Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Linear Extrapolation
Authors: Jiawei Zhang
Ziyuan Liu
Leon Yan
Gen Li
Yuantao Gu
Venue: NeurIPS 2025 poster ()
Keywords: Inverse problem
diffusion model
few-step
high order solver
Primary area: deep_learning

Abstract: Diffusion-based inverse algorithms have shown remarkable performance across various inverse problems, yet their reliance on numerous denoising steps incurs high computational costs.  While recent developments of fast diffusion ODE solvers offer effective acceleration for diffusion sampling without observations, their application in inverse problems remains limited due to the heterogeneous formulations of inverse algorithms and their prevalent use of approximations and heuristics, which often introduce significant errors that undermine the reliability of analytical solvers. 
In this work, we begin with an analysis of ODE solvers for inverse problems that reveals a linear combination structure of approximations for the inverse trajectory. Building on this insight, we propose a canonical form that unifies a broad class of diffusion-based inverse algorithms and facilitates the design of more generalizable solvers.
Inspired by the linear subspace search strategy, we propose Learnable Linear Extrapolation (LLE), a lightweight approach that universally enhances the performance of any diffusion-based inverse algorithm conforming to our canonical form. LLE optimizes the combination coefficients to refine current predictions using previous estimates, alleviating the sensitivity of analytical solvers for inverse algorithms.
Extensive experiments demonstrate consistent improvements of the proposed LLE method across multiple algorithms and tasks, indicating its potential for more efficient solutions and boosted performance of diffusion-based inverse algorithms with limited steps. Codes for reproducing our experiments are available at https://github.com/weigerzan/LLE_inverse_problem.

Improving Diffusion-based Inverse Algorithms under Few-Step Constraint via Linear Extrapolation
Jiawei Zhang
Ziyuan Liu
Tsinghua University
Tsinghua University
12aaei-230mmals,.. tsinghua. edu.cc
Liuziyua220malls tsinghua. edu. cn
Leon Yan
Gen Li CUHK
Yuantao Gu*
Tsinghua University
Tsinghua University gyt@tsinghua, edu. cn
yansc230mails. tsinghua.
edwiiaaaeeddecc.. edu. en
genl 10cuhk. edu.hk hk
Abstract
Diffusion-ban -based inverse algorithms have shown remarkable performance across various inverse problems, yet their reliance numerous denoising steps incurs high computational costs. While recent developments of fast diffusion ODE solvers offer effective acceleration for diffusion sampling without observations, their application in inverse problems remains limited due to the heterogeneous formulations inverse algorithms and their prevalent of approximations and heuristics, which often introduce significant errors that undermine the reliability of analytical solvers. In this work, we begin with an analysis of ODE solvers for inverse problems that reveals linear combination structure of approximations for the inverse trajectory. Building this insight, we propose canonical form that unifies broad class of diffusion-based inverse algorithms and facilitates the design of more generalizable solvers. Inspired by the linear subspace search strategy, we propose Learnable Linear Extrapolation (LLE), a lightweight approach that universally enhances the performance of any diffusion-bian based inverse algorithm conforming our canonical form. LLE optimizes the combination coefficients to refine current predictions using previous estimates, alleviating the sensitivity of analytical solvers for inverse algorithms. Extensive experiments demonstrate consistent improvements of the proposed LLE method across multiple algorithms and tasks, indicating its potential for more efficient solutions and boosted performance of diffusion-basel inverse algorithms with limited steps. Codes for reproducing our experiments are available at wttp:/uwwww.ooeeweeeeee////////e
Introduction
Diffusion models have demonstrated remarkable capability in modeling complex data priors 2 4]. which has led to their widespread application in solving inverse problems. Extensive efforts have been devoted the development of diffusion-bei based inverse algorithms 65 achieving impressive performance in numerous tasks including inpainting 15 super-resolution 16
deblurring [17. and compressed sensing [18
One major drawback of diffusion- based inverse algorithms is the need for multiple neural network inferences, resulting in high computational complexity. Directly reducing the number of inference steps often degrades the performance of these inverse algorithms. Therefore, enhancing the perfor- mance of diffusion-based inverse algorithms under limited steps has emerged critical research
The corresponding author is Yuantao Gu.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Nta
X4
Sumpler 4
Xnat
Corrector h2
Rort
Noiser V/
X11-1
Degalian
Nostin
Original dats flow
N
EC
N:A
LLE data flow
Figure I: The proposed canonical form of diffusion-basee inverse algorithms and the workflow of our LLE method. Our canonical form decomposes an inverse algorithm into three key modules: Sampler, Corrector, and Noiser, providing unified framework that encompasses a wide range of existing approaches. The LLE method learns linear combination of the current corrected estimate and previous results to obtain better estimation of the original image, therefore universally enhancing Liffusion-based inverse algorithms' performance under few steps.
direction, leading to several recent attempts such as shortcut sampling [9,200] learned correction processes (2T). and the introduction of conjugate integrators [22]. Recently, high-order order diffusion ODE have shown strong performance in the few-step sampling of diffusion models without observations. However, although conceptually these solvers can also accelerate inverse algorithms, two challenges remain in practice. First, not all inverse algorithms admit an explicit ODE or SDE formulation, necessitating a more general framework to unify their analysis and the design of generalizable solvers. Second, the prevalence of approximations and heuristics in existing methods often leads to substantial estimation errors, undermining the reliability of analytical high-order solvers.
solvers [23 24 25
26 27
To tackle these challenges, we begin with an analysis of ODE solvers in inverse problems from the perspective of posterior sampling, revealing an equivalent interpretation in the form of linear combinations. Based on the insights, we propose canonical form for diffusion-baus based inverse algorithms, which serves unifying framework for a broad class of existing methods, facilitating a more systematic understanding and more generalizable development of inverse problem solvers. Building on this canonical form, we design universal enhancement strategy for iiftusioon-aseed inverse algorithms inspired by optimal linear subspace search [27). Our proposed method, termed Learnable Linear Extrapolation (LLE), learns set of linear combination coefficients to boost current corrected estimate with previous results, thereby mitigating the unreliability analytical high-order solvers under large estimation errors or heuristics. For linear inverse problems, we further leverage the structure of the observation function by decoupling the coefficients associated with the range space and null space, leading to enhanced performance for LLE.
LLE is highly lightweight and can be trained with 50 diffusion generiated samples within minutes on single NVIDIA RTX 3090 GPU, which makes it practical and easy to integrate into existing inverse algorithms. We reformulate nine mainstream diffusion- based inverse algorithms into the proposed canonical form and conduct extensive experiments to validate the effectiveness of LLE in both linear and nonlinear tasks. The results demonstrate that LLE effectively improves the performance of these algorithms under few steps, providing novel and more general perspective to enhance diffusion-baser inverse algorithms in the future.
Backgrounds
2.1
Diffusion models
Diffusion models
consist of forward diffusion process that transforms a clean data distribution into a pure Gaussian distribution, and reverse denoising process that generates clean data from the noise distribution. this paper, we consider the setting of Variance Preserving (VP) diffusion. Given target data distribution (xx),, the forward diffusion process is modeled by the

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following stochastic differential equation (SDE):
dx, f(t)x,tt + g(t)dw,, E [0,
(I)
where Xo g(xo) and w; standard Wiener process. VP diffusion specifics the drift term f(t) and the diffusion term g(t) as
f(t) log v g²(t) d(1 σe) d log Vān (1-aa),
dt
respectively, where T3 is a predefined monotonic parameter controlling the noise level at each timestep, with ag 1 and ār 0.
The forward SDE can be exactly reversed as
dx,- = (J(1)x - o (1)Vx: lo (x:)) di + g())d,,
where W, is the reverse-time standard Wiener process, Xy No I), and Vx, log q (xx)) is known as the (Stein) score function [30] 31] Practical implementation aims to estimate the unknown score function Vx, log (xxx)) via a parameterized neural network EØ e t) as
(x(x..() -V1 Vx x, log qi (xt);
which can be learned via matching
32
33
Song el (4 demonstrate that the above reverse SDE shares the same marginal distributions as the following reverse-time ordinary differential equation (ODE):
dx; df
f(t)x; 2 V log (xx),
(3)
which facilitates the introduction of efficient ODE solvers to enhance sampling efficiency. Using first-order solver to solve the diffusion ODE 3 leads to the iterative formula of the well-known DDIM sampler
33, which is
a,,o t)
v at
2.2 High-order solvers for diffusion ODE
Numerous works have developed variety of fast ODE solvers for diffusion models 35
2 24 25
26 36 37 38 39 40). Recent studies have adopted exponential integrators to solve the diffusion
ODE follows
A(t) exp( (p((A))e(x())))- (AA)dd
X2
X,
Cxs
where X(t) = log / aa a.)
leverage the Taylor expansions of c)(-.-) estimated from previous score functions solve the integral more accurately. which reduces the dependency on small step sizes and enables efficient sampling with fewer steps. Despite differences in derivation and specific fornulation, these solvers share common structure in that x, can be expressed linear combination of the initial point and the scores from all previous steps 27 i.e.,
kt). High-order diffusion ODE-solvers 23
24
26
25
X, E span (XT, (x T).... cx
Thus 27
proposes the optimal subspace search method to optimize the coefficients of x, in the subspace shown in (4) aiming to align the trajectory of few-step solver with a target solver using more steps (e.g., a 999-step DDIM). The learnable solver achieves remarkable performance with few steps while remaining lightweight as only requires learning a set of linear combination coefficients per timestep. In this paper, we design a lightweight learnable method tailored for inverse algorithms to generally enhance their performance under few steps leveraging similar insights from the optimal subspace search approach.

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2.3 Diffusion- based inverse algorithms
Consider the degradation equation
(5) where A is the degradation function, is the observation, and By is the standard deviation of the observation noise. The goal of inverse problems is to recover the original data X from the degraded observation y. The ability of diffusion models to capture complex data priors has inspired wide range of studies exploring their applications in inverse problems from diverse perspectives.
y = A(x)) + oyn, n N
DDRM [ defines a new set of variational distributions to approximate the posterior ((xoly) and derives inverse algorithm for linear problems based projection operations. DDNM (6] directly employs the null-range decomposition and heuristically projects the intermediate estimation of xo during sampling onto the hyperplane corresponding to the linear observation. Posterior sampling methods [8] [0] attempt to estimate the conditional score Vx, log qe (x.l)) to sample from the posterior distribution (xxoly). DifiPIR [ proposes plug-am aun--ad approach by guiding the intermediate estimation of Xy with proximal point problem and further refining the reverse sampling trajectory. ReSample [2]. originally designed for latent diffusion, can be interpreted as optimizing the intermediate estimation via gradient descent to enforce hard data consistency and guiding the reverse sampling trajectory with a resample algorithm. More recently, maximum posteriori (MAP)- based methods such as RED-diff [11] have gained attention, which explicitly express the data prior (xx) using diffusion models and solve inverse problems by optimizing maxx; g(xoly). DAPS [14] adopts Langevin dynamics to enforce consistency with the observations and progressively anneals the timestep to achieve noise reduction.
Despite the effectiveness of diffusion- based inverse algorithms. they typically require 100 to 1000 steps during inference. this paper, we first establish a canonical form for these algorithms and then propose a lightweight and general approach to enhance their performance under few steps, achieving a better trade-off between performance and computational efficiency.
3 Analysis of ODE solvers in inverse problems
can be shown that the associated probability flow ODE for sampling from the posterior distribution p(xoly) is given by
dx; s(1)x - (1)Vx, logp(x.ly).
(6)
In the context of inverse problems, the primary focus to restore the clean signal. Thus we define xo(1,)) x + (1 a, Vx. log 1(x,()))
va,
To assist our analysis, we assume the temporal discretization for the numerical solver as ts to} S [0. 2] with is > f5-1 > ... > to = 0, and S denoting the total number of steps. Then the evolution of y)((,y) from li+1 to 4 can be described as the following proposition. Proposition 1. The evolution ())
v1- Oity VTn,1 xo(li+1.y)
VT- at (y
xo(li.))
d(F)xo(T,y)dd
VI Cttt
Var.
Vas,
(V1 at Vx., logp(x;ly) /1 Otii Vxiga
log (x 1x)).
(8)
Val
(*x)
22 2(1-a_)3/7-
where (bT)
The detailed derivations are deferred to Appendix Note that term (+) contains an integral involving xo(T,y). which can be approximated using standard k-th order Taylor expansion, i.c.,
S" (())xo(Tyydde 1:11
d'xo(tet1,y) dt! d+
(r f++1)
1! ((T)dT + 0((++1 - -1), (9)

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d'xo(ta) in (9) can be estimated using multi-step methods. In particular, de
The high-order derivatives
under uniform time discretization lf+1 1g h, Vj, the forward differences approximation 42 of the I-th derivative yields
43
d' ('x0(4+1,g) de
1 h j=0
Σ (-1))- () (1i+1+;.));
(10)
which implies that the Taylor expansion approximation for (+) is linear combination of previous steps St, for 2 + 1.
Term (**) involves the score both time steps 1 and éú, which can be reformulated as
Van.
Oit V xi, log pl Xt; (y) + Xt: -xo(ti+1,)))
(1I)
Van,
An estimate of (o(t,,y)
where
81, V Xo(le+1()) - V( (1--,,)( (1-
Vx log 1xx:(xx.113))
(12)
Note that & serves as the sample calculated by deterministic DDIM sampler. Consequently, the reformulation (IT) can interpreted as linear combination of coarse estimation the current time step I and the previous result at time 44+1- Based on the above analysis terms (+) and (*+), we arrive at the following conelusion.
Corollary 2. Approximating the evolution of xt(, from t,+1 to t, can be interpreted as computing a linear combination of an estimate of the current step t, and all previous results at t, for Vjj 2 sit
We note that many inverse algorithms introduce random noise, which makes them more naturally aligned with the SDE perspective. We provide a detailed analysis of the SDE case in Appendix B.3 and obtain conclusions similar to those in Corollary 2
This insight provides foundation for generalizing diffusion ODE and SDE solvers to a broad class of inverse algorithms. Conceptually, for most algorithms that are difficult to characterize via posterior sampling, we reinterpret Xt (f. y) as an estimate of the clean sample at time f corrected with the observation y. Meanwhile, the linear combination structure enables learning the combination coefficients in data-driven manner, thus avoiding the failure of analytical coefficients when the posterior score estimation is highly inaccurate or heuristic.
4 Designing unified solvers for inverse algorithms
In this section, we first establish canonical form that unifies wide range of diffusion- iiffussio-bbsed inverse algorithms and clarify its connection the analysis in Section 3 We then propose the LLE method, which is compatible with any algorithm adhering to the canonical form and effectively improves their performance in the few-step regime by learning a set of linear combination coefficients.
4.1 A canonical form for diffusion-based inverse algorithms
As illustrated in Figure our proposed canonical form decomposes each iteration of inverse algorithm into three modules. Below we detail these modules at step i; for any 1 < i < S.
(1) Sampler generates estimate of the clean data X,,s, from the noisy input xt, using pre-trained diffusion model:
Xo.t; 4 (xt.)-
(13)
(2) Corrector incorporates the observation y to refine the sampled result, ensuring consistency with the measurement:
* = (Xo.t. A.y).
(14)
5

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Algorithm Training of LLE.
Require:
Pretrained diffusion model 6g, discrete timesteps Ls, 15...... lo. inverse algorithm in
the canonical form Lt h tu.- reference samples (1) ..... X
(N), observation function A,
observation noise deviation dy. N do 2: Get observation y!") as 5
1: for
3: end for
(m) 4: Initialize (
(n) NO,.I);
n=1
5: for i S do
6: for n N do
7: Calculate Xo. h4:
tt:
x
A, y(m)):
8: end for
9: Optimizing Y6, --, as
19
10: for n==
N do
11:
Calculate X0,4. as 18
12:
Update x V (x0,1)
13:
end for
14:
end for
15:
return Coefficients ......... 7 S-i for S,...,
Algorithm 2 Inference with LLE.
Require: Pretrained diffusion model cg. discrete timesteps Is. 45-1, 1,,1,.... to observation y. observa- tion function A, observation noise deviation oy, inverse algorithm in the canonical form +.,- hr,, tt, optimized coefficients ",,,,....;, .,,......... R4 for = S....,
1: Initialize Xts ~ ~N,
2: for i= to i = - do
3: Calculate Xo.1, h, (PPr (xx,) A, y);
4: Calculate No.t, as (88
5: Update X Lr, r, (x
6: end for
7:
return Xto
(3) Noiser maps the corrected clean sample back to the next noise level for the subsequent iteration:
V (8o.t:)-
(15)
One immediate advantage of this decomposition is that (4) broadens the definition of xo y) in 77). which applicable wider range of inverse algorithms beyond posterior sampling, including but not limited to projection- based and plug-and- and-play approaches. Another benefit lies in that, by decoupling an inverse algorithm into discrete modules, we can reorganize these modules and directly obtain an estimate at 4 from the results at 44+1
S h, +., L Die. (St,)144). )
(16)
which provides an effective alternative the approximation term 11I). Consequently, we can construct unified solvers for any inverse algorithms conforming to the proposed canonical form, requiring only the specification of set of linear combination coefficients, which we propose to optimize in data-driven manner.
This modular decomposition is applicable broad class of inverse algorithms. In this work, reformulate nine mainstream inverse algorithms into the proposed canonical form, with detailed decompositions presented in Appendix E
4.2 Iterative learnable linear extrapolation
Intuitively, the proposed LLE method attempts to enhance the performance of inverse algorithms under few steps by iteratively learning linear combination of the corrected sample (14) and

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all previous results at fz. > i. to obtain better estimate Xu,,, Below, we introduce the training objective of LLE at timestep t; inductively.
Consider a reference training set generated by the diffusion model X = {x000 x
X
where N is the number of reference samples. Suppose we have obtained the estimates from the ((a) previous S - i steps, i.e., 0 k i , S, == 1,.... N. Then we replace each & 0, with ((x) 00.4.41 in 16 and compute estimate step as
(n)
(n) 0
hi, (Pr.
V 1:41
(17)
where y(") represents the observation of X coefficients, denoted as .t..0..... -,, and update the estimates at step 4 as
(n)
Now we optimize set of linear combination
S-i-1
(n) 0.4
+
Σ f=U
(n) 0
(18)
;
Unlike unconditional generation tasks that often require sampling long trajectories as the training targets for learning fast solvers, inverse problems provide explicit supervision in the form of known ground-truth reconstructions. This eliminates the need to sample or store complete trajectories, significantly reducing training cost. As a result, the optimization objective of LLE is formulated as:
min E0..u (x())) x...)
(19)
7c 74..5
where the loss function is chosen as
C 0,4 xf") = CMSE
x0.1, xx(") + WLLPPIPS (x0.t) xf")).
(20)
LISE is the Mean Mean-Square Error loss that measures the pixel-level discrepancy between samples, while LLPIP represents the Learned Perceptual Image Patch Similarity [44] loss that quantifies the perceptual discrepancy between two samples. w is introduced in our loss function (20) as a weight
to balance the MSE loss and LPIPS loss. Smaller w encourages the algorithm to minimize MSE, resulting in higher PSNR, whereas larger biases the algorithm toward minimizing LPIPS, leading to higher perceptual similarity.
Once all coefficients are optimized, we fix them and perform inference in exactly the same manner as the data flow the training process. The pseudo-codes for the training and inference of LLE are presented in Algorithm and Algorithm 2 respectively.
4.3 Decoupled coefficients for linear problems
For linear inverse problems, we propose using decoupled linear combination coefficients estimate the null-space and range-space components of a sample with respect to the observation matrix separately. The rationale is that observations impose strong constraints the range-space component, while the null-space component remains more flexible. Consequently, decoupled sets of linear combination coefficients would perform better to capture the differences between these components.
Consider the observation equation y = Axy + n, where A is the observation matrix. The range- space and null-space components of a sample with respect to A are denoted as xll = A Ax and x (I- A A) x, respectively, where A represents the Moore-Penrose pseudoinverse of matrix
(n) A. At timestep 4; optimize set of coefficients " netgr j O,..., S to estimate X as (n)_L
(x) = 8
(n)ill
+ X
(21)
where
S-i-1
S-1-1
0,44 --.
(n),1l 1
(n).ll
Σ
(x).ll 0
(a),1
Th.s-
(1)
(a).L
N0.88-"
(n).1
==0
1=0
(22)
The optimization method and objective remain consistent with those described in Section 4.2
7

