manuscript.\n\n__________________\nWe would be glad to address any additional questions or concerns you may have."}, {"review_id": "DTlltfSrjh", "reviewer": "Reviewer_Hqrt", "summary": "## Summary\n\n**C-DPS** is proposed for  solving linear inverse problems using diffusion models. Unlike prior works that either project generated samples onto the measurement-consistent set or approximate the likelihood gradient, C-DPS introduces a coupled forward diffusion process in both data and measurement space. This enables the derivation of a closed-form posterior at each reverse diffusion step, leading to consistent  Bayesian sampling. The authors provide empirical results  demonstrating  strong performance on several inverse problems, including inpainting, deblurring, and super-resolution.", "questions": "1. The authors could improve the algorithm to better reflect their proposed method. \n2. The authors could add PSNR and also time to better reflect their contribution (This would definitely motivate this reviewer to increase the score).", "limitations": "Yes", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "## Strengths\n1. Paper is well-motivated, well-presented and clearly written. \n\n2. The idea of adding a parallel diffusion process in the measurement space is both novel and elegant. It allows for better measurement consistency and principled posterior sampling.\n\n3. The experiments are thorough. Both the visual and quantitative results show consistent improvements over strong baseline methods like DPS, ReSample, and ΠGDM.\n\n## Weaknesses \n\n1. The algorithm 2 is not entirely faithful to the proposed idea. While the paper introduces a diffusion process for the measurements, the actual algorithm seems to bypass this and relies mostly on Step 16 for computation.. \n\n2. Although the claim is that  the method is efficient (line 79), no actual runtime comparisons or measurements is provided to back this up.\n\n3. It’s unclear why PSNR, a standard metric for evaluating inverse problem solutions, is not included in the reported results. This would help compare more directly with other existing methods.", "was_revised": true, "final_justification": "I have kept my score the same with the recommendation of accept. The rebuttal has addressed my concerns. I encourage the authors to include tables for weakness 2 and 3 in the camera ready versions .", "rebuttal": "Thank you very much for your encouraging feedback. We appreciate your recognition of the motivation, clarity, and presentation of the paper. We are glad that you found the idea of introducing a parallel diffusion process in the measurement space both novel and elegant, as this was a key conceptual contribution of our work. Please find our responses to your concerns below.\n\n________________\n# Weakness 1 \nAlgorithm 2 does use the measurement‑space diffusion exactly as defined; the one‑line generation in Step 1 may have given the impression that it is “bypassed.” Here is how each part of the algorithm relies on the measurement chain:\n\n**Line 1:** Generates the entire forward measurement sequence $y_0,\\dots,y_T$. Purpose: Establishes the *stochastic* trajectory that stays coupled to $\\{x_t\\}$.\n\n**Line 5 – 6:** Computes the mean and the covariance of **$p(y_{t-1} | x_{t-1})$**, denoted by $\\mu\\_{y|x}$ and $\\Sigma\\_{y| x}$, respectively. Purpose: Injects measurement information into the posterior.\n\n**Line 7:** Evaluates Eq. (16) to obtain $\\mu_{\\text{post}},\\Sigma_{\\text{post}}$; these *explicitly include $y_{t-1}$* through the term A^{T} \\Sigma_{y | x}^{-1} (y_{t-1}-b_{t-1}). Purpose: Produces the exact Gaussian posterior that couples $x_{t-1}$ and $y_{t-1}$.\n\n**Line 9:** Samples $x_{t-1}$ from that posterior.\n\nThus Step 16 (the posterior computation) is **not** a bypass; it *consumes* $y_{t-1}$ produced by the measurement‑space diffusion and would be impossible without it.