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Table 1: Results of DDNM, DDRM, HIGDM, DMPS, and DPS on noisy linear tasks (ay 0.05) on the CelebA-HQ dataset using 3 and 5 steps. The better results are bolded.
PSNRt / SSSMS// / LPIPSL
Steps
Algorithm
Strategy
Deblur (aniso)
Inpainting
4x SR
CS 50%
27.80/ / 0.758 0.319
16.64/0.442
0.4992
27.09 0.773/ 0.296
16.55 0.441 / /0559
DDNM
LLE
28.08 0.784 0.291
24.38 0.552 0,433
/.770/ 0.299
17.29/ 0.473 0.520
27.68 / 0.795 0.277
16.68 / 0,448 /0.44)
26.71 0.764 0.277
16.58 0,495 / 00,78
DDRM
LLE
27.69/ / 0.795/ 0.271
24.53 / 0.625 0.406
27.49 0.761 0.295
17.07 0.527 / 0.,47
20.39/ / 0,536 0.520
19.57 / 0,533 00.495
50.73/0.5552 / 0,500
16.79 0.472 / 0.555
HIGDM
LLE
21.73/ / 0,588/ 0,489
20.72/ / 0,571 0.,883
21.54 0.583 0.490
17.30 0.487 / 0,554
14.02 / 10.174 0.747
11.91 / O. 100 008002
15.55 / 0.235 / 0.76.
12.45 O. U.2 / 0.791
DMPS
LLE
14.10/ / 0.177 / 0.744
11.97 / 0.104 0,800
18,76 0.253 0.776
12.52/ 0.105 / 0,789
23.59/ / 0.650/ 0.415
23.57 / 0.555 /0.497
25.49 006447 / 0.528
14,30/ 0,359 / 0.636
DPS
LLE
24.59/ / 0.675 / 00,05
27.51 / 0,748 00,366
24.57 0.666/ 0.465
15.83 0.468 / 0,591
29.63 / 0.819 0.299
22.76 / 0,550/0.431
28.97 0,818/ 0.262
18,20 0.474 0.491
DDNM
LLE
29.82/ / 0.831 00,339
26.35 / 0.659/ 0..66
29.02 O 806 0.252
19.41 0.536 / /0.41
29.29 / 0.826 0.244
23.21 0.71770,286
28.58 0.811/ 0.247
34.47 0.599 70.389
DDRM
LLE
29.36 / 0.827 00,235
26.09/ / 0.780/ 0,263
28.64 0.795 0.249
19.41/ / 0.632 / /0.363
22.46/ / 0.653 0.441
22.63 0.695 00,376
21.93 0.662 0.416
16.32/ / 0.518 / 00541
5
TIGDM
LLE
25.95 0,734 0.347
25.21 / 0.735 0.336
25.67 0.731 0,338
19.33 0.591 /0.448
21.207 0,404 70.562
14.7170.IG4 70.733
22.95 0.49% 0.541
14.85 0.165/ 70.719
DMPS
LLE
21.63/ 0.406 0.558
16.45/ / 0.186 0..90
24.66 0.625 0.437
15.57/ 0.180 0.699
23.94/ 0.680/ 0.369
2 0.617
/0.434
26.07
0.675 0,470
17.12/ 0.515 / 0,481
DPS
LLE
25.56/ 0.722 0.326
27.42 / 0.796 0.343
26.63 0.758 0.315
17.73/ 0.536 / 0..68
5
Experiments
In this section, we validate the effectiveness of LLE across various inverse algorithms and tasks.
5.1 Experimental setup
Inverse algorithms and hyperparameters. We consider nine mainstream diffusion-basee inverse algorithms with distinct formulations and design principles, including DDRM (5]. DDNM [6] HIGMM (7.. DMPS 10) DPS 8, DifiPIR (9. ReSample [12), RED-diff [1] and DAPS 14) All
these algorithms are decomposed following the canonical form described in Section 4. with the details provided in Appendix D
We evaluate the performance of all algorithms under total steps of S = 3, 4, 5, 7, 10, 15. Due to space constraints, we report part of the results in the main text and defer the complete results to Appendix H For the training of LLE, we fix w 0.1 and utilize the schedule-free Adam W [45] to optimize the coefficients on 50 diffusion genneratd samples. More training details supplemented in Appendix
Two datasets are used: CelebA-HO 466 and FFHQ [47]. The checkpoint for CelebA-HQ is obtained from [48) and we follow [6] to use the IK test set. The checkpoint for FFH is obtained from R and the evaluation follows 12 14 with 100 randomly selected images from the validation set.
Datasets and diffusion checkpoints.
Inverse problems and metries. We consider five inverse problems in this work: (1) gaussian deblurring (anisotropic), (2) inpainting (50% random), (3) 4x super-resolution (average pooling), (4) compressed sensing (CS) with the Walsh-Hadamard transform (50% ratio), and (5) nonlinear deblurring. Except for nonlinear deblurring, all other four tasks are linear and we apply the decoupled coefficients method proposed in Section 4.3 The primary metrics include peak hignal-to-nokee ratio (PSNR), structural similarity index measure (SSIM) [4). and Learned Perceptual Image Patch Similarity (LPIPS) [4). Results of Fréchet Inception Distance (FID) (50) are also included in the Appendix. All of the experiments are performed on a single NVIDIA RTX 3090 GPU.
5.2 Main results
Table I presents the performance of five algorithms on four noisy linear tasks from the CelebA-HQ dataset with 3 and 5 steps. Table [ shows the results of nonlinear deblurring task on CelebA-HQ and FFHQ under both noiseless and noisy settings with 3 and 10 steps. The LLE method consistently improves the performance across different tasks and algorithms.
Figure 2 illustrates the PSNR and LPIPS metrics for four algorithms on linear tasks from FFHQ with steps varying from 3 to 15. LLE consistently achieves superior objective and perceptual metrics
8

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Table 2: Nonlinear deblurring results of DPS, RED-diff, DiffPIR, ReSample, and DAPS using 3 and 10 steps. The better results are bolded.
PSNRT/SSIMt /LPISI
Steps
Algorithm
Strategy
FFHQ
Celeba-HQ
By 0.05
8y
Ty 0.05
15.75 / 00440 0.644
15.75/0.402
/0.647
16.09 /0.408///////// 0.575
16.09/ 0.407 0.577
DPS
LLE
16.94/ 0.440 0.590
16.93/ 0.440/ 0.590
19.06 0.479 0.574
19.06 0.475 /0.580
15.96 / 0.2002 0.766
15,867 0.119/ 00768
18.9370.384 0,651
18.81 7.3170.35100.63
RED-diff
LLE
16.71 / 0.241 00744
16.61 / 0,223 0.745
20,50/ 0.417 0,610
20.31 0,381 / 006610
20.16/ / 0.358/ 0.637
19.28 / 0,330/ 0.649
24.05 O. ...48// / 0,473
21 69 0.393 / 0.551
DiNPI
LLE
20.56 / 0.436 0.595
19.13 0,300 0.646
24.73 / 0.633/ 0.428
21.70 0.402 / 0.546
20.11 0,41999 0.607
18.64 0.260 0.675
23.37 7 0.5367 0.491
20.75 0.3370033..580 0,5584
ReSample
LLB
20.33/ / 0.426 / 00,044
18.86/ / 0,271 00.63
24.40 / 0,598 0.457
21.03/ 0,353 / 0,577
20.10/ / 0,413 0,590
18,97 0.225 0.642
25,37 00.55 0.397
22.60/ / 0.422 / 0,509
DAPS
LLE
20.21 / 0.424 00.81
19.10/ / 0.291 0.633
25.68 0.677 0.,82
22.55 0.423 / 0.516
20.39 / 0.542 0.447
20,30 0.536/ 0.466
23.76 / 0.675 0.351
23.74 0.673 00,30
DPS
LLE
21.18 / 0.581 /0.401
21.18/ / 0.580 0..01
24.49 0,.70/ 0.302
24.47/ 0.702 / 00.302
19:41 0.367 /..58
19.26 0.330 0.657
22.13 0.522 0.535
21.85/ 0.462 / 00.522
RED-diff
LLE
20.85/ / 0.484 0.602
20.89/ / 0.491 0.578
23.22 0.627 0,479
23,77 0.567 / 00.479
0.148 0.554
23.53 0.5552 /0.421
27.53 0.672 0.366
27.14 00663 00.34
10
DifIIR
LLE
24.27 0.649 00.442
23.58 0.571 00.410
29.98 0.835 0,282
27.44 0.695 0.327
25.35 0.699 00.81
21.61 7 7..348 70.567
29.89 89 0.333 0.237
23.06
0,407 70.507
ReSample
LLE
25,59 0.719 0,378
22.30 0,392 0.550
30,73 0.861/ 0.221
24.06 0.457 00,471
25.56 / 0.699/ 0.345
23.35 0,453/ 00.447
29.54
0.8TO 0,240
25.71
0.544 0.364
DAPS
LLE
25.80/
/ 0.731 0.343
23.37 / 0.453 0.446
30.55 0.849 0.223
25.71 0.547 / 0.357
Dubler (amime) (c, 0)
Inpointing (ry = 0.05) ReSample
4x SR (cy 0.05)
50% (o. 0)
RED diff
DAPS
DDRM
1032 240
-031 usu
PSNR
250
0335
13-
0.45
12.5
.028
12
11
12
14
12
Stops
Stops
Steps
Stops
Original PSNK
-
LLE PSNK
Original LPIPS
LLB LPIPS
Figure 2: PSNR and LPIPS metrics under various steps for four algorithms on the FFHQ linear inverse problems.
under different steps. Some qualitative results are provided in Figure 3 and more visualization of the results is postponed to Appendix ||
A notable observation is that LLE tends to show more significant improvements when the original algorithm performs suboptimally, while achieving comparable or slightly better results when the original algorithm already performs satisfactorily. This aligns with our expectations since LLE searches for better solutions in a linear subspace spanned by previous steps. When an inverse algorithm's 's trajectory is close to optimal, LLE is expected to approximate this original trajectory.
5.3 Ablation studies
In this section, we present the ablation studies of the decoupled coefficients for linear problems and the effect of different w in LLE.. We defer more ablation studies and analysis of the results to Appendix D including the visualizations of the learned coefficients, the training and inference time of LLE, as well as the cross-dataset and cross-task generalization results.
Decoupled coefficients for linear observations. Table 3 compares the performance of LLE on linear tasks using decoupled coefficients versus a single set of coefficients. All the algorithm uses LLE and steps. Decoupled coefficients consistently achieve better results, validating the advantage of handling the null-space and range- space components separately.
Table 3: Ablation of the decoupled coefficients.
Algoritmm
Task Deblur (aniso)
Decoupled
PSNRT / SSIMI 7 LLPPST
DiEPIR
[f2 U)
x
33.05/ / 0.893/ 0.189
52,81 / 0.887/ 0,194
DDNM
Inpainting
20.74 74700 367 70.542
(ay = (.05)
7
22.56 0.424/ 0.501
DDRM
4x SR
0.05)
/
x
25.90 0.736 / 0.311
25.64 0.703/ / 0.342
BIDM
(5,
CS 30%
/
X
18.47 0.567 / 0.523
18.39 0.5537 0.336
9

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IIDMM -LS IIGDM
DPS Onigg
DPS
3 Staps
Sleps
5 Stops
Steps
10 Stops
Steps
Xa
Figure 3: Qualitative comparison of restoration results with and without LLE on the FFHQ dataset. The top two rows present the results of IIGDM on the noiseless compressed sensing task, while the bottom two rows present the results DPS on the noiseless inpainting task.
JEGDM- 4x
DPS- US 60%
Effect of w. Figure 4 presents the PSNR and LPIPS metrics of the LLE method when vary- ing the hyperparameter w in the range of 0 to 0.5. Both tasks are noiseless and the algorithms use 3 steps. Existing works [[]] 13) have ob- served that objective metrics (PSNR) and per- ceptual quality (LPIPS) often trade off against each other in certain tasks. Figure 4 indicates that LLE provides straightforward mechanism to control this balance by adjusting w.
PSNR (3
LPISS
1.71P5 (1)
Origina
11.
Figure 4: PSNR v.s. LPIPS for varying w in LLE (from bottom-left top-right: = 0.5, 0.4, 0.3, 0.2, 0.1, 0.075, 0.05, 0.025, 0.0).
Conclusion
In this paper, we design a unified canonical form for existing diffusion-bassd inverse algorithms and propose the LLE method to enhance the performance of these algorithms under few steps. Extensive experiments validate the effectiveness of LLE across nine different inverse algorithms and five tasks on two datasets. LLE learns improved solvers for diffusion- based inverse algorithms in a lightweight approach, providing novel insights for the design and acceleration of diffusion-brid based inverse algorithms in the future.
Linitations and Future Work. One limitation of the LLEE method is that its search space is confined to the linear subspace spanned by all previous steps. Future work could explore the introduction of nonlinear methods to further improve the algorithm's 'ss performance while maintaining lightweight and universal as LLE. Additionally, the canonical form proposed in this paper offers a flexible framework for combining and designing inverse algorithms. Research could focus on leveraging this framework to design more effective algorithms in the future. Moreover, we note that the CelebA-HQ and FFHQ datasets used in this paper have demographic imbalances, which may cause our results to fail to generalize to underrepresented groups. A more diverse and balanced dataset is needed in the field ensure fairness across factors such as ethnicity, gender. and age. more representative diffusion model trained on such data is also preferred to better support real-world applications.
Acknowledgement
Jiawei Zhang, Ziyuan Liu, Leon Yan, and Yuantao Gu are supported by NSAF (Grant No. U2230201) and Grant from the Guoqiang Institute, Tsinghua University. The authors would also like to express
10

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sincere gratitude to Zhenwei Zhang, Cheng Jin, and Wenyu Wang for their invaluable feedback and suggestions, which significantly improved the quality of this paper. Jiawei Zhang would like to extend special thanks to Jiaxin Zhuang for his continuous efforts in maintaining the availability of our computational resources. His dedication was crucial in ensuring the successful completion of all experiments. This work would not have been published without their support and contributions.
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Appendix
In the appendix, we provide additional discussions and details that are omitted in the main paper due to space limitations. Appendix A reviews existing methods for solving inverse problems with diffusion models under few steps, highlighting how our work is complementary to these approaches and has the potential to be combined with them. We also discuss the relation of LLE to non-stationary solvers and its possible synergy with learnable timesteps. Appendix B presents the derivations of the probability flow ode for posterior sampling, the main results shown in the main text, and also the analysis of solvers for inverse algorithms in the SDE form. Appendix [rrovidees additional experiments on the ImageNet dataset as well as additional nine baselines, further demonstrating the generality of LLE on more challenging tasks as well as its applicability latent-space and consistency model-based methods. Appendix D supplements more analysis and ablation studies omitted in the main text due to the detailed decomposition under the proposed canonical
space constraints. Appendix D elaborates form of all inverse algorithms adopted in our experiments. Appendix Dopendix/Fjitrree inttroduces our tailored design of LLE for DDRM and DDNM on noisy linear problems, which specifically adapted to better align with both algorithms. Appendix describes additional experimental details, including inverse G
problem settings, the hyperparameter configurations of the inverse algorithms, and the optimization methods for LLE training. Appendix Hrroovdds comprehensive results across CelebA-HQ and FFHQ datasets using 3, 4, 5. 7, 10, and 15 steps, covering four or five tasks (depending on the algorithm's ability to handle nonlinear problems) in both noiseless and noisy scenarios. Appendix presents additional qualitative results.
A
More related works
A.1 Other methods for few-step diffusion-based inverse algorithms
Recently, several works I19
have attempted to design algorithms capable of solving inverse problems with diffusion models under few steps. We provide a detailed explanation of these methods and describe their differences from this work.
20 21 22
Chung et al. [9] point out that it is unnecessary to sample from pure Gaussian noise for solving inverse problems. Instead, starting from lower noise level is more efficient by initializing with a noised version of the degraded image. Liu et al. (20) adopt a similar idea and refines it by using DDIM inversion obtain the noisy image from the degraded observation as initialization. Both methods can essentially be seen as utilizing only a short segment of the diffusion sampling chain close to the clean image rather than the full sampling chain. In our work, we default to the use of the complete sampling chain for inverse algorithms. The proposed LLE method can also be applied to these methods, which only requires adjusting the sampling interval of timesteps (s,..., to and using the corresponding initialization methods. Moreover, it can be verified that the algorithms in [19 and 20 can both be decomposed into our proposed canonical form.
Chen et al. [2 introduce deep data consistency (DDC), which employs an additional UNet to learn the data consistency process. We note that this is equivalent to using UNet as the Corrector in our canonical form, making LLE directly compatible with DDC. Chen et al. (21) also observe that existing methods essentially focuses on the data consistency step, our proposed canonical form systematically modularizes the diffusion model-based inverse algorithms into three components, providing more comprehensive framework that fits well with our LLE method.
equivalent to imposing constraints on Xo,t- While (21) primarily
Pandey et al. 22 propose enhancing the efficiency of inverse problem solving by integrating the Conjugate Integrators method from diffusion sampling. They design a specialized Conjugate In- tegrator for IIGDM, named C-IIGDM, which transforms the solving process into space that is more flexible and more friendly to inverse problems, thereby reducing the solver's 's dependency on small step sizes. The solver in C-IIGDM remains first-order, and the concept of LLE can still be applied to introduce higher-order information to further improve its performance under few steps. Additionally, the observation- based correction step in C-IIGDM can also been transformed to process in the original space by using projections between the original and transformed spaces, making well-suited for our canonical form and LLEE method.
In summary, existing few-step diffusion- based inverse algorithms are complementary to the proposed LLE method, and all fit within our canonical form. Combining LLE with these approaches has the
22

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potential to further enhance the performance of inverse algorithms under few steps, and we leave the exploration of these combinations for future work.
A.2 Relation to non-stationary solvers
Recently, formulates non-stationary solvers as general family of iterative schemes where the update at each step is expressed as linear combination of all previous estimates and their corresponding velocity fields (or equivalently, score functions). Let xt, denote the current estimate at timestep E and with the predicted velocity Vt,, then non-stationary solver can be written
9
b Maya
(23)
J=L
where the coefficients {a... bt, } are adaptively determined and can vary across timesteps. From this perspective, our proposed L.EE can be regarded as data-driven realization of non-stationary solvers tailored for diffusion-basee inverse problems. By learning the combination weights directly from data, LLE adapts the solver dynamics to the empirical characteristics of the ideal trajectory.
A.3 Combination with learnable timesteps
Recent works 12 53 54 55 have proposed to improve the performance of diffusion solvers by learning the timestep discretization. Specifically, the timesteps can be parameterized and optimized under the supervision of teacher solver's trajectory obtain an optimal scheduler. We note that this approach and our proposed LLE represent two complementary perspectives for enhancing few-step solvers. Incorporating a learnable scheduler into the LLE framework could potentially lead to further performance gains for diffusion-basee inverse algorithms in the few-step regime. We leave this as promising direction for future work.
B Derivations of the main results
B.1
Probability flow ODE for posterior sampling
The derivation is similar to the probability flow ODE for diffusion sampling without observation in 56 Recall the reverse time SDE for posterior sampling
dx, (t)x; g² (t) xx, log logp(xxly,t) t)) dt + g(t)dW, xT p(xTly, T),
(24)
where dw, denotes the reverse-time Wiener process. Denote T - I, then we can rewrite (24) as
dx, (f(r) Vr) logp(x,ly, dr g(r)dw,
(25)
Its corresponding Fokker-Planck equation follows
@p(xly,7) = Vx ((f((()))=))) (((())x-((r)VV) (og()x/))))x p(x),, 2 Vx (+
(26)
with the fact that the marginal distribution of x at time 7 is p(xly, 7). Note that
Vxp(xly, 7) = p(xly, V log
(27)
Thus 266 becomes
1 (-) 1 log((x)xx)))) 0p(xly,T) Vx (fl=)x-
(28)
which is exactly the Liouville equation for reverse-time probability flow ode
dx = [(())x - 7 g log p(x),,)))) 1) dt.
1
(29)
23

---

B.2 Derivations of Proposition| 1
By definition, we have
Varxollyy) - Xt.
Vx x, logp(x;ly)
(30)
1-0 a4
Thus 66 can be rewritten as
dx
2(1 20001-555000
1
(1)
x,
1 2 (1)
Va=
x)
(31)
dt
The
Given Xt,41 the solution to the semilinear ODE
(31 is known to [57
X1, = C(1,, ((11,144))x4.11 l"d
2( (T) Orr
1
x xo(T,y)dr.
(32)
a+
where
((1,8) = exp (J. (1((116) 2(1 2(1- at) (T()) ) dr)
=exp
(33)
Note that
f ((T)
1
{ 2(1 aar)
d(1 - -a)T dr = log
v - a₁
+
g (P( dr ).
(34)
2(1 (k,)
dr
VI a.
Thus 32 is
0+h (M.xooli.g0 - - Ott;) Vxx, log ((x;,))
I Cte
Votti1Xole xo((i+1,)) - (1 - -tes Vxi,1 log logg(xe,11y)))
abit
pl 1 (T)()
Var
ati Uum 2
(1 -, yo(t,
(35)
Rearrange 35 yields
Xo(li,y) Ot.
The Vxi, logp(x;,>)) - v11 -bii Vxn... log logp(x(..1l))
(36)
VoG
/T -
Ott
vaur xo(li+1.y)
-T-au, hh g var
(1 (1- (1-a,)377 33/2 Xo Xo(T,
Vor
Vau. Sems 2
B.3 The case with SDE
Empirical evidence has shown that the injection of stochastic noise plays a critical role in the effectiveness of diffusion- inverse algorithms. Accordingly, extend our discussion to cover the SDE-based formulation in this section.
Considering the SDE for posterior sampling
24 we may follow 24 to solve the SDE as
Xx, = E(tá, ((12)441))x40) - f"" E(ta, ((),,))9(())
Var
Xo xo(T.y)dr + f" E(tt,))((7ddw. (37)
a
where
E(4,,7) = exp (J. (I(T) + 1- a-
=exp
(T) dr )r).
(38)
We have
1.¹ ((() + T- ( ( - dr = /"
1
1
d(1 - σr))
d dlo var dr
(39)
Tr dr
dr
1
Tr Va,
= log
(40)
ar Vai,
24

---

Then the Itō integral is computed as
E(t,, ()g())), = 1- ar ((th,t41)e, e~,()
(41)
8241
Vāa.
where
x2
(11,1441)
a₂
(d(1 - a+) - 2( - log aar)
(1
- [-a,)
d(1 a+) 1 da,
141
0+)
a+
a+
1
1
(42)
01111
Cre
Thus with 33 one have
Tte
Oft,11
Xo (f
xo xo(t++1,y)
1 01s11 Ct
Otti (" n(T))x(T.y))d 1 Ot
1
1
1t
aie
1 11-08 -11+1
1
14
Cita
(aate art Vxi,
log logp(x;,ly) -
-Vot V X1s41
log pl Dgg((x44111 ly)).
(43)
04
where T(T) (I-a,)* 11= 9 (r).
Similarly, the integral of xo(T. y) may be approximated by the (informal) Taylor expansion, with high-order derivatives estimated from previous-step results, An estimate X₁, can be sampled using standard DDPM update, i.c.,
(1- (1-
Otti4 -1-tt
1 - at, Vati Xtal13 Oc4p1
Cte
Ofte41
)r)
&1. N
X0 +
1
Ott;
041441 Var,
C'ty
(44)
If the sampled noise is exactly the same c in 43 then ! + ② in 43 is
Cr
N1, (1 Ot
Vx., log p X1, ly)
X X0(4++1,8),
(45)
04
Vol,
Otti41
which leads similar conclusion Corollary 2
C Additional results and comparisons
In this section, we present additional experimental results on the ImageNet dataset, as well as the performance of LLE nine additional baseline algorithms.
C.1 Results on ImageNet
Here we supplement the results of IIGDM, RED-diff, DPS, and ReSample on the ImageNet dataset [58] with 4 steps and 10 steps. Following [12 [4]. we sample 100 images from the ImageNet test set for evaluation. For noisy tasks, we add Gaussian noise with
with respect to data range [-1, 1]. The ADM pre-trained checkpoint in 4 is adopted. The results are shown in Table which demonstrates that LLE consistently improves performance across all methods the few-step setting on ImageNet. This confirms that LLE is effective and broadly applicable more complcx datasets.
25

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Table 4: More results of IIGDM, RED-diff, DPS, and ReSample on ImageNet.
PSNR / SSIM 1/LPIPS
Steps
Algorithm
Strutegy
Deblur (noiscless)
Inpainting (noisy)
SR (noiseless)
CS 50% (noisy)
23.56/ / 0,727 70.294
18.75 /0.641
17.4170.37300.57
14.60 0.60//.21700.777
IIGDM
LLE
30.14/ 0.875 0.196
19.41 0,,45 0.619
20.53 00t.//0.244... 0.531
16.73/ 0.359 0.650
22.57 / 0..89/ / 0,490
18.00 0.357 /0.592
24.8170.711 0.419
17.10 0,409 / 0..57
RED-diff
LLE
22.7 / 0.602 / 0.486
21.69 / 0,524 0..92
25.19 / 0.726 0.359
17.77 0.468 / 0.531
15,90 / 0.268 0.681
22.92/ / 0.672 / 00.421
17.10 0.358 / 0,567
15.30 0.300 / 0.633
DPS
LLE
18.42/ / 0,388/ 0.622
24.41/ / 0,690 0.410
20,33 0.517 0.533
18.03 0.474 / 0,556
25.89/ / 0.746 / 0.355
20.49 / 00,440 (0.521
23,18 0.651 / 0.441
17,90 0.439 / 0,550
ReSample
LLE
26.18/ / 0.757 00,340
23.28 / 0.561 / /0.443
24.98 0,712 / 0.369
17,80/ / 0.454 / 0.529
26.22/ / 0.789/ / 0.2899
21.99 / 0.609 /0.487
21.49 / 0.532 / 0.503
16.21 / 0.360 / 0.681
HIGDM
LLE
29.92/ / 0.856 / 002166
23.69 / 0.611 / 00.471
22.76 / 0.596 0.450
18.61 / 0.509 / 0.540
23.41 / 0.620/ / 0,458
22,95 / 0,539 00.449
25.75 2 0.749 0.349
18.49 0.466 / 0,517
RED-diff
LLE
23.42/ / 0.623 /0,422
25.07 / 0.669/ 0,386
25.92 / 0,754/ 0.309
19.22/ / 0.513 / 0,488
10
19.55 0.426/ 0.608
24.65 7 0.711 00.34
21.28 0.5207 0.512
16.61 7 004413 7 70.612
DPS
LLE
20.33/ / 0.463 0.555
27.17 / 0,780/ 0.317
22.42 0.585 0.460
19.52/ 0.590 / 00.633
26.49 0.765 0.331
25,26 0.625 /0,667
25.27 70.721 0.353
19.60 0,5197 70.499
ReSample
LLE
26.72 0072//0.70
0.772/ 0.317
25.78 / 0.651 0.349
25.61 0.736 0.329
20.02/ 0.541 0.444
C.2 Additional baselines
To further validate the general applicability of LLE, we include nine additional inverse algorithms
as baselines, including five pixel-space methods (LGD [59]. DCDP 60), MPGD [6]] FPS 622
and SGS-EDM [63]), three latent-space methods (Latent- DPS 12), PSLD 644, and STSL 665
and one consistency model-based method (CMInversion (66)) For pixel-space methods, we follow
the same validation settings on FFHQ as used in the main content. For latent-space methods, we perform evaluation on the FFHQ-256 dataset using the pretrained LDM checkpoint provided by [67]. For CMInversion, we adopt the pretrained consistency model from [68), which was trained for class-conditional generation on ImageNet-64 with & loss. We randomly sample 100 images from class O ("tench"") in the ImageNet validation set and resize them to 64 x 64 for evaluation. The corresponding results are presented in Table 5 Table 6 and Table 7 The results indicate that LLE consistently improves the performance of all nine baselines in the few-step regime, further confirming its generality and broad applicability.
We now elaborate on how latent diffusion-baasee and consistency model- bosel-based inverse algorithms can be incorporated into the LLE framework. There are no conceptual difficulties in extending the canonical form and LLE from pixel space to latent space. The observation model with LDMs can be formulated as
y - A(D(zo)) on,
(46)
where D(-) denotes the decoder of the VAB, A(-) is the observation operator, and zo is the latent. This corresponds to a nonlinear inverse problem with respect to 2o, since the composed operator A D is highly nonlinear. Thus the theoretical foundation of linear extrapolation in LLE still holds, as it does not rely on specific assumptions about the observation function. The only modification required for training LLE in the latent space is minor adjustment to the loss function (20). which becomes
(m) C < (D((2 (0...) ). D (z0"))
(m)
(47)
since LPIPS is computed in the pixel space.
Regarding consistency model-based methods, though consistency models do not provide explicit
sampling trajectory, they can always be viewed as one-step efficient samplers. Inverse algorithms built upon consistency models (e.g., CMInversion (68)) also follow a paradigm, which naturally fits into our canonical form and can therefore benefit from the LLE framework well.
comrectionnnoiiinng
The detailed decomposition of all nine algorithms above into our canonical form supplemented in Appendix E
26

---

Table 5: Additional results of DCDP, LGD, MPGD. SGS-EDM, and FPS-SMC on FFHQ with steps.
PSNR1 / SSIM 1 //PPPS
Steps
Algorithm
Strategy
Deblur (moisy)
Inpainting (moisy)
4x SR (moisy)
CS 50% (noisy)
26.00 / 0.619 /0428
24.75 0.558 00436
24.16 0.555 /0.778
17.70/0.461 /0.535
DCDP
LLE
26.12 / 0.681 00378
25.54/ 0.396 / 00,41
25.24 / 0.707 / 0.350
18.16 / 0.471 0.524
22.81 / 0,645 00.43
21.77 0,548 / 00513
24.95 / 0.680 / 0,439
15.84 70.451 0,513
LGD
LLE
24.59/ / 0,695 0,376
28.25 0.768 / 00,39
25.55 0.705 / 0,402
16.52 / 0.503 0.499
25,29 0,705 /0.380
27.33 0.753 0,309
24.93 0.562/ 0.579
17.36 /0.437 /0.565
MPGD
LLE
25.43 / 0,704/ 0.366
27.76 0.768 / 00,298
25.42/ 0.703 0.360
18.02 0.446 0.543
27.72 / 0,776 0.308
19.71 / 0,416 / 0,538
26.53 / 0.738 0,333
17.72 / 0.482 0.530
SGS-EDM
LLE
27.67 / 0,782/ / 0,307
24.44 0.511 / 00,453
26.95 / 0.762/ / 0,309
18,40 / 0,511 0,486
25.11 0.517 /0.475
17.52/ 0.188 0.686
24.62 0.524/ 0.474
16.58 0 0.224/ 0,666
FPS
LLE
26.26 / 0.617 0.428
18.63 0.229 / 0.643
26.14 0.672/ 0.400
17.66 / 0.384/ 0.569
Table 6: Additional results of Latent- PSLD, and STSL on FFHQ with 4 steps.
PSNR t /SIM 1/LPIPS 1
Steps
Algorithm
Strategy
Deblur (noisy)
Inpainting (noisy)
4x SR (noisy)
CS 50% (noisy)
16.42 / 0.428 / 0.633
15.43 / 0.442/ 0,628
15.30/ 0,369 / 0.682
14,51 /0,364 / 0.665
Latent-DPS
LLE
17,08 0,441 / 0.609
17.83 0,492 0.652
15.49 / 0.399 0.648
14.70 / 0,366 0.646
17.55 0.468 0.630
17.71 /0.47// 0.627
17.72/ 0.471 0.630
15.06 00.438 0.647
PSLD
LLE
18.10 0.477 / 0.607
18.55 0.490 / 0.599
18.57 / 0.489/ 0.597
15.24 / 0..33/ 0.640
16.08 0,,78/ / 0.655
16.87 / /0.4444000.33
15.70/ 0.368 0.663
15.31 / 7,383/ 0.629
STSL
LLE
16.97 0,392 / 0.616
17.50 / 0.407 / 0,598
16.27 / 0.360 0.628
15.72 / 0,384 0.621
D More ablation studies and additional analysis
In this section, we supplement the ablation studies and discussion that are omitted in the main text due to space constraints.
D.1 Visualization of the LLE coefficients
Here, we present and analyze the visualization of the learned coefficients from several algorithms trained on the CelebA-HQ dataset. Figure Figure Figure 7 and correspond to DDNM on noisy inpainting, IGGDM on noisy compressive sensing, DPS on noiseless super-resolution, and DiffPIR noiseless anisotropic deblurring, respectively. All visualizations are based on 7-step inference. The vertical axis indicates the current step in the inverse process, and each row represents the linear combination weights applied to all previous steps. The left subfigure shows the coefficients for range space, while the right subfigure shows the coefficients for null space.
Different algorithms yield distinct patterns of the learned linear combination coefficients, which is attributed to their underlying different architectures and designs. At the same time, several shared characteristics emerge across methods, aligning well with the design philosophy of our framework.
First, observe clear difference between the range space and null space coefficients. This supports our hypothesis that learning separate coefficients allows the algorithm to better capture the varying strengths of the observation constraints the range and null spaces. Notably in DDNM, the range space coefficients are almost entirely concentrated along the diagonal with values close to one, while off-diagonal elements remain near zero. This is consistent with the design philosophy of DDNM: DDNM's correction step explicitly projects the sample onto the hyperplane defined by the observation, thereby enforcing a hard constraint in the range space while preserving greater flexibility the null space. Some non-zero off diagona coefficients in the range space are attributed to the observation noise. The other three methods do not employ explicit projection operations. Consequently, these methods impose relatively less stringent constraints on the range space compared to DDNM, which may result in estimation discrepancies that L.LE could compensate for by learning a more effective linear combination.
27

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Table Additional results of CMInversion on ImageNet 64 64 with steps.
/SIM L/PIPS
Steps
Algorithm
Strategy
Deblur (noisy)
Inpainting (noisy)
4x SR (noisy)
CS 50% (noisy)
12.69 / 0,204 70.613
12:71 0,29/ 0.649
14.65/0.311 /0,559
14,80 0.242 / 0.619
CMlaversion
LLE
16.50 0.283 / 0.522
17.64 0.383 0.566
18.57 77.41000.477
16.58 0.268 0.580
Range Coefficients
Null Coefficients (1)
Xoea
Xayz Xoea
10.000
1.00
-0.17
0.98
Current Step
X514 x425 P5te X110
0.00
-0.00
1.00
Current Step
-0.13
0.09
1.17
0.8
-0.00
1.00
-0.00
0.00
-0.00
-0.00
1.00
0.01
-0.00
-0.16
0.05
1.12
0.4
-0.00
0.00
-0.00
-0.00
0.00
1.00
Xyu
0.01
-0.04
-0.03
0.11
0.96
Xo xy0
0.05
0.05
0.05
0.05
0.05
0.05
0.73
0.00
0.00
-0.02
0.05
0.13
0.12
0.0
0.03
0.03
0.03
0.03
0.03
0.03
-0.11
0.91
Xo
0.00
.0.01
0.02
.0.06
.0.02
0.03
00.09
1.17
X994
Xa52
X110
#iss
X426
K2ss
X542
X934
Xe52
Xr1n
*sss
X234
X142
X0
Referenced Step
Referenced Step
Figure 5: Visualization of learned coefficients of DDNM the noisy inpainting task.
Second, we observe that the learned coefficients are not always dominated by diagonal elements. For instance, in IIGM at step 770 and DiffPIR at steps 142 and O, the model assigns larger weights to earlier steps rather than the current one. This suggests that LLE may, in certain cases, favor earlier estimates over the most recent one. Such behavior reveals an implicit form of regularization, akin to "carly stopping", within the dynamics of LLE, echoing similar strategy reported in prior studies [69]. This demonstrates LLE's ability to dynamically assess the reliability predictions and turn to past estimates when appropriate, thereby enhancing robustness.
Moreover, a consistent pattern across different inverse algorithms that the LLE coefficients tend to concentrate around the diagonal and its immediate neighborhood. This suggests that more recent steps exert stronger influence, while earlier steps contribute less as their estimates become increasingly outdated. According to Equation (TO). elements farther from the diagonal correspond to higher-order derivatives of the sampling trajectory. Since most sampling trajectories are expected to be smooth, the high-order derivatives gradually decay to zero the order grows, which results in the weight matrix being dominated by the diagonal and first off-diagonal elements. Such behavior is also intuitive, since earlier predictions are typically less accurate and thus less informative for refining later outputs. Building on this insight, one possible improvement to introduce a limited bock-back window for LLE, which could reduce both computational and memory overhead during training and inference. This modification may further improve the scalability of LLE, particularly in inverse problems involving longer sampling trajectories. We leave deeper investigation future work.
D.2 Cross-task and cross-dataset generalization
Although conceptually LLE requires training with the same prior distribution and inverse task as inference, we also evaluate its generalizability across datasets and tasks. Table S reports LLE's performance with cross-dataset training between FFHQ and CelebA-HQ. Table shows the cross- dataset results between FFHQ and ImageNet, and Table 10 presents performance with corss-task training.
We observe that LLE trained on one dataset generalizes well to another, even the more challenging ImageNettto-FFiit cross-domain setting. Cross-dataset training achieves comparable performance to training on the matching dataset, and occasionally even surpasses it. This indicates that L.E possesses certain degree of cross-dataset generalization capability.
28

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Range Coefficients (I)
Null Coefficients (1)
1.2
1.2
Xpo
0..04
0.79
1.0
Xesz
0.01
0.78
1.0
Current Step
KLD Xien Kax Xraa xy1s
0.01
0.77
0.16
Current Step
-0.00
0.77
0.09
0.8
0.01
0.05
-0.09
0.6
0.01
-0.06
-0.07
0.09
0.6
-0.00
-0.04
0.07
0.13
0:81
Xy4 Xoes Xan
0.01
-0.03
0.03
-0.02
0.92
0.4
0.2
0.2
-0.00
-0.00
0.00
-0.02
-0.10
-0.08
1.20
0.00
-0.04
0.03
-0.07
-0.17
0.07
1.17
0.0
-0.0
Xo
-0.00
-0.00
0.00
0..01
0.02
00.21
0.10
1.10
0.00
-0.04
0.05
-0.01
0.01
-0.99
0.14
1.14
-00
X710
X126
X2L0
X508
X384
Referenced Step
Referenced Step
Figure 6: Visualization of learned coefficients of IIGDM on the noisy compressive sensing task.
Range Coefficients (I)
Null Coefficients (1)
X001
Aggz
-0.03
0.78
1.0
Xnaa Xys
-0.00
0.77
Current Step
X584 X:35 x5ee X11
0.02
0.02
0.78
0.8
Current Step
-0.00
0.00
0.77
0.8
0.08
0.00
0.00
0.00
0.05
0.06
0.81
Xou X:s
0.00
-0.01
0.03
0.10
0.78
0.4
-0.00
-0.02
0..00
-0.01
-0.00
0.87
0.2
0.22
Ne xya
0.00
-0.00
-0.00
-0.00
-0.04
-0.15
1.19
-0.00
0.00
-0.01
-0.02
-0.07
-0.20
1.17
0.0
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.14
1.13
0.00
0.00
0.00
0.01
0.03
0.02
0.06
1.08
-0.
Ma52
X110
Kisss
X426
X284
x242
X094
XV10
K568
X234
X0
Referenced Step
Referenced Step
Figure 7: Visualization of learned coefficients of DPS the noiseless super-resolution task.
When the training and testing tasks differ, LLE performance generally degrades. However, some cases, the cross-task trained LLE coefficients still outperform the original baseline without LLE, For instance, LLE trained on super-resolution or compressive sensing can improve performance inpainting. This suggests that certain inverse problems may share similar underlying structures. potentially leading to similar optimal coefficient patterns. Future work could further explore these shared structures enhance LLE's generalization across tasks.
D.3 Sensitivity to the hyperparameters of original algorithms
It is well known that some inverse algorithms require careful tuning of hyperparameters for each task and dataset, for example, the step size Ç in DPS and the regularization weight À in RED-diff. Small perturbations in these hyperparameters
often lead to significant performance degradation.
We note that the linear combination mechanism LLE can be interpreted as equivalently involving learnable scaling factor, which may reduce the sensitivity of the original algorithm to its hyperparam- eters. We conduct an experiment analyzing the performance of DPS and RED-diff with and without LLE under varying hyperparameter settings on the noiseless compressed sensing task from FFHQ, with hyperparameter or ranging from 0.2 to 0.8. The performance is shown in Figure 9 The results demonstrate that LLE mitigates the influence of hyperparameter variations and consistently achieves better performance, indicating that LLE can potentially reduce the reliance on precise hyperparameter tuning.
29