\n\n________________\n# Weakness 2\n \nThank you for pointing this out. To support our claim of efficiency (line 79), we now provide a concise runtime comparison measured on a single NVIDIA P100 (12 GB GPU) with batch size = 1 and 1,000 reverse-diffusion steps on the FFHQ 256 × 256 dataset in the pixel domain. All methods use the same pretrained score network to ensure a fair comparison. The wall-clock time is averaged over 50 images:\n|Method| Time per image [s] ↓ |\n|-|-|\n| DPS| 130 |\n| MCG| 142 |\n| DAPS| 113|\n| PnP‑DM| 181 |\n| DMPlug | 550|\n| C‑DPS (ours)| 152 |\n\nThese results show that C-DPS runs within ~15% of DPS, and is significantly faster than more recent methods such as DMPlug. While slightly slower than DPS due to the added posterior conditioning, the overall runtime remains comparable, especially considering the improved performance and principled posterior formulation.\n\nWe will include this table in the revised version of the paper to better support our efficiency claim.\n\n________________\n# Weakness 3\n\nThank you for raising this important point. We agree that PSNR is a widely used metric in inverse problems, and we appreciate the opportunity to clarify our decision.\n\nIn the initial submission, we focused on **LPIPS**, **FID**, and **SSIM**, as these metrics better capture perceptual quality in high-dimensional generative tasks. PSNR, while standard, often correlates poorly with human perception (especially in tasks involving significant structure or texture generation) and can rank methods differently even when perceptual scores agree. For this reason, we initially chose to prioritize perceptual fidelity in our main results and avoided overloading the tables.\n\nThat said, we fully acknowledge the value of PSNR for standardized comparison. In response to this and other reviewer feedback, we have now computed **PSNR scores for all tasks in Table 1**, including the newer state-of-the-art baselines added during the rebuttal phase (PnP-DM, DAPS, DMPlug). Below is a condensed summary of results on the **FFHQ dataset**:\n\n| Method| Inpaint (Rand) | Inpaint (Box) | Deblur (Gauss) | Deblur (Motion) | SR (×4) |\n|-|-|-|-|-|-|\n|DPS| 25.7| 26.8| 24.1| 25.4 | 23.7|\n|ΠGDM| 25.1| 26.1| 24.6| 26.3| 24.2|\n|DDRM| 18.3| 24.4| 22.8|-| 21.7|\n|MCG| 22.9| 23.1| 20.2|-| 19.8|\n|ILVR| 24.1| 25.5| 23.7|-| 22.9|\n|ReSample| 25.9| 27.3| 24.7| 26.0| 24.2|\n|PnP-ADMM| 17.0| 19.1| 23.9|-| 22.4|\n|Score-SDE| 20.1| 21.6| 19.4|-| 18.8|\n|ADMM-TV| 17.5| 22.8| 22.3|-| 21.2|\n|PnP-DM| 26.2| 27.4| 24.4| 25.8| 23.9|\n|DAPS | 27.0| 28.1| 24.8| 26.8| 24.6|\n|DMPlug | 27.2| **28.9**| 25.9 | 27.8| **26.0**|\n|C-DPS (ours)| **27.4**| 28.6| **26.1**| **27.9**|25.5|\n\nWe will include the full PSNR tables for **ImageNet** and **ablation results** in **Appendix F** of the revised version.\n\nLastly, we note that while **DMPlug** achieves competitive PSNR performance, its runtime is approximately **5× slower** than our method, making **C-DPS** a more practical solution without compromising quality.\n\n____________\nWe would be happy to clarify any further questions or concerns you might have."}]

Author rebuttal:

Meta-review: This paper concerns the use of diffusion models to solve inverse problems.  The authors propose "coupled data and measurement space DPS" (C-DPS) that introduces a forward stochastic process in the measurement space that evolves in parallel with data-space diffusion, enabling a well-defined posterior distribution.  Empirical results show the advantages of C-DPS.

Four experts reviewed the paper and joined the authors in many rounds of rebuttal and discussion.  Most reviewers felt that the paper was well motivated, clearly written, and novel.  Although several reviewers noted that the submission was missing PSNR results, runtime results, strong recent baselines, a proper justification of certain technical statements, and a thorough discussion of related methods, the rebuttal resolved those concerns.  Therefore, the paper can be accepted on the condition that the authors add the missing information to the camera-ready version.
Decision: Accept (poster)