---

Range Coefficients (I)
Null Coefficients (1)
xpp4
1.0
1.0
0.05
0.95
X5os Xaz xsaa xt1 Xpo
0.08
0.79
Current Step
Xn2
0.00
0.86
Current Step
0.00
0.08
0.80
0.8
-0.00
0.05
0.14
0.81
0.00
0.05
0.11
0.79
Xpaa Xis
-0.00
0.02
0.08
0.11
0.79
0.4
0.00
0.03
0.09
0.08
0.78
0.4
0.2
0.2
0.00
0.00
0.01
0.04
-0.17
0.90
0.21
10 xtc
0.00
0.01
0.03
0.06
-0.19
0.86
0.19
-0.00
0.00
0.01
-0.05
00.18
0.09
1.15
0.17
0.00
0.00
0.01
0.04
-0.18
-0.06
1.16
0.10
2563
X284
Referenced Step
Referenced Step
Figure 8: Visualization of learned coefficients of DifIPI on the noiseless anisotropic deblurring task.
Table 8: Cross-dataset generalization results of LLE between CelebA-HQ and FFHQ on the noiseless tasks. Testset denotes the dataset used for inference, while Trainsel indicates the dataset used to train the LLE coefficients. We also include results without LLE for comparison, denoted by "_ in the Trainset column.
PSNR1 SSIM /LPIPS
Testset
Trainset
Deblur (aniso)
Inpainting
4x SR
CS 50%
39.99 00.922 0.090
27.83 0.755 0.332
30.97 0.877/ 0.188
18.98 0.619 0.450
CelebA-HQ
CelehA-HQ
40.20 0.964 0.087
30.93 0.887 0.221
30.90 0.874 0.182
20.15 0.640 0.415
FFHQ
40.28 / 0.965 0.086
30,70 / 0.883 0.226
30.89 0.874 0,182
20.13 0,643 / 0,413
39.82 0.964 0.087
26.34 0.719 0,380
29.99 0,865 0.210
18.29
0.488
FFHQ
FFHQ
40.08 / 0.966 0.084
28.71 / 0.838 0.292
29.94 0.863/ 0.206
19.21 0.605 0.463
CelehA-HQ
40.00 / 0.965 0.085
28.84 / 0,838 0.292
29.92 0.863 / 0.206
19.19 0.599 0.466
D.4
Training cost, inference time, and momery footprint
Table presents the training time of LLE for DDNM, DPS. DifiPIR, and DAPS noiseless inpainting tasks from CelebA-HQ. LLE requires only 2 to 20 minutes for training, depending on the task, the complexity of the original algorithm, and the number of steps. This demonstrates the lightweight advantage of our LLE method.
We also compare the inference time of each algorithm with and without the LLE module. Table 12 presents the inference time on the CelebA-HQ I test set for the inpainting task. The results indicates that incorporating LLE introduces negligible additional computational overhead during inference. The efficiency of LLE stems from the simplicity of its operation, which only involves computing a linear combination of previous samples without introducing any complex transformations.
Regarding the memory footprint, here we further report the GPU memory usage during inference (in MB) of DDNM and DPS with and without L.EE in Table 13 We measure the CUDA peak allocated memory, and we account only for the memory used by model parameters, data, and additional storage required by the algorithm. The results show that LLE introduces only few several dozen MB of additional memory overhead, even under the 15-step setting. This consistent with our expectations: during inference, LLE only needs to store a few 256 x 256 RGB images from previous steps, typically around 0.7 MB per image, which is negligible compared to model weights or other algorithm-specific overhead (e.g., back- paek-propagttion in DPS).
D.5 Analysis with default steps
Though our primary focus is on performance under limited steps, we also present the 100-step results of DDNM on CelebA-HQ in Table [4 reference. These results further indicate that LLE
30

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Table Cross-dataset generalization results of LLE between FFHQ and ImageNet on noiseless tasks. Train task denotes the task used to train the LLE coefficients. We also include results without LLE for comparison, indicated by "-" in the Train task column.
PSNR1 / SSIM L/PPSS L
Testset
Trainset
Deblur (aniso)
Inpainting
4x SR
CS 50%
39 82 / 0.9 0,964/ 0.087
26.34 0.719 0,380
29.99
0,865 00.210
18.29 00,59 / 0.488
FFHQ
FFHQ
40.08 0.966 0.084
28.71 0,838 0.292
29.94 0.863 0.206
19.21 0.605 0.463
ImageNet
39.94 0.965 00006
28.30 00.18/ 00.317
29.94 0.863 00.008
19.01 /0.608 0.470
36.48 / 0,949 0.099
23.35 0.628 0,416
25.74
0.754 0.302
17.86 0.535 0.486
ImageNet
ImageNet
36,52 0.949 /0.098
24.87 0..22 0.364
25.76 0.752 0.297
18.35 0,541 0.478
FFHQ
36,53 0.949 0,099
25.14 0,759 0.341
25.70/ / 0.750 / 0.296
18.20 / 0.527 0.485
Table 10: Cross-task generalization results of LLE on noiseless tasks from FFHQ Train task denotes the task used to train the LLE coefficients. We also include results without LLE for comparison, indicated by "_" in the Train task column.
PSNR1 SSIM t LPIPS 1
Train task
Deblur (aniso)
Inpainting
4x SR
CS 50%
Deblur
40.08 / 0.966 / 0.084
24.53 / 0.655 0.432
29.90 / 0.862 0.231
18.06 / 0.578 / 0.509
Inpainting
39.04 / 0.960 / 0.094
28.71 / 0.838 0.292
29.56 / 0.854/ 0.217
19.07 / 0.602 / 0.467
4x SR
39.32/ / 0.959 / 0.095
27.70/ 0.774 0.339
29.94 / 0.863/ / 0,206
18.73 / 0.602 / 0,475
CS 50%
38.89 / 0.957 / 0.098
27.90 0.799 0.324
29.46 / 0.854/ 0.213
19.21 / 0.605 / 0.463
39.82 / 0.964 0.087
26.34 0.719 0.380
29.99 0.865 0.210
18.29 0.591 0.488
is particularly effective when there is a large performance gap between the few-step setting and the optimal setting. For example, in the inpainting and compressive sensing tasks, the difference between the few-step performance and the default-step (100 steps) performance is substantial, thus LLE provides significant improvements, with PSNR gains ranging from 1dB 8dB. In contrast, for deblurring and super-resolution tasks, the performance gap between few steps and 100 steps is relatively limited, and correspondingly, LLE yields smaller improvements. This is expected as LLE is constrained to searching within the linear combinations of previous predictions; thus, when the few-step trajectory already approximates the optimal performance, LLE tends learn trajectory similar to the original one.
D.6 Performance of LLE with more steps
Here we supplement an additional experiment comparing DDNM with and without LLE under a larger number of sampling steps (25, 50, and 100) on the FFHQ dataset, as presented in Table [5] To avoid the need to store an excessive number of past results when the number of steps is large, we limit the lookback window 10 steps. The results show that as the number of steps increases, the performance gap between methods with and without LLE becomes smaller. This is consistent with our expectation as LLE searches for an improved sampling trajectory within the linear subspace spanned by previous samples. When the original algorithm takes more steps, the discretization error reduced, and its trajectory already closely approximates the optimal one. In such cases, LLE tends to follow the original trajectory more closely, resulting in smaller gaps. Therefore, LLE brings more significant improvements few-step regimes, which aligns with our primary objective of enhancing the performance of inverse algorithms under a limited number of steps.
D.7 Effect of Optimization Iterations on LLE Performance
To assess how sensitive LLE is to the optimization of Eguations (19), we further investigate its conver- gence behavior. We optimize LLE for (25, 50, 100, 150, 200} iterations and report the corresponding performance on the FFHQ noiseless task using the 5-step DDNM. The results are summarized in T6 We find that LLE achieves consistently strong and comparable results across a wide range of iterations, indicating that it converges quickly and shows no significant overfitting even with extended
31

---

DPS
RED-diff
17.5
18.5
0.700
0.675
PSNR (1)
17.5
0.65
-0650
16.5
17.0
-0.625
PPPS
0.60
16.5
16.0
0.600
16.0
0.555
15.5
-0.555
0.550
15.0
0.50
15.0
-0.25
0.2
0.5
0.7
0.8
0.3
0.5
0.7
Original PSNR
LLE PSNR
Original LPIPS
LLE LPIPS
Figure 9: PSNR and LPIPS for varying DPS and in RED-diff noiseless compressed sensing
task on FFHQ using 3 steps.
Table 11: Training time (minutes) of LLE for noiseless inpainting task on CelebA-HQ under different
steps. Experiments are performed single NVIDIA RTX 3090 GPU.
Algorithm
Steps
Steps
5 Steps
Steps
10 Steps
15 Steps
DDNM
2.0
2.7
3.4
5.4
6.7
10.9
DPS
2.2
2.9
3.6
5.8
7.3
11.8
DiffPIR
2.2
2.9
3.7
5.9
7.3
11.9
DAPS
2.6
3.7
4.7
7.1
8.6
14.3
training. Theoretically, this behavior is expected because LLE learns interpolation coefficients by minimizing a weighted sum of the MSE loss and the LPIPS loss. When the LPIPS weight set to zero, the problem reduces to a standard least squares formulation, which is convex and therefore free from local minima. Although the LPIPS loss introduces non-convexity, sufficiently small such that the overall loss landscape remains smooth and easy to optimize, These findings confirm that LLE's optimization process stable and computationally efficient practice.
weight is chosen to be
D.8 Diversity of the solution
conduct an additional visualization experiment comparing the diversity of IIGDM and DPS with and without the proposed LLE. As shown in Figure for typical image restoration tasks, the posterior distribution is generally narrow, meaning that the plausible solutions tend to concentrate around the ground truth. Consequently, the ideal output should remain close to the reference image rather than exhibiting large variations. LLE effectively stabilizes the output and maintains high reconstruction quality.
Detailed decomposition of the inverse algorithms
Here, we present the specific forms of DDRM [5]. DDNM 6), DPS 8, JIGDM 7 RED diff
DiffPIR 9. DMPS [10), ReSample [2], DAPS [14], DCDP 60 LGD 59 MPGD
61 FPS
62 SGS-EDM
63
Latent-DPS [1], PSLD (4), STSL 655, and CMInversion 68 within the
canonical form proposed in this paper. For DDNM, DDRM, DMPS, IIGDM, FPS, and PSLD, we
consider linear observations,
Ax + nyn, n n~~N(0 I).
(48)
For the other algorithms, we consider general observations
y A(xo) + toyn n 10.1).
(49)
32

---

Table 12: Inference time (minutes) of LLE for noiseless inpainting task on the Ik test set of CelebA- HQ. Experiments are performed on single NVIDIA RTX 3090 GPU.
Algorithm
Strategy
3 Steps
4 Steps
Steps
7 Steps
10 Steps
15 Steps
2.0
2.66
3.3
5.24
6.57
10.47
DDNM
LLE
2.0
2.66
3.34
5.35
6.68
10.74
5.07
6.76
8.48
13.46
16.75
26.92
DPS
LLE
5.13
6.84
8.49
13.61
16.96
27.21
5.18
7.14
8.66
14.27
17.54
28.18
DiAIIR
LLE
5.45
7.14
8.76
14.35
17.79
28.31
13.2
17.29
20.57
36.7
43.58
73.36
DAPS
LLE
13.3
17.53
21.14
36.88
43.66
76.02
Table 13: GPU peak memory (MB) of DDNM and DPS with and without LLE on the noiseless inpainting task on FFHQ
Algorithm
Strategy
5
10
15
590.25
590.25
590.25
590.25
590.25
590.25
DDNM
LLE
597.50
599.37
601.50
608.25
612.75
626.25
2388.78
2388.78
2388.78
2388.78
2388.78
2388.78
DPS
LLE
2396.41
2398.41
2400.41
2407.29
2411.79
2429.04
E.1
DDRM
The Sampler of DDRM is a single-step DDIM, i.e., tweedie's formula, which is
+ 1 (x.,)M(())) X1, Tie (x,,,
(50)
Voa
Consider A as a diagonal matrix and 8k as the k-th singular value. A general matrix A can be equivalently transformed into diagonal matrix using Singular Value Decomposition. The correction process is defined clement-wist wise as
x
0, /1 at.1 5
Xu.t. + /1 1/ / 1- y¹ xod,
Dr
h DDRM (Xo.., (Xo.r A,y)
(1 - m/) )x)x, + thy
Cle-1
(51)
The Noiser combines noise addition which is similar to DDIM sampling:
Ctt D..e + /1 1/2 V ar
E (x
(i)
8k
0,
&DDDRMM WE..DRM(SOO.. (Ro.1;)
1 + 17 -
8k
ak Dit
+ Ctim
Te 07
2
o7
8k
(52)
where NN( I), 7) and 77 are hyperparameters. x" denotes the k-th element of the vector x, and the same notation applies to other vectors.
E.2 DDNM
DDNM is similar DDRM, where the Sampler is the single-step DDIM (50) Considering A diagonal matrix, the Corrector is a modified projection operator as
hy. (Xa A,y) = Xo,t. + Σ. A (y Axo,t:)
(53)
33

---

Table 14: Comparison of DDNM with few-steps and default steps CelebA-HQ Dataset.
PSNRT /SSIMT /IPISS
Condition
Steps
Strategy
Deblur (aniso)
Inpainting
SR
CS 50%
27.80 / 0.758 00319
16.64 0.442/ 0.492
27.09/ 0..73 / 0.29
16.55 / 0,442 0.539
3
LLB
28.08 / 0.784 0.291
24.38 / 0.552 0.433
37.84/0.7700 / /0299
17.29 / 0,,47 0.520
28.98 / 0.795 0.285
20.27 0,510/ 0,457
28.35 0.803 / 0.276 28.65 0.792 / 0.270
17.39 / 00444 / 0,511
LLE
29.27 / 0.817 00256
25.29 0.592/ 0.407
18.43 / 0.499 0.474
29.63 / 00.81 0.259
22.76 0.550/ 0.431
28.97 0.818 0.262
18. / /0444/ 0.491
5
By
0.05
LLB
29.82/ 0.831 /0.399
26.35/ 0.659/ 0.366
29.02/ 0.806 0.252
19.41/ / 0.536 0.441
30.19/ / 0.838/ 00.33
25.93 0.627 00,88
29.52/ 0.833 0.222
20.28 / 0.554 0,434
7
LLE
30.29/ / 0.841 00.255
27.69/ / 0.719 0.328
29.33 / 0.818 / 0.233
21.32/ / 0,608 0,385
30.45 / 0.845 0.227
28.19 0,716/ 0,333
29.07 0.821 / /0.241
21.30/ / 0.636 0,379
10
LLB
30.50/ / 0.845 0.214
29.13 0.789 0.279
29.08 0.808 / 0.2235
22.42/ 0.674/ 0.341
30.50/ / 0.846 0.217
30.57 0. 821 0.259
29.39/ 0.829 0.222
23.39 / 0.744 0.296
15
LLE
30.52/ 0.846 0.206
30.74/ 0.854/ 0.228
29.17 0.812 / 0.227
24.04 / 0.769 0.271
100
29.69 / 0.829/ / 0.203
33.06 / 0.904/ / 0.171
29.21 0.823 /0.219
30.42 / 0.882 0.13
Table 15: Performance of DDNM with and without LLE under 25, 50, and 100 steps on FFHQ.
PSNR /SSIM //PPISS
Condition
Steps
Strategy
Deblur (aniso)
Inpainting
4x SR
CS 50%
41.14/ 0.972 0.064
34.33/ 0,941 0.109
30.92/
0.885 0,146
25.01 0.837 / 0.220
25
LLE
41.35 0.974/ 0.061
34.23/ 0.941 0.110
30.93/ 0.885 00147
25.61 0.841 / 0.215
41.57 00.74 / 0.056
35.48/ / 0.953 0.077
31.14/ 0.889 00.12
28.94/ 0.897 / 00.158
By
50
LLE
41.71 / 0,976 / 0.005
35.33 / 0,952 / 0.079
31.15 / 0.889 0,142
28.81 0.897 / 00.158
41.70/0.974 00005
36.29 / 0.960/ 0.058
31.41 0.883 /.114
32.15 0.927 / 0.122
100
LLE
41.86/ 0.976 0.055
36. 0,959 0,060
31.37/ 0,883 0.139
31.56/ 0.927 0.122
where E,, = diag (1, ...., An } is diagonal matrix with elements defined as
1,
b
1-
(54)
8/
8k 0.
The Noiser is defined element as
0t._, X + VI n/ cx(x) 12) not, c, Sk 0,
T DDNM (20,1)) (Xo,))
T the d + no
0
Dt, By
Sk
Thu xt. + Vol
6
01 1
/rt_1 By
o
(55)
where E ~ (0,1)) and of, = VI-aa
E.3 DPS
We rewrite the update formula of DPS as
X Vou. [xo,1 Gu./oo Vx., ly A(x,xx))))11)) cie cxco(xi, 44)
(56)
where No.c, is the estimation using Tweedie's formula as (50). Thus the Sampler of DPS is single-step DDIM. The Corrector is
DPs. A, = X0.6
Vin. Se V x1, ly - A(xo,,,))
(57)
where Go is the learning rate hyperparameter. The Noiser is exactly the same as DDIM sampling
V Dr.s..(
V Sun, + tie + (260 (z),
(58)
34

---

Table 16: Effect optimization iterations on LLE performance.
Train steps
Deblur (noiseless)
Inp (noiseless)
(noiseless)
CS (noiseless)
39.82 0,964 / 0.087
26,34 / 0,719 0,380
29.99 / 0,865 / 0.211
18.29 0.591 / 00.487
25
39.93 / 0,965 0.086
28.21 0.821 / 0.316
29.93 / 0,866 / 0.207
18.87 0,597 0,4476
50
40.12/ 0.967 0.085
28.61 00834/ / 0.303
29.95 / 0.863 0.208
19.13 / 0.602 / 0,468
100
40.08 0.966 0.084
28.71 0.838 / 0.293
29.94 / 0.863/ 0.206
19.21 0.605 / 0,443
150
40.10/ 0.966 0.084
28.77 0.840 0.290
29,91 0.862 0.204
19.23 0.603 00.461
200
40.14/ 0.967 0.084
28.82 0.842 / 0.288
29.93 0.863 0.204
19.25 0.604 / 0.462
IIGDM HIDDM OOig.
DPS Orig.
DPS
Sample
Sample
Sample
Sample
Sample
Sample
Figure 10: Visualization of reconstruction results under different random seeds.
where
C1 11
at
atc
Co C:,_
cj:
(59)
with 1) as a hyperparameter and (0, I). Similar transformation is applicable to all posterior sampling approaches, including IIGDM and DMPS. The derivation is omitted hereafter.
E.4
JIGDM
The Sampler of ITGDM is single-step DDIM as 50). The Corrector is
T
2 I Th:
(G.. A,y) = Xo.t, + +V /a.//r... -y- Axo,,)
AAT
A
OxL:
(60)
where Pt, -v--a... The Noiser is the same DDIM sampling (58)
at.-
E.5 RED-diff
The Sampler of RED-diff is a single-step DDIM as (50). RED-diff differs from other algorithms in that it uses an optimization process to update Xo. Here we consider only the gradient descent update, where the Corrector RED-diff equivalently given by
b RED RED air , A.y) &0,tum + (x0,.. 80,1047
AV Wintttt |y A (80.) (61)
35

---

where ç is the learning rate, and À controls the trade-off between the prior and likelihood terms. However, this formulation fails reconstruct reasonable results under few steps in our experiments. We find the issue lies in the likelihood term, so we adjust it as follows to improve its performance unders few steps
(x, y) 80.141 + E (o.t; 80,4+11
AV Xo.d yy - -xxx,)))l))
(62)
where we use the Xu,t; to calculate the loss for the likelihood term rather than X0,011- The Noiser of RED-diff is directly noise addition as
(r,,RDD-a = =Vor Vou._, Ko,e + vI VI-aun Tenic c~ c~,().
(63)
E.6 DiAIIR
The Sampler of DiftPIR is a single-step DDIM as 50 The Corrector solves a proximal point problem
Di, (x0,e, y) arg min y A(x)) - ||x - Xo,e, 11
(64)
where Pr = / (1 [e) and À is hyperparameter. The Noiser of DifIPR is modified version of (58
V 4. DiMPIR (80...) Var__, So, + +1/1 0i.-, 6 v1 1/2 11-10--- 04-1
Xt. V Ciey Ro.11
V1
Ctty
(65)
where they calculate an effective ég (x,, using the corrected sample 80,4, and ~ N((,I).
E.7 DMPS
The Sampler of DMPS is the single-step DDIM (50).
Consider the SVD of the observation matrix
A = UEVT, then the Corrector
h DMPS (M....X............. y, -0,11
at.
Vx., logp(y(>x,),
(66)
Var.
where 0tt Tt / , and
V Xx, log ((y)))))
VE
I
1 -
1-Ti
Tt 5
Uhy- 1 VVXX))
(67)
Var.
Ct.
Val,
The Noiser is as 58
E.8 ReSample
ReSample 12 is originally designed for latent diffusion, thus we set the Encoder and Decoder in ReSample to identity mappings for pixel diffusion models. The Sampler of ReSample is the single-step DDIM as (50). The Corrector an optimization process which termed hard data
consistency as
ReSammle (ReSample(Xo.t:: A,y) = arg min ||y - A(x)
(68)
with the initial point of the optimization algorithm set as init Xu, The Noiser of ReSample is the ReSample method, which is
oi at. So,t. + -āt- X
of (1 a(,-.)
E C~ I),
T + 1 -Ot;
2 +
(69)
where
07 = 11 11
Tt
(70)
Crt,
Cre
and
x, v a1, X0.1, + CiE + Czc0 (Xi, tt).
(71)
with
C1 = "/1
Rz
1-a 1 1-30
De.,, and 02 1- or. 7 is hyperparameter.
p,,-1
36

---

E.9 DAPS
The Sampler of DAPS is k-step DDIM sampler, i.e.
DAPS = DDIM,
(72)
The Corrector follows Langevin dynamics process, with the iteration given by
0.t. T/s, V x6.1, (1rGa x0.1, /2/ + ((x.)).)) - -x12/203) + Ej
s+1 O.
(73)
where E, ~ (O, I), and rt, is heuristics hyperparameter, which can be set as rt, vI ar, The Noiser is direct noise addition as (63)
E.10 DCDP
The Sampler of DCDP is either single-step DDIM as (50) or k-step DDIM as (72). The Corrector starts from &o,t, and performs fixed number of gradient descent steps to minimize ||A(xx)) yl The Noiser direct noise addition as (63)
E.11
LGD
The Sampler of LGD a single-step DDIM as 50 The Corrector is
n
exp (-ly (xl11)
h LGD (X y) X0,4;
V Cr, Vx log
(74)
11
where St the step-size, x(i) q(xo|xx,)), ). and lx(x) lly A(x) 1|2 g(xo|xx,) can be approxi-
mated as N N(xo,,, r?. I) where ri, = 0 / + of with
-4::11
a2
of 1 0t
(75)
at++1
The Noiser as 158
E.12 MPGD
We only consider MPGD without manifold projection. The Sampler is single-step DDIM as The Corrector is
550p
h br. -x0.t. - Se, Vxa.,, lly A(xo,,.))
(76)
The Noiser is as 58
E.13 FPS
For simplicity, we only consider FPS with particle size The Sampler is single-step DDIM as (50).
The Corrector is
0t Xo,t; + of "or... A7x)
Fes A, = Var.
PPPS t
(77)
c(1 am.,,)
where
EPS
obart A ATA)
(78)
2
c(1 0t:.1)
and e the DDIM hyperparameter. The Noiser is
(v- co(x,,,t)))) + ozāa..
L Pr,
Va,,, So.s.
FPS En
AT Cxcys
F
c,
(79)
37

---

where E e~ (0, and Ey,t, is defined recursively
Ytut Vouuy + ve(1 ai,)) A
Ey.t, = v(1 c)(1 (i.,) v1 04141
(80)
Yt, Vany + cydss eu ~NO,I),
(81)
with Yts = Ae
E.14 SGS-EDM
The Sampler is k-step DDIM as 72 The Corrector
SGS EDM Xo.t; y, = A-
A
A
ah-1 X0,t
(82)
0y
1 Ope-1
where
a1
A
(83)
0t:-1
The Noiser is
c, SGS-EDM
V /t,_, X0,t, Va A e~N().
(84)
E.15
Latent-DPS
For latent-space methods, we denote E-)) and 2,, as the Encoder, Decoder, and latent variable, respectively. The decomposition of Latent- MitS is similar to that of DPS, with the Corrector adjusted as
1 O1t
h LatentDDDD (No.t. (X..Leen-DDSSX............ A.y) X0,,,
Sc. Vxe, lly A(D(x)...))
(85)
E.16 PSLD
The Sampler is single-step DDIM as (50.. The Corrector
h h,,,.LL(ZZ,.) A, 20.4.
The Vz,, 21, lly A(D(Zo,,)) ))1
aa
Mh Valg Zo.r, -E(ATy -I-AI"AA (=)""))D(zo.:))]l
(86)
Von:
The Noiser is as 538
E.17 STSL
The Sampler is single-step DDIM
50
The Corrector is
h (Zo.;: = 20.4:
ar. Ta. Vz, Ily A(D(Zo..)))1
E T (En(Zz, + Eg, Eo 1)),
(87)
d/1
Ott.
where df the dimension of the latent variable, and E; ~ NO,, The Noiser as (58)
E.18 CMInversion
We denote the consistency model as ge(s))) The Sampler of CMInversion is a single consistency model inference, i.c..
Le CMInversion (xx,) = go (x;: t4).
(88)
38

---

The correction process can be recursively defined as
Xt.,k Xo,,,o, + v1 Cte Di.kk Ko,4,, = (x) bi);
(89)
(90)
d
D1,,+1 = - -
||A(xo,t. yll?, ((,,),
(91)
dnu,k
where Xo,t,,0 X0,t, and Dt.,1 ~ I). Then the Corrector is
h CMInversion (D.t2 A,y) = X0.4,.K.
(92)
The Noiser is direct noise addition as
& MMInversion (Ko,t;) V 101--18,11 + vI d e~,().
(93)
F Specific design for DDNM and DDRM on noisy linear tasks
DDRM and DDNM are specifically designed to take aware of the observation noise in noisy scenarios. In particular, the variance of the additional noise is reduced such that the variance of the observation noise adds the additional noise equal to the desired variance of the next step in the diffusion model. This implies that in noisy linear tasks, the expected ground truth of the output of the Correctors in DDRM and DDNM is not noiseless image, but rather noisy image perturbed by the observation noise. Therefore, we adjust the ground truth for the LLE learning in noisy scenarios for DDRM and DDNM better align with the characteristics of these algorithms.
The implementation is straightforward by simply passing the noiseless image through the Correctors of DDRM and DDNM, i.e.,
(xo"), A, y((n))
t. DDRM hr. DDRM
(94)
XD (n) A, y((m))
B.DDD h4, DDNM
(95)
The objective for the LLE training then
En-u( (I (x0.1. (n)
min
(96)
-= S-8
min
Enout En.ee
X
X ...ND
(n)
(97)
Teu 2 -=0
where the ground truth in 19) is replaced by unchanged.
94 and 955.. The remaining training details remain
G Experimental details
G.1 Inverse problems settings
All the linear inverse problems follow ( while the nonlinear deblurring follows g We describe the details of all these problems to maintain completeness. For 4x Super-Resolution, we use a 4 x 4 average pooling operation for downsampling. For Inpainting, we apply a random 50% mask to all pixels, where the mask is generated using the same random seed to ensure fairness. For anisotropic deblurring, we use Gaussian blur kernels with standard deviations of 20 and I in two directions, respectively. For compressed sensing, we use Walsh-Hadamart transform with downsampling rate of 50%. For nonlinear deblurring, we adopt the neural network from [70] as the observation function. Unlike (8), we follow [44 by using deterministic observation equation generated with
fixed random seed to ensure fairness.
For the noisy tasks, the noise standard deviation is multiplied by 2 to account for the data range of [-1,1] 1). i.e., in the implementation, we actually add Gaussian noise with a standard deviation 0.1 to the observations.
39

---

G.2 Hyperparameters for base inverse algorithms
We carefully tune the hyperparameters of the inverse algorithms ensure they achieve satisfactory performance under few steps. The detailed hyperparameter settings are as follows.
DDRM. We use the recommended values from
S
i.e., 1j 0.85 and = 1.0.
DDNM. We use the recommended value from 6 i.e., 0.85.
DPS. We find that the learning rate form recommended by 8 is too small for few steps setting. Therefore, we follow the learning rate from [12], i.e., Cr, = Cyar, which demonstrated superior performance with fewer steps. The specific value of is adjusted for different tasks, as shown in Table The DDIM hyperparameter 1 set to 1.0.
Table 17: Tuned learning rate Ç for DPS.
Deblur (aniso)
Inpainting
4x SR
CS 50%
Deblur (nonlinear)
CelebA-HQ
0.5
1.0
6.0
0.1
0.1
FFHQ
0.5
1.0
5.0
0,5
0.1
IIGDM. We set the DDIM hyperparameter 1.0.
RED-diff. We set the learning rate g = 1.0. The weight À is adjusted for different tasks, shown in T8
Table 18:
Tuned weight À for RED-diff.
Deblur (aniso)
Inpainting
4x SR
CS 50%
Deblur (nonlinear)
CelebA-HQ 0.5
0.5
7.0 5.0
0.5
0.2
FFHQ
0.7
0.4
0.5
0.2
DiffPIR. We the DDIM hyperparameter 7) 1.0 and A 7.0, which performs well across all tasks. For the optimizer of the proximal point problem (64). we use the schedule-frea Adam [45), which offers better stability than SGDM in some tasks under few steps. The learning rate is set 0.1 and the number of optimization steps is 50.
DMPS. We specify the DDIM parameters as set 1) = 0.85. The weight carefully tuned task-by-tass and step-by-step.
02 ---1 77 /1 Ct and
ReSample. We set the DDIM hyperparameter r 1.0 and y 100. The optimizer for hard consistency (68) is the SGDM algorithm, with a learning rate of 0.01, momentum of 0.9, and 50 optimization steps.
DAPS. We use 100 iterations per timestep with the step size as nr no (6 + t t//((1 6)), where 0.0001, = 0.01, and 7' 1000. For noiseless linear inverse problems, we find that replacing the gradient term in with
steps for the DDIM sampler and perform the Langevin dynamics for
Ax yo
(98)
xa.e, 217r
leads to better results under few steps. For nonlinear problems and noisy scenarios, we follow the original setting and set oy in (73) to 0.02 instead of 0.05 for better results, which is consistent with the observations in 14
DCDP. We use Tweedie's formula approximation in the purification stage. The optimizer SGDM, with the learning rate of 0.01 and momentum 0.9. The number of optimization steps set to 50 for Super-Resolution and 25 for all other tasks.
LGD. The particle number is fixed 50 in all experiments. The step-size Ç is set to 0.5
40

---

for deblurring, 1.0 for inpainting, 5.0 for super-resolution, and 0.1 for compressive sensing.
MPGD. The step-size is set to 0.5 for deblurring, 1.0 for inpainting, 5.0 for super-resolution, and 0.5 for compressive sensing.
FPS. The DDIM hyperparameter e is fixed as /c = 0.85 for all tasks.
SGS-EDM. We use 1-step DDIM (i.e., Tweedie's formula) the prior step.
Latent-DPS. The step-size is set as Çt. = 0t S, with C = 0.01 for deblurring, Ç = 0.5 for inpainting and super-resolution, and Ç 0.005 for compressive sensing.
PSLD. We fix 7r 0. and set T)t 8.0 for deblurring, 10.0 for inpainting, 25.0 for super-resolution, and 9.0 for compressive sensing.
STSL. We fix 0.02 and set the number of Monte Carlo samples to = 1. The is cho- as 0.01 for deblurring, 0.02 for inpainting, 0.1 for super-resolution, and 0.015 for compressive sensing.
CMInversion. We fix the correction steps K = 5. The step-size is set to 0.3 for deblur- ring, 1.0 for inpainting, 3.0 for super-resolution, and 0.095 for compressive sensing.
In this paper, we default to sampling evenly spaced timesteps [0. T). For all algorithms, we initialize from standard Gaussian noise. Specifically for RED-diff, this is equivalent to using the random initialization method proposed in [13), which provides better stability. Starting from intermediate steps like the original setting of 7 does not affect the applicability of our method, as discussed in Appendix
G.3 Details of LLE training
For the training of LLE, we generate N = 50 reference samples using 999-step DDIM sampler with the corresponding diffusion model. We fix the weight of PSNR and LPIPS as 0.1. The Schedule-free free AdamW 455 is employed with the epochs set to 100 at each timestep f and warmup steps set to 50. Gradients calculated directly using the full batch. In most cases, the learning rate set to 0.04/S, where S denotes the total number of steps used by the inverse algorithm. For ReSample and DAPS the nonlinear deblurring task the FFHQ dataset, we adopt dynamic learning rate that at timestep 1,, the learning rate is set to 0.2 SS. is worth noting that we have conducted minimal learning rate tuning for different algorithms, tasks, and steps. Such general learning rates have already yielded satisfactory performance. In practical applications, further tuning of learning rates and other optimization hyperparameters could potentially further enhance the performance of LLE.
We design adaptive initialization method for the learnable coefficients at each timestep. Before the (nn learning stage at timestep t, we compute the loss for (nn)) and Xo.t:.1 1 n = 1, I,...,,,, N, and determine the initialization strategy accordingly. For linear inverse problems, if
En-ut1.......
(2(m)
80.1.11 x(")) 2 E
(x(m), (20,1,xx0"
(99)
then we initialize as
21l Te.s-: = - =.
(100)
Otherwise, we initialize as:
Tt. ft..Si--1 = --
(101)
All other parameters are randomly sampled from (00 10 10- Intuitively, this initialization allows the coefficients to initial with smaller loss, thereby improving optimization efficiency.
For nonlinear inverse problems, we further introduce a soft initialization to prevent the algorithm from prematurely converging to local optima. Similarly, if
En-.U(,.....
(n) 88..111:
(m) 2 E (n) (m)
(102)
41

---

we initialize as
-4,S-i = 1.
(103)
Otherwise, we initialize as:
14,9---1 Olty St,S-i
Ott,
(104)
other coefficients are randomly sampled from (0, 10-9). This approach allows LLE to follow
the original trajectory in earlier timesteps, while searching for better linear combination coefficients more aggressively in later timesteps.
H Complete quantitive results
Here, present the complete results of all algorithms the CelehA-HQ and FFHQ datasets using a total of 3, 4, 5, 7, 10, and 15 steps. PSNR, SSIM, LPIPS, and FID are reported on both noiseless and noisy tasks. The table index is illustrated in Table T9 for convenience.
Table 19: Index of tables.
DDNM
DDRM
LIGDM
DMPS
RED-diff
DiAPIR
DPS
ReSample
DAPS
CelebA-HQ FFHQ
Table
20
Table 22
Table 24
Table 26
Table
30
Table
32
Table 28
Table 34
Table 36
Table
21
Table 25
Table
Table 27
Table
Table
33
Table
29
Table
53
Table
37
42

---

Table 20: Results of DDNM on CelebA-HQ Dataset.
Condition
Steps
Strategy
PSNEE/SSIAHHLPPSUUIDD
Dehiar (aniso)
Iopainting
4x SR
CS 5056
27.50/ / 3192/53,22
16.64; 00.4544 O 114923/73.3
27.09 (,7732/ 00.2%1 / 660.11
16.55 / /0.445500 0.5444 194.7
LLC
28,08 0,7842/ 0.2909/ 43.53
24.38 0,5523 0,4329 /64.6
/7.4// 00,2989 / 46.29
17.29/ 004733 0.5197/ 155.4
98.8 0,7948 0 08.28557 47,36
20.27 1055 0.4568//7 /7../ 15
28,35
/0.503 /0.2755 59.34
17.39
LLG
29.27 / 0,8172/ 0,2558/
42.83
25.29 0.5915 0,,473/6611 61.119
28.65 /0.7921 /0.2700/ 33.87
0.4994 00.77 42 96.63 18.43/
29.63 00.63/00.55 0,8192 /0 (025500 45,43
22. 76, 00,550 /0.4310/73..25
28.97 0.8182) 0..269//56.20
18.20/ 0.5742 /0.1914/ 77.195
dy-
00.2344 45.10
26.35
55.84
/0.2522/37..0
(0.535) 0.4410/
01.19/0.8577/1 (0.23777159 50.72
25. 93 /D.6200 (5,55000 ( 29
(U5325 /0.2418 /54.00
20
/0.5536/00 /044
00.2447 /44.66
(0.7188/ 0,3279 /5063
29.33/0.8117/ /0.233//////
0..3652/ 54.22
30.45/0.846// 0.24
51.61
21,30/ 0.6356/
30,50/ 0.8454/ 0.21377 42277
29.13/ 0.78% (0,7789 (45,50
0180775 00.2353 33.47
22.42/ (167EN 0.3408 44.97
3IL50/ 0.5461
47.013
20.57/
/46.30
00222444 /47.5H
23. 04755 2961
30,52/ 0.8463 0.2055 440.88
30.74; 0.8540 (0.2281 338.23
/181200/ 0.2336
33.42
24,04/ 0.7687 0.2706 31,03
39,4470,5577/ / 1004
11..00
19. 39/ 0.425) 0.5946 137.9
50,20 /O 5603 /0.2413
50
17:27 0.5188 5521 252.3
LLE
39.56/ 0,955N 0.8986
11.72
26.31 0..199 (1.5790/ / 77834
30.29/ 0.5022 /0.2509/
55..77
1749 / 015185 0.53N5/ 229.1
39,80/ 11SKIN / (1,0331 19.790
24.11 / 0.61E7 /1.4427/ 88.72
56.76
(.8728 2026143 /51.54
18.
0.5751 / 4056 187.6
LLE
39.97/ / 0,9623 0.0914/
10.00
29.13/ 0.8419 (0,2540 / 66.00
30.71 /US770 (0.1979/
48.91
18.9 0.5840 0.4695 154.7
39 S9 119K2ES / 010097/
18.80
27 XS / 0,7550 11.3322/ /6666
30,97
(US766 /0.1876 46.
18.9% 0.6699 / AAIN. 149.3
1.F.E
40.20/ / 0,9640 / 0.0865 8,770
30.93 0.8868 /0.2206/ / 552.78
50,90 /0.5745/ 10.1223/ 43.52
20.15/ / 0.6395 0.4147 118.1
40.24 0,9643 / 0.0864/ 7
26. 20 0.6668 /0.5722/55.. 55.35
31.17
0.5802 / 1700 99.58
21 64 / 0.7 0.15122 99.41
LLE
40.47/ / 0,9662/ 0.8815/7.390
32.56 0.9159 00,6688 / 33.79
51,08/ /U8872/ / 00.163 38.15
22.43/ 0,7270 0.3275 77.24
41.42 U.9K06 / (1,0015 7. 0.50
33.54/ / 0.9262 (1.1440 30.86
$1.26 (U8IS 0.1553 /
22.68/ / (U.791 / 012222 08.34
12F
40.70 / 0,9677 / (U9773 6.500
33.63 / 0,9294 0.1336 / 29.31
51.16 /0.8722///// 0.1564 33,90
23.81 / /L.744/ 0.277N 57.84
40.55 0..56/096676611 / 0776/ 5.880
34.57/ 0.9402 00.1122 /2.22/22.114 14
31.32 /0.8288/ 00.1506 30.91
25.25/ 005500013//0222660 23:33/03444/43366
12F
40.78 / 0,96514 / /U0074//5.8330
34 /5.55/000 0.9708 00,1225/1195
51.28 00.812/0.1
0.1507//0 /30.79
26.27 / (18316 0.22222/ 38.99
Table 21: Results of DDNM on FFHQ Dataset.
Condition
Steps
Strategy
Deblur (amiso)
Inpainlig
4x SR
CS 50%
0.6341 0.4177/
139.1
/
/0.350 (L.557 (67.9
25.78/0.7527/ /0.3314/ 122.7
16.27 0.333) 7 227.6
0,6859 0.3836/ 127.7
22.56 10,4243
145.0
0.7548
0,4100/ 00.500 /20.0
0.6671 3935/ 134.8
19.59 0.3669 (1.566)
156.1
/0/7704
16.72/ U. 3040 /0.335 189.2
0,4473/ /155.1
/0.2810/
00,424/
00.561/
0.015
28.72/ 0,8014/
70 0.3526/
0.2802/
99.94
23.05/ 0.5000 0,4737
3804
150.5
133.8
27.8/0
0.7977
/0.2924/
00.2650 196.46
0.300
0,4430/ 0.5400/ 202.4
0.8011 (1/2659/
0.4379 (1,4955/
(022679// 10660)
/156.1
13..
20.42/ 0082255 0L2490/ 93.16
34.57/ 0.5386 (0,4461
/134.
81.81 0..2403 184.1
0.4635
0.51000 171.3
29.79/ 0.8337 2497/ 102.0
62
54177 0.4336 134
28.08/ 0.5005 /0-2767
100.8
19 0.4332 3850 145
1...F
008387 / /0.2317/ 58.46
26.84/ 0.6701 00.3022
114.7
/O 50.3 00.2509/
187.98
0.5701
0.4212// 127.5
20,98/ 0.8443/ 2370
28.61 707061 7033576 1169
28.64/ 0.8168 10 1022566/
20 6374/00
3868 123.5
LLE
29.57/ 0,8417/ 0.2207/ /11.90
28.45 0.7891 0.22009/
105.1
25.27 /L.8114/ 0.2346
81.45
21 0.6919/ 0.3325/ 101.9
34.02 0.9572/ 0.3035/ 22.09
19:31/ / 0,A262 (1,0029 190.9
28.78 (USU94 /O 10.30S3 1168
17.32/
0,5089 /OO 57555 342.3
F...
39.18 0,9590 / 0L1013 / 78.74
24.74/ 0.6379 (1.4222/ 141.7
29.14/ (U.332 /0,2832/ / 111.
17.40/ D 0.5075 0.5666 326.1
19 S4/ 0.9618 / 0.0017 /2668/0000//22..0
21 / 5873 / 152.9
29,60 (5,5500 /O 2404 / 104.2
17.55 14 5308 788
LLE
39.75 / 0.9637 0.8899 /2.41
27.35 / 0.7873 0,3474/
129.1
29.06 008551 / /0,2309 99.66
18.19/ 0,5557 0.5136 2.51362553.2
29,82/ / 0.9633 / 010871/ 20.42
26.34 / 0,7199 21000/0..7900 1..7798/ / 130.33
29.99 0.8646 (0,2107 / /92.55
18.29/ 0..9910///.0075 /246.0
L.F.E
40.08 / 0.9662 / 0L0843
20.55
28.71 / 0.8334 (0,2929 / /13388
29.94/ /686627/ 00,2060 99007
19.21 / /L6946 0.4626 200.0
40.1670.344 / GB03/ 16.09
20 96, D. D.8666 ( (0.6055/ 100.2
30.00 (U8723 0.1825 / 76687
19 / /..653/0.44 40977 179.
L.F.E
40.57/ 0.9697 / 8J0763/ 17.00
30..9; / 0.8827 (0,2331 / /91.22
50.23 /0.0000/// 0,1802/ 75.35
20.42/ (L6671 0.39599 166.6
40.56.70 70,9991 /00074/ / 13.78
31. 75 D. 9349/ D. 1956 777
30.50 (U8774 /0.1657 / 66.08
20.51 / 00.7103/ 0.3554 152.5
LLC
0,9713/ 0,0708/
14.16
31.80 0,9072 0.1910/73.11
30.44 /.5754// 0.1645/
63.59
21.45/ 0.7174/
0.3424 137,4
0.9709 0.0691/ 11.69
33,05 0.9266
0,14477 /8.25
30.69/ 0.5813 0.1527 /4.188
22.89 0.7816 / 28122/ 106.6
11.95
33.05 0.9264
/5390)
0.8806 0.1518
23.30 (1.722)
0..774/
19.63
43

---

Table 22: Results of DDRM on CelebA-HQ Dataset.
Condition
Steps
Strategy
Dehiur (aniso)
Inpainting
4x SR
CS 50%6
27.68 0,7950 0 (0.2775/ 55.45
16.68/0.d888 00.094 / 0.4403 / 54.28
26.71/0.76.9 00.6.39 0.22772/
45.38
16.58 / 0.6953/004177
90,78
LLC
27.69 0,7951
0022113/
52.88
24.53; /.665 00,461 /66.05
27.49 /0.7614
0,22999
41.97
17.07 0.5269 00,698 111.6
28.75 0.8159 / 0025561 1,5..99
20.48: 0.6418 0.3332 / /42.95
27.95 007766 00.2603/522 52.16
17.46 0.5547 /0.255/ 90,66
44.18
25.01 0..755 (0,2935) 40.65
0.40148/
700.1
0.05
29.20 818255/ 2444444 /2.22
0,8265 29,36/
00.3449 /4593
26.09
(0.701 /0.3632/
(0.2560 44.73
(U8114
/0.24/00.27//// /51.07
/0.2486/
18.
005965 38838 71,
00,3300 0.36630/
29,82/ /315515///0
26.85
0.66666/ 00.322//0 60:01
LLE
0.8366
/44.7
28.18; 0.8237 (0.2.904 433.39
29008 0U8811 10.2537 32.77
21.21/ (U,951)/ 0.34190 40.95
(..441 (2.52157445 /4.772
29-97 0.8351 (1.2229) (45.0
(UXINI
(.77155 3017
1.B.E
00.841 /42119/41.27
29.70/ 0.8507 0.2099 /41.61
29.00/ 5043 0.22345
332.1
22.40/ 0.7369 0.27799 38.51
BUL31 2141 45
30.69/ 0.8GT8 (4.1996/411.0 94
/45.37
23.05 O0,
1.B.E
30.35/ 0.8440 0.2075 40.39
30.95 0.8737 1901 /39.44
29.1 1/ /0.4094// /0.2270// 33.30
24/04/ 0.7782 00.2472/ 34.88
3M.20 / O 031351/ 19.74
19, / 0.2282 (1.59591 135..4
SOL1G / CU85564 /0 10.24229 (22.40
17:25 005599 / SSUS 242.5
LLE
38.15 / 0,9493 0.3090/ / 14.32
25.99/ 0,7057 (0.5885 / 198.5N
50.27/ / 0.5019/ (0.2527 /60.90
17.4/ 0.5184 0.5361 221.6
3M.45/ 11944A 0313 18.47
23. 26 / D.50OS /1,77000 55.01
S0L62 (L8644 10021533 (-57.49
18.08 / 0.5666 / 5040V 188.5
1.F.E
38. 0,9506 / 0.1092
15.37
28.01 0.8199 0.3148 78.35
50.62 /6.6699 00,2072/ /54.45
18:91 / 00,5509 /.447 40 154.5
5
LLE
3N.51 0.9520 / 0.3064/
38.57/
0,9301 /O 1281 / 17.89
14.63
26. 09 / 6997 / 0..3819 /81 43
29.74/ 0.8657 0,2556 / 66.80
30.78 (U88719 19/ 00.1337 / 550.44
36.83 00.8731 /0. 2000 /52.68
18.85 / 0.8006 / 00-6658 156
19.97/ / 0,6287/ 0.4275/ 124,0
T5.75 / 0.9521 / 12482 16.89
29. 0.8505 11,2645 F.3.44
51,12/ (U2766 00.1836 00505/4475 47
21.01 / /U,881 0.EMSS 109.
1.F.
3H.ES/ / 0,9555 / 0L1009/ 13.20
31,33/ / 0.8975 0,2002 / /5.400
50.88 /0.87 / OL) 177777 / 45.17
21.58/ (L77441 0.3318 89.23
1R / 0,9535 /O 1221/ 16.17
31, 70 0,8994 O. 1917 F48.81
31,13/ 0.8765 10.1723 10/1//33000 /3.007
21 21,77 / 7239 /O 3402 89, 77
F.F.R
38.97 / 0,9563 / 0L0993/ 12,67
32.45 0.9138 0,1142 /41.39
50,94 /0.7700/ 0.1676 41.16
22,83/ IL7433/ 0.30%N / 73.65
79.01 7 0,99548 1193/
33. 22 /.9233 F.. 1439/45555 1390/35.00
08
31,20 0.8799 / 1636688// / 39.23
23.55 / 0,7785 / 11.2800/61 44
LLE
39.07/ 0.9571 0.0983/
33.41 0.9256
0,1361 32.66
30.96 0.8758 0.1609
38.41
24.58 0.7923 0.2610 57.46
Table 23: Results of DDRM on FFHQ Dataset.
PSNR // (PXP)//SMMPPPPPPPPFFPI. /FID
Condition
Steps
Starteyy
Debiur (aniso)
Inpainting
4x SR
CS 50%6
26.38 O, 0,7630 O 0048/113.2
0o:3977 7 0,5204/ / 160.3
25.27
6966 /0,348 / 1058
16.44/ O. 0.4506 /0.5110/ 185.5
0,7656
0.2972/ 105.2
23,63 0,54403
00,618/
142.1
25.90/
0.736 0.3107
17,01 0.5104 0.555 /228.4
0,7933 2796/ 105.5
20.32/
(1,0033
129,0
/0.2912// 101.2
(.5367 0
152.4
27.59
0.2686 96,91
24,89; 0,7194
00,336/
124.5
0,7689
0.2755
0.5662
0.4381
145.2
(126677 101.3
66 0.6689
110.5
FO.7553
10.2770
17.62 0.S713 /2113/0.4444
142.8
by
00.2540 19.188
0.7535
163.8
/0.2564
00,631
/0,4001/
125.7
2510/44
7600 0.2776 100.0
/0.0407 /91.05
19. /0.6373/0
3692. 121.2
28.95/ 0.8237/
0.2382/
26.86
0.795S 00,2200
94.19
0.8051
/0.2347//
0.6508
0.3565/
196.2
29.51/ 0.2466/
N3 011114
59.04
0.7974 10/26636/
19.78/ 0.5701
3892/ 115.5
1...R
70 03.831
(0.2276/ 79.88
38.41: 0.8305 (0,2997
16.32
27.68
0.2448/ 18489
0.6874
0.3197/ 98.19
29.79/ 008405 7380/ 87. 70
0.8553 002007/
0.8138 /
20
1...R
79.68 0.8385 (.2215/
778.1
29.97/ 0.8634 (..1956/ 78.99
28,09 /O 807004 00.2315/
78.48
21.59/
(L7343/ 0.2824 87.52
07.53/0.9377 / 0.1536/ 41.15
19.35 0.4268 (1.011/199 191.8
28.84/ (.815T O (0.3057) /11222
17.02/ 0.5058 O 075759/ 351.5
LLE
37.07 / 0,9309 0.1268/
33.44
24.63/ 0.6399 0.4417
143.1
29.08/ 0.5366 00.285/
115.6
17.30/
0.5068 0.5670/ 332.6
37,88/ 0,94199 / 0.1478// 38, 74
22 67 / 0,50377 (1,4908/
138.9
29.58 (L8527 / 10.2542/ 110.4
17.70/ 0.5525 / 0355455/ /290.5
1.F.E
37.36 0.9409) OL1216) 34,50
26,89 / 0.7763 /0.3610/
133.7
29.62/ (US543 0022537
109.0
18.17 (U5529 0.5166 255.6
5
LLE
38.040094497 /O 1450) 96.85
37.62 / 0.94445 / 0.1168 /3,00
25 0S / 6661 /D.4736/ 140.8
27.77 0.8126 /0. 003256
125.9
29.56 /0.5604/ / /0,2510/ 99.36
20.54
(.8614 /0.2256/ 101.0
18:23/ / 5576 / 5022 253..
18.93/ 01 0,5906 0.4778/ 223.7
38.30 0.94463 / 00.1405 /34.06
27. 95 0.7999 / 00.295 /121.4
30,13 0.8687/ / 00.2017/
89.48
19.17/ / /1////////.////2 0..4735/ /207.5
LEE
37.99/ 0.9488 / OLIOR5
28.14
29.35 / 0.8575 0 197.9
5000 /086679/ / 0..2018/89.77
19.97 / (L6474 0.4191 176.6
38.53/ 0.39455 /1 1367/ 33.84
20 3S, 0.8633 ( (.2651 / /100.3
30.0/ (U8742 00.187//777 (05117/77..0 777.48
20 / 0.6809 / 3..9911/ 172.6
LEE
38.15 0.9510 / 0L1047
26.50
30.61 / 0.8869 00,321 / /92.88
50.25 /0.7755/ / 0,1810/ 76.35
20.58/ (1,8900 0.3727 1539
35.77 / 0.9506 / 1331 30.91
31 65 /.9007
/ 93
30.4 RU879T 00.6673 66.16
21. / 0.730/ / /0.386/ 141.0
LLC
38.41 / 0.954L/ 0,1006
25.90
31.82/ 0,9057 0.1190
30.34/ 0.8761 / 0..168
64.45
22.20 0.7394
0,3218/ 124.8
44

---

Table 24: Results of IIGDM on CelebA-HQ Dataset.
Condition
Steps
Strategy
Dehiir (aniso)
Iopainting
4x SR
CS 50%6
20.39 0.5361 / 00.5197/ 100.3
19.57/0.5333 0.4947 / 116.5
20.73/0.5519
/1199/..00 / 104.5
16.79/ 0,4721 /0.5566 114.8
LLC
21.73/ 0,5877 0,4894/
115,0
20.72 / 0,4829 / 199,4
21.54/0.5833 0.5833 / 0.489 119.7
17.30/ 0,4871 /0.53%/ 126.4
0.5735 / 87.53
21.85; 0066750/ 0.4126 84.91
19.94/0.5582 00,7443 76.88
16.47/ 0.5005 / /0.5650/91..0 99.76
77973
23.61 0.6805
81.92
(U6795 0.39933
18.71
0.4722
0.6550)
004410/77..8
22.63 /D.694 /1.376)
49
/O/6620 (1.41)0000411//// /65.21
16.32
70.5//0 /0.540//
0.05
25.21 0.7349 0,3358 /7.90
0.7308 00.332/ 666.60
0.4490
25.75/0.7411/ (2.32
0
24.78
/0.3304
15:05 00,651 /0.5566/
LLE
28,29/ 0.7956 00.724/552
2X,12/ 008055 00.2730 55.70
27.20 0.7851 00.2715 48.56
21,07/ (U67433 0.3622/
78.07 (0.2511
29.98
(1.2477/ /55//555.. 63
27 70.9993/ /0.2000/551.5
15 015215
1.B.E
29.28 0.8179 00.2312/443333 43.38
30.15/ 0.848/ 0.2338 /50.99
28.64/ 0.8098 00.2358/ 49.96
22. 0.7278 0.33200 55,
31 0.876
/50. 18
16. 12200.500 105008
1.B.E
29,80/ 0.8293 / 0.2134742..0 44.07
31.77/ 0.8763 0.2113 /47.69
28.84 /0.8145 /0.2230/ 44.47
23,89/ 0.7831 0..2788/ 49.05
31.13 / 11.844 (1 (3.2244 (.7 23
24.41 / 0,7827 (1.57733 /96.26
26.35 /L.7557 400-3303 /88.66
18.24
015220 D SHIF 213.2
LIE
33,83/ 0,9119 0.8735/ /3.16
25.57/ 0.7843 (0.5228/ / 77.66)
27:71/ / 0.7988 00.3578/ 75.76
18.54/ 0.5507 0.5414/ 181.7
29.49/ 0.88655 / (0.2219) 36. 99
26. 36 0.8115 (1.3054/ 42
26,1 /O.76998 3919110///
76.50
18:00/ 0.S3IN
57747 195
12..E
34.78 0,9313 / /.1534//277777
28.03/ 0.8450 /0.2539 /67.84
28.61 00.8254 /0,29444 68.51
19.72 / 0.6657 0.4738// 128.5
5
32.95/0.92199 / 1697/ 26 36
27. / 0.8365 0.2774 72, 99
27.78 /0.5077/ 00.3293/ 655.5
17.
0.5269 0O55223/ 196.6
LLE
35.38 0,9357 / 00,339 26.30
29.47 / 0.8712 00,3196 / 66.17
29.47/ 018441 / 00,2602 / 62.044
20.18/ 0,6389 0.4475/ 118.2
35,95 0.9437 / 1255/ 20,86
30.299 / 0.8859 (1.19933/ / 115
29.44 FU84446 10.29999 /53.06
15.53/ / / 0.SKE1 218.77
1.F
36,80/ / 0,9505 / 0L.142/ 24.89
31.76/ 0.9838 0,1173 /44.733/47.55
30.75/ (U8709/ 00,2995 51.95
21.70/ / /L7147/0.3571 82.28
37,87 0,95533 1015/21,19
33, 32. 0,9900 15165 30
50.65 80.8734/ /O 1947 04477/45
15.83/ / 5135 /O 5260// 181
1.F
37.83 / 0.9578 / 0U0992/ 23.67
33.44 0.9922
(0,1531 / /39.85
31.27 / (L8810 0. BL1838//
45.55
22.58 (L, NL7606 OL.IS5// / 66.55
38.44 0,9619 / 0.001//////////0 09.33/3.20
34 69 0.9343 O. 1318378 /3.88
31.11 7O 8826 /0.775 /42.07
16. (6,35/ / 5568 /0.4567/ 141
LLE
38.39 0,9618
0.8905 23.48
34.69 0.9341
00,1355 34.85
31.29 0.5834/
0.1684
40.51
24.46 0.8165 0.2444 51.69
Table 25: Results of IIGDM on FFHQ Dataset.
PSNR / PPNEPSSSMM PIPS
Condition
Steps
Strrteyy
Debitr (aniso)
Inpainting
4x SR
CS 50%
20.72 0,5570 / 5370/ 188.9
19. 19.23/0,52500 O..00 0.5 / 178.7
20.85
115661/0.1115 /0:5154/188 1829
16.81 / 00.4767 /0.571/ 186,0
0.5110/
20.51 0.5686
0,5008
174.1
0.5746 0.5112/ 1788
17,11 0.4906 0.5566 187.3
U. 0.5333 015315 181.1
21.
0.620/
151.9
19.82/0.57M4. (0,39999 1598
16.25 U.4837 0.5922 179.9
149.6
22.45 0.64833
0,4300
144.7
0.4324/
18.11 0.5407 0.5097
164,0
20135 0.6145 01 0.5000//
91: 0.6700 (1.342
141.5
FOL6284
(0.4667 145.3
0.4725 /06145/ 199.
by
0.3777/
24.35 0.7088
138.7
0.7093 00.3757/
18.53
00,8355/
70,6556 0-433//
113..2
/06689) /0.4114/
13.43/0.416 0/416/
6818//22
27.27/ 0.7752/
0.3095/
26.32/ 0.7620 0,3289
124.4
0.7585 0.3112
0.6378/
0.4331/ 110.99
26.13 0.7565 0.3158/ 132.1
29. 37 0.8139 11.2902/ 120.7
25.46
0.7564 (0.3229
13,89/
0.6474
/229.7
1...R
28.51/ 008856 0.2648/
100.3
38.43/ 0.8104
10.2766
116.8
0.7855/
0..266/ 1029
20.34/ 0L6834
0.3883/ 117.0
78.07 0.7771 7658/ 104.3
0.8527 0
24
/ 167769
5747/
1...R
20.34/ 0.8226 0L2472/ 97.73
30.25 0.8503 0.2511/
186.2
27.89 0.7965 0.2598/977.77
(L7952 0.2756/ 99.09 24.59/
31,75 / 0.900N O 0.9999/ 66.86
24.06/
0,7579 0.3526 169.9
26.78
(.7766 /0.0022/ 15010
18
(.5500 /O 5430/ 325.8
LLE
35,39/ 0 0.1358/ /44.66
25.5 56. 0.7971
0.5204/
148.8
27.39 0.5020 003478/ 142.6
18.47/ 0.5673/
0.5226 290.6
26.67/ 0,8061 / / 108.7
26. 0.8131
/ 140.4
24.15 0L7334 /0.4250/
167.7
17.49/ 0.5334 0.5552 325.6
1.F.E
36.21 0.9477 0L1197/
33.77
27.25/ 0.8406 (1.2629 / 119944
28.26 / (U8254 /0,3002 / 128.6
19.16/ (L6139) 0.47113 237.2
5
LLE
36,96/ 0.9531 / 0.0085
0.8006 / 0222101 105.
33.22
28.77 0.8707 / 002228 / 111.4
2K-48 8.110//////00 /. 0.2526 131.4
25.81/ 0.8383 / /0.2710/ 1189
25.31 /O 76.76 3935 155.6
15.94 / 0.5051 5858/ 345.
19.4 / 0.6391/ 0.4487 212.0
30.26/ 0.9213 / 0.1882/ 105.7 .7
29.37 / 0.8829 / 00.2083 / /94.56
25.42 /0.7816 0,33499 149,1
13.57/ / 0.4492/ 0.63554 / 333..8
LIE
37.71 / 0.9610/ 0U9893
37.10
30.42 / 0.8938 (0.1899/ / 888.2
29,97/ (L8646 / 00,2211/
98.01
20.25/ / (L,984/ 0.3756 165.7
36, 16 0..0720/72//// / 1118/ 50.10
32.37 0.9161 (0015553/ /63.8
28.07
8088 / 0025533/ / 111.5
14.23/ 0 0.4501 /O 5940) 317.6
LIE
39,26 0,9684 / 0U0712/
19.95
32,25 0.9135 15,559 59
30.61 (U8785 / 0,1858 / 76.62
20.67/ (L7351 0.3322 141.
40.37 0.9732 / 0701 19.31
34.18 0.9332
/
29.02
/0.854/// 10.1954/8000
14.17 / /222/0..575. 272.
LLC
39.99 0.9/24/ 0,6622/
19,22
33.99 0.9317
0,1324 51.07
30.93/ 0.8861 / 0..1744/ 63.55
25.61 0.8493
0.1984 /.241
45

---

Table 26: Results of DMPS on CelebA-HQ Dataset.
Condition
Steps
Strategy
Dehiar (aniso)
Inpainting
4% SR
CS 50%6
14,0270, 0 0 (074666 205.9
11.91 /0.100 0.8018 / 296.5
/5.53/ 0.7632
2011
12.45 00.1025/0 R..102/0.7985 325,
LLC
14.10/ 0,1771 0,7440/ 197.7
11,97 0.1042 0,8000/297 /97.8
18.76
18.76/0.2530
7.77
1860
12.52/ 00,151 /0.7887/320
0,,3221
/ 100.0
13.41 (0.144) /0.7624/ / 187.2
19.33/0.2655 0..645 106.5
14.09 / 0.1508 /..7//00/200000 209,0
13.55 00.149
/18.11
00.154/ 0..784
21, 40,6666 19.5619/90 198.57
14.71
/1:7525
151.1
22.95 40.4978 10.5412/71.1
14:85 00.651 /0.7194 186.
0.05
0.5565/
06166 0.4902
140.8
0.6989/ 169.1
25.127 0.4462/667..
134.7
(0,3960/
0.22331 /
LLE
25.84/ 0.5759 (L4355 66.91
20.12/ 0..3374 (0.5615
19663
25 4L7065 0..355 55504
17.36/ 0.3030 0.5940/ 116.2
(0.449)) 56.07
16, (7117 70,0006 116.55
27.86
14.91
/0.153 // 192.9
1.B.E
26.54/ 0.5773 0.4051 53.03
23.01 9.4236 00,991// 63.78
27.72 0.7528 0.2996 41.67
18.49/ 0.3584 0.5461 99.12
70.7234// 3005 35.61
(USINS 10.2491 /36,43
18.47/ 3153
102.4
1.B.E
28.93/0.7459/ / 0.28733 33.97
25.36/ 0.5664 /..713
28.22 /O, 70.7946 0.24666 37.68
20.52/ 0.4838 0.4500 66.70
15.35 /1,2104 0171015 117.9
12.
/ 0.1432 / 226-1
18:32 C 0L34416 O 0071147//
132.2
13:45
1006 / 7576. 357.4
LIE
15.52 0,2196 0U6920/
112.9
12.81/ 00.1478 (0,7799) / 217.4
22,05/ / 0.4104 00,7775
112..7
13.55/ 0.1656
0.7568 361 A
19.18 0122966 (16666)) 16. 10
14.5% 0.2144 (1.7242/ 155.9
60 F CLA564 7 (0.5600/
Ho.40
15.19/ 0.25531 / 0866 264.5
1...E
19.43/ 0,3044 / 0.6022/
16655
14.67 / 0..222 00.7233 155.5
26.73/ (L.988 00,4053
62.17
15.28 0,2590 0.6826/ 272.1
5
22 / 0.3533 (0.5259 65, 94
16.25 0.2265 / 133.3
26. 18 /0.6359 1 10,4429 66.76
16.08
0.5067 / 6454 241
LLE
22.49 / 0,4037 / 0,5117/ 66.85
16.43/ 16.45/0.27760 0.22762 00,6700 / 113.11
28.34/ / 0,7947 / 0,,3071 559.01
16.15/ 0,3994/ 0.6432/ 250.4
26,099 U.SSS1 / (1.391 / 50,899
18,06 0.3399 (1,5113 / 1126.8
28.91/ /0.7913 0.3168 /52.36
17.29/ / U14052 OL (LSMMO 228.5
12F
26.55 / 0,5731 / (0.377) 49.45
23.47 0.5990 10,4406 / /79.94
29,09/ /.88329 00,2516 58.33
18.64/ (.5121 /0.1187/ 168.2
75.767 006663366/// /O 32801 45
20.67 00446400.00..0 00.5119//9111111 /911 88
29.80/ /0.81 /0.2560/ /4.62
17 40/ 0 0,4532/0 /2.///.5555 0055574 228.
1.F
32,23/ / 0,8288 / (0.1899 23,23
26.72/ / 0.6933 00,379 / 599777
29.89/ / 0.8461 / /0,2129 43.73
19.71 (15756 0.4638 128.8
31.22 0,7975 / 2026/ 24.48
76 04 6668 0.3702 1661.5
30.53 /O 5494 /0.2115/37..4
20.24/ O. 5970 /7700/0.00 /0.4531/ 132.2
LLE
34.47/ 0,8791 0.3482/
28.52 0.77743
0,2997 53.96
29.95 0.5537 0.19999 /
21.54/ 0.6484 0.3996
19.74
Table 27: Results of DMPS on FFHQ Dataset.
Condition
Steps
Strrteyy
PSNR (/SIMM: /FISS
Debbar (aniso)
Inpainting
4x SR
CS 50%
14,42 0, 1843 /0 234.4
12.43/0.1155
0.9997 /30.09
15.56
/3.2302/ 0.755//
355.1
12,30/0.1 //330/0.14///.70000 7917 3634
0,1854/
0.7358
232,8
12.48 0,1206 0,7952 312.7
18.3 0.2575 0,2697
12.86 0.1175 0.7878
0,2081 (.6SU//
13.75
252.2
18,09 /O2262 7,703 33295
14.22 U.1648
/.74488 283.6
168.9
13.81 0,1600
/
250,0
0.885/ 0.558
14.33/ 0.1670
0.7435
288.7
(1.9076/ 164.0
14.75 /0.164 (1.7474 235.0
FOLSSTT 10.5601
U.1651) O
295.1
by
14,89; 0.1668 (0.7436
225.5
0.5955 0.4643//
0.1672
00.7374/ 291.5
0.51540.54///11 0-484/
/0, 1964
F /243/210.1
24
70.634 127.4
15.767 1984 7096 252.6
0.5244/ 0.4820
19.55 0..366 00,5993/
170.00
0.6791/ 0.3879 1236
16.71/ 0.2682/ 0.6476/ 200.0
28.48/ 0.4009 (14740)
10.59/1 0,1555
191.2
10.3122/ 111.0
U.1522
299
1...R
25.65/ 0.5404
(L4420/ 120.0
0.4007 22.42/
(0.5249
1338.4
26.76 0.7269 0.3353/ 1036
(.3411
0.5819/ 168.2
27.85/ (335/0.0033)
3497/ 103.0
3387
145.5
27.64/
0.7774 10.2848/955
17
2008 6097/ 176.6
1...R
28.06 10.7155 (L3350
97.64
34.35/ 0.5227 0.,5533
127.4
7.77700/ 0..776/911.
19,07
(0.465
0.5091 143.7
15,90/ 0.2429 (.6847/ 171.1
13.25 / 0.1577 / /26.7
18.07
(L.3122 (0.0952) / 259922
13:75 0.1125 / 0.7603/ 396.3
LLE
16.06/ 0.2551 0.6728/
103.2
13.28 0.1032/ (07730
/26.3
21.77/ 0..0.35 0.7009
200.1
15.03/ 0.1778
0.7557/ 404.07
18.97/ 0.2910 0.6285/ 152.4
14.85/ / 0,,2277 /.7218 230.0
2095/
0L3849 10.6343 267.3
15:24 0.2645 0.08855 354.1
1.B.E
19.78 / 0.3274/ 0L9919
140.9
14.92/ 0.2299 (0.7200 / 225.8
25.88 ELSSHS/ 00.4420/ 130.7
15.35 / 0.2667 0.6916/ 363.0
5
LLE
24.53/ 0.5214/ / 0.4315/
21 20.4 3670 / 55301
138.1
107.3
16-21 / D.26600 0.6800 /244.8
18.54/ 0.3406 00,6133 210.6
27.34/ 0.7527 / /0.3560/
25. 14 /0.5672 (046677 / 120.0
117.7
16:11 / 5034 6664 342.5
16.79/ / 0,,442/ 0.6318/ 245,4
25.21 0..2477/ / 00.4269/ 105.0 0
17.72/ 0.3332/ (1.465 / /20.9
27.89 /O.755/ / 0.3399) / /11..1
17.00 / 10.3812/ 1..312/0.6157 /309.3
LIE
29,57/ 0.7431 / (L27700 76.39
22,69 / 0.5557 (0.4724/ 169..
28.43/ / (US195 00,2899
1090)
17.85/ (1.6339 0L5500 /L5500/251.0
76.87/ 00.5788 / 3125 2255/83.200 83.26
30 / 0.4261 /0.5654/ 167.7
29. 15 8088885 / (0.266 /98 24
17.30/ 0.4539 0.58177 326-9
LEE
32,51 / 0.8513/ OLIM05 48.60
36.10 / 0.6880 00,8819 / 125,5
29.21 (U8438 / 00,2383
92.21
18.95 / (15680 0.4871 /22.6
92.11 111170000 708833 /O 19167499.69 60
25. 48 0.6675 0.3957 / 123.1
29.96 8548 00.2520/91.00 12.
19.53/
504000 / 48166/ 214.9
LLC
34.62/ 0,8978/ / 0.1398/35.35
27..4 0,7519 0,3360
/13.8 113.8
29.36 / 0.8557 / 0..111 88128
19.91 0.6231 0..397 177.
46

---

Table 28: Results of DPS on CelebA-HQ Dataset.
Contriin
Steps
Delix (anle)
Tagaining
Deller (collinon)
23.50 1O 404152 99A1
23:57
FOSSTT
(AAFS) K5.40
35.49/
DUSARA EUSERO 92.53
16.30
BLRSS TO 1064
191.
16M0TOAAAE ( 1216
9,455 73.57
27.51 / 6.3664 64.57
34.57 (6/661
$1.79
1553:
04679
19.N NWW.0044753/00777
4
111
25-18
23.35
(07063 (8,3399 / 60.57
(1.020 /03995 55.33
28,55 (8.7639/ 8.3297) 90667
/
004500 $1.52
15:34: 9.7196 (L360)) 67.81
15.21
F0A558/ FLA94 723
1656:
0,5018 0L.102/
$22.7
1355
21 (0.579 (84464/ 13.46
19.22 20.500 FO (0-4555 98.45
5
11E
35.56/ (0.2188 9.3334 54.62
76
156.3
27
25
(0.7964 0.3835 44.76
3607 FOUSTE
36.63 6,7583 /6763
(0.110) 55.01
17:73/ 0,530 10 4040
17.12/005100
(40
192.0
22.38
(0,066) 00,3130 63.75
75,
(0:229//
23-11
2705 FUUN2201 114275
(04201)
10
25
70.2764/ 16.75
(0:2531/
32.73 (02845
25.00
FO-7791 (02555)440
0,1968 41.58
27. 17.70/60726// IL F1L505 01 33
29:26/ 668253
(0.3000
38.15
20A1 0.632 0..708
10.00/ OLSONS 2013
770177
54.23
24
23 073772 57.00
(0.7024
15
LLE
26.27
27.25
(L77MS
2305 20.2566/
(8,2393
/42.00
14448
36.20
33.73 6.9146 0.1715) 38.92
(ON0///810//0000 022112 40.08
27:20 107285 11.1577
29.32/ B.226/ 16.1111 20.0
22.66 0,T03 (0.11.0/
19.50 0L5506 01,2000/ 09.78
55.22
24 SO FO. 2L22 (0.3255 50.55
25.75 (0.7376 9.27339 46.55
/0.659
70.111
753.2
24,00
3631 (67724/
1450/0..00
11910
1619/0000)) 885514/ 1201
11E
3471 00.635 9.3344 77666
28.74/ (9A399) 0,299) 66.00 60
37.56/
6,7979 (0.577/ 77.64
1653 0L4730 (9.0899/
117..1
19 (0.57) / 199.9
/0.300//
24.2 FOMMZS (AA42)
36:44 TL3266
71.57
19.20
/O.455)
LiE
38-52
74,40
0
11.
25.0 T 5839 TO 00.305// 59:78
26.55 707772 06377///01117777 07.00
31
51.35
17:55 (08211) 103862/165
103545
10
IE1E/05150 199 TO 0009
4671
21 IS 0058 70,000/ 76.50
/002915////
54.70 10 2029 78.200755.00
31,5 PO8R6P / 02256:448.5 40.50
35.40/ (UBAES 11.3009 56 45
17.58/ 05576 TO 45744 104,
22 FL FO.3855 56.33
L.E
26.00 A0
(L7309) 18 222900 50.35
33 0.9275 8.1539) 30.17
30-49 9:3640 (.2114) $3.00
18.80 0,5T78 O142HL 554.00
23 (LAG42 ( 9.3637 56.40
10
11B
37.00
(0.7099 (6.3499 46.43
70.2770
34,33 /0.339 (.111) 31,00
35.37 (81455
0.1842/ 30.43
51.07/
30.75
/OS71L// (962)-2933 39
6.8764 (L.167 (4677)
21442/ (.630) (0.3702 77.12
19:00/ 0.608 4020 55
24.01 / 2.24 0.3015 56.36
76 0756 3556 0150/55..5
36.29
(0093)) 36.58
10.57 0L551
24.01
L1E
(0.033)
31.00/
(0.31.3/ /5..4
Table 29: Results of DPS on FFHQ Dataset.
Caslito
Sleps
(ANIMMALPUS
Telfer (anbun)
Tegainiing
Ex. SR
CS 50%
Telter (ronllnnar)
3
12.
(0U799 A177
11513
27.01 /8.421/ 128.7
0.4700
24.31 (L463//
/66631
153.7
18.50 0.5466 03.54/
(0.4033 /0.900////00000
5881 CO
25/00/WARTE DI ATW6 1-4055
12 1106 7D 5288/
5
2005
5
1112
23 (298 704114/1455
(17131 (8.455
311.5
28,77
22 12.8//055555/.B8
/6.7633 8,3352: 112.5
0-4443 1543
25.55
16.33 86742/ F13616
138.
131.
19.56 (LBT75
15.00
0.4535 (D 2071
/350.3
285.8
19.20 02200/05/////09133 13964
19 / 18.5266/800 (.487) T6L.
7
123:
73.51 (0.000 70 4017
25.56
1063
116.
29,01 988244 8.2799 192.4
26097 8.7351 /122996
/6/7700//1 1866
99394
175.1
26166 ELETT4 (8.1008
15
161.7
335.1
20.22 (8.545) B.A4TS/
19 110110 5773 0571/////1 157
10
111:
26.0
(0.7457 /8.2008/ 95.11
31,10 (99992) 062251/1111111111 94.38
27.94 9.9866 (L2631
99.24
30.56 (L7103 (0.2411
2L18 (0.991 99.911/ 121..
73 201
101.2
20 (1
/0-4411
3
13.
23.53 5609 TO 10/2238/ 1587
24.01 /0.0066 (8.8133
1659
28,41 93356 0.2155 122.5
25.00 /0-713///441111 196
25.00 /17755 /63362/
37.13 6.7996 IN 11115//
14277
145.2
T8AT) 0L9675 (8.5224/
18.5 1 5T09 TO 5230 333.8
20044
I5 /O J032 6855 23L
16 I 4013 /0.900// 200LS
4
181:
22
2470
(17104 0.3764
70
129.7
38,19 030536 8.2337/ 196.5
23.75 (O/671) DJ401T 168
ST BUSESK IL28641
25.21/
133-1
1263)
1616: ELEE39 (0-42922
1L555 (O 5655// 125-7
237.1
1892 /O -60 06-30005500 JKLS
IS /0.8K25 /8.5178/ 1953
7y-
5
(O.T185 /.5444/
38,5 /.865T/ (.3182) 1090
18,38 BJNNS/ (13551
11788
19.41 BL6591 (0.487/
19.7 /0 5181 /553/
2003
26
EL31002 187.1
11E.5
IS
(O /O-0E15
10
26.10 00.299//
7346 5608 TO 7513/
101.
34.14/ /03079/
30 17 110000////77720000
6.1242: 51.41
111.6
25 50/ 90/558551// IB 12200
39:90/ 6.8611 (1.047) 94.51
195 54
20L7T 0.7351 (03022
18.5 B 1801 TO 19A00 317-4
$41.4
21.18
NN T9 /O 5017 101770 TEDSS
(9.9999
15
121:
23.71
26.56
(LTA654
118719 70:277
(8.7777
/ S0LLD
126-3
35,91 /88875)
54.71
/0L1405 50.25
BJ8866 26.40
36.49
10.34
DUSP 1L3919 76.12
0-5714/11221111111 (1.10)
25.61 EL5693 (1.1664) /72422
15. 13.16.1533 165211 2E00 /212.2
21 I5 2555 FO:4425 134.0
22,06/ 18.31 119.5
Table 30: Results of RED-diff on CelebA-HQ Dataset.
Conditio
Stepe
Smrlegy
PSNNESSIMM (FID)
Jepanting
Delter (cHnlinca)
3
L11
25.27 37/47/7002 (L7110 0.3555 / 46-47
24.91 AEaI 703227/ 53,35
24,00 9U222 6-4866 63.45
FOLSR55 FO..99//01111 D0153/ 122.5
15:45 FILS/ER FLanes 19318
36.92 8.7730 ILETEE 41.64
12.00 OLSTE 0L4664 93.57
16/8/BARC TO SESMIS 22566
20.31 /OURELS (LALS 95.67
ISSL FOASEE 111.
4
L1E
20
0.726/ 0,335 44.15
23.33 (8638)) 0.4399 77.53
24.81
/67266/ 0.3331
$3.77
17.33 0.300 10 (84883
134.9
2L19/0 2L49/0
/ 9,5688 $9.30
TTy
3005
(241 /0.333 /4190
22.17
(OJ26)
(L3014)
35:48 FUS934
20.20
7
54.5// (8.2962/
MI (677864 /0-51.00 11000
25.77
23.24
FOSSAN (JA838 19.25
(3737)) 6.3199 46.77
34,77 /67054/
25.18 FUANDN T1L160 RS K3
56.32
20.28/ 0L6306 (0.3726
IM FLANN (12 NON2 156.1
55.38
21 ME /.0 99515
10
L1E
27.02 /0.3319 /0.2999 15:43
26.59 59/
012474 (8.2652/
33.28
25,53
26,33 0.7547 / 8.3030 43.30
F0.9983 041144 61.45
15.36 /ULSST4 (U5229 663
(6883/
(L3602) 06 63.00
2115 0L5600 01.413
2035/00524 TO 454000 L1551
43.09
23.77/
21 /0.4619 F0.5415 79155
(1507/ 9.4757 $8.52
155
1.E
27.10 002541115567//1001
70
/ 36.89
28.41 (0,7663 6.3646 36.14
(03756)
35.51 FO5548
24.51 6.709)
(03726 05335
(65.5
21166 0L6897 (8.
0L543
(0.443/
/47.6
24 (0/317 (U.317/94334 75.50
25.
70.5237/ 56.46
/0.3338/
19.77 (0/2217
(0001:1 128.7
10:38
10.35- 160
13.13 003335 /0.537
LiE
(0.
0.7443
(1 1826 (0.5E/1/ 25.24
19JM
(OA1611 (1004/1 185.
25122 JUS43K//1 112106 172 AM
1E 1L5127 (1 2W35
/234-A
19.00 IU C1L0 (32223 IIK
1.E.
0,3037
57.36
24,45
667370
6,407)
39:34/ 668664
(0.388
A5
9.5164/
175.
/108/9.359
$5.50
5
LEE
(L7W03 /8 (828975/ / $1.49 26.00
06.17 /0 70.297/5572
755.05
0.9964 0.3138 T660 26387
23.77/
04736: 04.30
20:29 6 112477/66573
3000 (08569)
1.3155
54.30
75
18.55 0L5130
19.35 QUSTTS 01.877
(0.5075
76.21
L8F-3
20
21/22
(0.555500 (0.5920/ 182.5
015293 9.5536 96.71
11B
36.6 (L7708 0.6666 /ATAS
26,00/
98861
/ 6.2633 T3.61
36.31 6.836 In 111995
$3.55
26779 BUST39 (0.3307 /61.24
31 52 / /8.93) 102 3
45.25
23.45
70.205
0334500 13.00
30,30/ (88550) 102100
42644
13
LiE
/
62.35
47.76
15
11.
37.70 (4.3
NO (O 75477 TO
16.16
AN11
25.45
31 /8/0933/
(62959) 55.0 60
(.196)
51.13 D.8756 IN 19641/4143
668766/ OLIMRS 40
3377: 0.7666 (0.7188 (6.33
LB I
6836 TO /9337
34.71 (1136
23 54500055000011111
47

---

Table 31: Results of RED-diff on FFHQ Dataset.
Contriin
Steps
Delixr (enleo)
Tagaining
Deller (coelinon)
3
25.18 18 0U884 FO /
0,303
23/44 (9.5099 6,659) 140.2
18,01 F0.395 DI606: P9L9
14.96 FU5636 TUS 143.8
/6/370/
199.6
17.00:
16580 BLADSS TO DAN7
352.1
242.6
15 rO (0.770 7ESLO
4
L11
25,00/
45
(07065 ( (938699/ / 100.5
(877 78.851 / 1127
22.56 0.9964 / B4585/
19
7O00772522000 06352: 196
164.1
25:99 RA948 IL3631
34.23
1622
1264
17.47 0.497S 0L5433
12.0--4885 (O (1
2296
365-3
16.55 (0.9913 (8,7411 268.0
16.30/0
1992 F0.053 352.3
5
11E
25.91
/0.711 / 107.0
7098
23,85
(0/203 0.4037 133.5
33.10
66676/ 10,3633 137.6
18.46 0,5915 10 (94229) (13.22
18.35 (0,3275 (0,822) /317.5
70.2157
26.20
22,2
34,97
145-1
10
LLE
25 7202 3006
6033313 95.36
24.22 7O.565 (A5199 136
(.736) 0,3499 117.4
33.38
34.55 (0S54N IU55NN
00.3880
134.2
149.1
10000 (L480 1O 5220 220.5
0.6394
123.5
TO TO 5503 70/572/331
9.5779/
15
LLE
26.96 (257 /0.0612/ 105.5
26.77 (07599 (9.3009/ 99.77
25.21
27.27
FOU042 0.3939, 115.5
0.7637 / 8.3223: 106.5
25.83 (U5655 1L5367 131,6
25.25 R.795/
(LANC1
135.2
21.7 0L9884 01.5446
20,26 0,512 404221 155.5
107.0
20.05 /O 01.6119 (0.336 208.1
21.75 (L.5412 9.5345 ISLS
18.24 7.24/0/066/
19:00
(025770
1256
57511 331.3
15.0//0
250L
11E
35.60 (0.714 /0.3024
/ 121.3
34.57 1863E 0.4363
143.1
0.8411 (02477/ 1134
17.30 0,5313 (8. (8.545)
226.6
16.71 /0
0934666 0.74440 274LE
10.551 /0.3355
LB04
/04165 (0633)) 3060
18.39
/0320000/0000 FL35501 1410
0L505 10 5515
16.23
/O:2534
LiE
17.93
36.5
7O03 TO /
1878
/UA6/0511///SSSE 199
2810 RAATS/ 102810/
1218
18 I 5009 TO 5505 228.6
I7 7O 3963 70.7055/227 627
11..
36.30
25,15/ (4744 6,3859/ 147.3
113.8
18.T
26.00
10 007343 F0.2915 / 19.6
22.73
FOM0LE 048933 1682
19.33 B:846S 103855
118.7
18,00 (L50P) TO 12,2203 2600
18.2 /O 035008 +O.673 22263
L.E
26.1/ (U7798 (0,2913/ / 101.7
26.34 0.7773 8.3677: 142.9
1LT4: 0L8544 1O 0.23488/
$46.4
19 0.44.3 0.6400 217.4
10
11B
37 20
(0.7000 00.2902 96.56
7854 76 256776777 (57.51
37 888366
25
70.7173 DJ4665) 1663
0,3199 131.8
36:04
FUBE0S /02454/ 106.7
6.9690 (1.072 99.72
DAST/ 0L692) (0.3664| 135.5
4755 2211
20.85
19
( 0.8835 /8/033/ 199.1
/0.2040/
(83162
0.555)
L1E
/836666
/82.99
(0,3090/
0..7710 09.523
Table 32: Results of DifIPIR on CelebA-HQ Dataset.
Casliton
Sleps
Telfer (arbu))
Tegainiing
Ex. SR
CS 50%
Telter (ronllnzar)
3
12.
33/0//00
5754 /0 5532 50.45
24.53 8.5565 76.36
6634/67221/ /12447 /4.71
17.52 /
21. 21.00/0.000 00./4
(9.464 /6.31
1675//67700 II 1961 26
18 /D /185-7
21.85 /8.97/
2005
5
1112
17.17/ 6269 76.3578/ 45563
(L7637 /0.2900
24.35
27.31 88/95: 8,3451 46.96
(6.51
28:86 /07655 / 1122655
2/A3/ 8:379) (12429
37.90
34 96
19.36
18.07
0.515) (0 2057
0.5484
(04540)
1143.6
/84.4
24 0.5187 /0.4003
24.56 /0.5/ (131 /845533 E5,37
13 (0:0092
3185/
2K.77 /.7932/2/11 (0:167//44
/300
2010000570
01576 (0 4767 (43
75 (1 5611 12--346///
123:
25,00
23.55
28,55 8.7626/ 8.2883) 37.54
3T.47/ (171//00//////33 /1.390 '4.9
20A9 FLE287 (8.0311
/43.78
26.11 /0.0S4: 88,7947 448.49
10
111:
(0.998 /8.2155/
/0-220/36331
12.43
29.47 998174 8:2365/ 20.15
21.33
/1//3/////////// (12421)
01841
(03115) /45.70
27.44
/8.329 /8.63
29 (L.PPA /0.3118
38,19
(1.719// T3
37.89 8.7941 ILL254 (9)
22 11B1 (D "010 /41441
75 RID4
10.54
31.30/0/771/
3
13.
74 79 7936 70 2397 /0775/33377 33.75
(88799 (8.1967
24.81
25.85 0.2013 6,3956) 79636
18 09 1O 0032 (6615/ 166
ML19 BUSSRT/ 113396
TOOS /1 2535500/2255/0000 2513
85
17.67 0L4602 IO 0018/
17.50 BL8528 (B.0HES /376.6
275,9
24 /O 5086 7O (OU775 61 26
24773 /LATE /84279/ F1.54
4
121:
54.25 (L9104 8.1674
70 20.0000
22.43
33.0 (0.5475 0040
27,90 039957 8.3237/ 72.44
MLSS BJ8701 (1:1000 55 I7
10L53 IL2051
95.57
18,30 IASTE /1.5009 204,8
IS EL4TH5 5007 2311-5
25 OO 1996 (0-4722/ T5.03
24 /0,202 (8.0055-8877 88,7755 44.29
7y-
5
13.
(09131 /8.665// 22.48
21,45 90555//
63.57
3047 /08704 /L3901
(45.08
18S1 0L5911 /0.5352
27.77 (1.2044 99,3533
25.0 103571
MLSS BU8769 IL
19
AMTS/
RSTS /0.-000
10
34.9 (.9219 06.154
57.60 10 5357777770 7O 7053 /3.71
20.71
32.34 0.9115 6.1730 30.62
31 A8 /0
10.37 /03744/044//000 (L164) 36.20
138677 In TE90 31.11
20.53
28 BLSTI TO 0613/176
111.2
20.96 0.3344 00.2244/ 20.94
77 53 /O 6776 TO TEKT/ 10 TO
15
121:
52.30 K255 70.2372
16.28/
(L9269
08.1472
19-44
222.14
31.35 088288/
11.16 7000300/011100 0L1030//111///511 52.02
/
8.5450 30.55
31.15 /.386//
1144 (05756/
(L1572/ 32.61
21.85 FLA252 (8.4003
21.52/155156 (O 41431 /1COL
58.51
22.15
30.601 (18875 08,2933 36.15
(1) (18774 (O:5N4) 749.
Table 33: Results of DifiPIR on FFHQ Dataset.
Conditin
Stepe
Striegy
PSNETASIMU (FID)
Delia (ssis0)
Japanning
Delter (cHnlioca)
3
111
22.10 16/
22.29 10 36339 885628 1882
1 5588 (05753
156.
19A2
23.33/ B.998 / 6-48541144
FOLSESS DD3187 183.
14L1
25.49/ FLSISS 138.1
(087900000000000 (LARC)) 1139
17.28 94155 F0.0088/ 385.2
18412 FUIEET MOOLE
19.29 FO FQ.049 186.
266.3
4
L1E
35.33 0.5969 43233
131.
24.04/4.5744 04353/ 13L.
36.37
0.777/ 0,3974
91.33
17.55 04414 (0.555)
244.5
20
TTy
3005
23.84/100 5464 /0413)
23AL
702033
184 187 106.0
36-92
11.50-
21.27
/9.559// 141,3
7
37.63 /0.244/////2
27 10 D0IS 10:5556 /95.4
25 (0/933)
25 MN FO:S0N D- 112M
6,3458
1OL.7
3673//07547/
0137788/91139
1916
15.00 IL SEDEO (O 00C2S/
22 /O (8.7788) 120 g
10
L1E
$2.90 (0, (09) /0.2545//// 15192
21 33/
(L7790 18 (82690 75.12
28J77 8.7606 0.9940 96.15
27.48 FO/7227 033304 90.12
1674///771/ (0.7007 (83 (3.40
107515 (037944
95.00
19.60 GLE205 0..9550
10.55/ TO A2462 L.554
$31.4
23 (0.5%0 (9.485 /117.2
23 1L.58/00.244--22
1103
155
L.E
27.70 1 /0.3633/ / 78.34
28,98 908244
28.00
6,2478 77.55
55.55
2600 (07500/ (0.3677
99.92
DAN1 06987 (0.3031/
5541
99.02
SI (0/006 9.3533 16LS
3675
(11331 (0.2203
18.99 (UCC1)
113179
12.01
16 02.16/02003
(O.333/
LiE
(0.459
11 20222S5/ 29.57
23.17
105001/ 1L.1 1 16L6
30.33 BU8199 111260 110.1
12.25 13 GARS (1 MARRS 111.
23 /U 20T5 /OAL/S/ 17N3
1.B.
33.79/
66.1766
24:53 (
6.733/
1363
39:49/0000
103.2
0.5345
261.
21
5015/9,5453
5
LLE
3412/
$1.0
009908 1011077 / 51.00
52222 /022200/5550
15.63
24.45
21
/OAPPS/ 04412/144 143.8
0.9034 / 8.3301 125.0
19-39
008584/ 102210
(L25000 1030
97.14
18.00 0.4697 (1.7788
(..56L//200
291.2
21 (0.475500591 FFLS
22 A2 (LSSLS 9,5143 135.5
11B
3432 1
081633 / 50.33
29.44
98633 6.7666 107.3
29:55 /U8666/
18.1 IL 4986 18 (0.5301
248.5
23 DU ( 1267
10/
102135 15.10
23,55
7033507
822057 102.0
30.30 NU813R
70.5480/
LiE
(L3767 72.771
19.16:
24.27/
15
11.
/
31 (8/0059
51.30 20.8998
162066/52 52.10 10
6.1933 75.06
ML54: BURTNG IN 16811185 6l
(0.100 89.14
2IL12 O OL5NN6 173.9
SR IN STAL TO 46007
118T.3
25 58
24 /18/1 110.
FO 5037 70355367 1566
48

---

Table 34: Results of ReSample on CelebA-HQ Dataset.
Contriio
Steps
Delinr (eelno)
Tagaining
Telter (coelinon)
3
54.58/ O 466S 005055
0.613 9,4144
55.32
24.28 (0.5389)
18.53
(66200 126
6,467:
05.64//6811/
22:53
FOSSIS 2E5E3R3333311 13
57.00
17.57 04164 (0.777/ 2204
16.57
BA40G TO 5922
/27.//2444
21/0
20 75 FO 5855 /0.5943/////// 58030
(0.57% 68.11
4
L11
26.50 (LA002 (0,3933 / 55.40
24.55 10 484422 7O (8/9255 /18.9
23,21
25,68
0.5865 / B-8844 57.30
92.45
9:7266
(L3172)
44,4)
12400/44457 TO (U.5AA
18.77
1 64601 1 OLO0IE
11854
22 M0 0.9954 -8.233 66..20
21 78 1L./800.367 FO.5511 753.3 15
5
11E
26.22 10,048 003383/ / 15.56
70
26,07
(O/265 0,3690 49.35
34.1
25.49
/67864/////////// (0.400 59.60
17.17
20.1:
0L513 (0 094589 /95.3
23 39 00.4344
/8,438// 59.33
114775 /04778//
25:93
(OA177
35.39 F00350/
00.77
21.27
(0.5134)
10
24.00/ 10.4207773
12.50
21.51
TO (0.433))
6.3365 40.00
37 667247/ 10:3455
26.33
FU6507 (1A555
44.12
33.51 0,550M
TO 30993
/6428
531.44
24
IN TO AATO 50555
9.776 57.65
15
LLE
26.54 54/
12/
10.4055 00/9907/ (554)
0,59133 (8,3641/
55.67
25/22
28.28
(OA554 0.3256: 19.41
0.6688 0.38599 37.52
27.24: 8,7218/ 10.351 43.08
27.18 (USSSP) BLA30 (5.19
25,7
26.18
0L5585 00.3615/ /48.58
0L5258 01.4255
443.2
24.11/ (0.462 9.4709 $7.19
23 2) (L.-150 (0.9999 57.65
70.25677
733,68
(0.4000 DuG0E5/
23.49 70,340// (0.3527) (3.700
(55555
/2515
23 /0.559 70.4935 06.63
1.E
24 24/
(0.4459 00.3264/ 33.86
24.21 (0.7144/ 0.3837/ 77666
28.40/ 6.8364/ (0.7704 7L.9
17.00
0L5343 (0.5336
2231.7
34.0 (..9%%
0.4574 61.64
/0.133//222
23.25
(A4133
3504
(L3072/
19,20
23-71
LLE
/
28.54/
(07459
0
1..
50 188675 20.2044//370
/6.1990
26.00 76.755/ 75.57
30.17 (03737/ 6.2470
17.66 DUB: 26 23.56
668350 (0.303
30 BL504 TO 2564 1550
23 7364 70 3457
50.75 10 70.1063 26:41
30.04
FO80500 82436: 55.04
29.21 7G8435 10.342 /63.76
22.33/0
196 TO 00.6225 (96.4)
28 FO 78755 (0,2000 35.4]
L.E
50.65 (0.59L (B.189T/
25.14
31,45 0.M51 8.19644 47.63
19A8: 9:550// 102308
55 45
23.38
0.7425 013259/ 76/04
29,07 0.8296 - 9.2634/ 31,42
10
1.B
50 56
86780 1996
5623
(611837
725.55
24.38
33.16
32.
70.916 (1155))
8.1616 36.30
36.16 6.8666 (L.996 (91.6
30/04
(03552 /0:14115/5333 30.00
25.06/
25.50/
0.7963
(0.3735 556991
3058/ 71 05
30.73
1 8617 /8,2300/ 24.42
8529 02.255//57.74
/0.1868/
(BL365)
30.68/
L1E
34.24/
/8.355
37.17
(0867// 0..1918/47.0
(0.2223/
(0,8775 /9.194/
Table 35: Results of ReSample on FFHQ Dataset.
Caslito
Sleps
JANIMAALUSS Ex. SR
Debfee (anbun)
Tegainiing
CS 50%
Telter (roniinzar)
3
11.
7 /221///
23.38
10.3000 B.6778
141.1
35:06/66677 FL4190 124.7
17.16:0.4106
19.6// /4776
9199
56/86/005H55/ STAUL 146-7
1247/86611 96/4/
19.97 t0 /RATTR//// 166.1
2005
5
111:
24.18
(L8117 /84101 1157
4622 70.5100/111 141.7
24,33
25.95 /SA339
/4.33/.5577//bb555 (-865///55//11111 127.
8.3987
113.4
24-58
15,82 BURST/ (ILTIW
0U6001 /0.481
125.9
10L17 0.3133 (0,4921
18.25 0.4668 (0 5325 233.6
174.4
21 (0.399 (BARRR 151.
20 25 /0 /070// 1496
7
123:
26.27
(LLAS1 (8.9154
70 9933
116.0
26,85 /8A76// 8,3737/
76:01 /0: 10-0000/// 10000/1111 111
161.4
16.38 8,707// IL3T16
/18326 /11119/11111
131.0
1134
20AD) IL5484 (0.6301
/159/7
(339.
21.77 6L5677-8.5777 /35S
10/0 5T7 554// 1455 G
10
111:
26.30
(1,015 (8,4115
27.48 (BA35) 8,3529 90.96
26.3 8.7T9 FL3600
1998
21.9 0L5531 (0,003)
$34.5
22,90 9.3911 (8.5495 1360
TRAS
110-3
/O TAANS/
WIT 5156/ 135.4
3
13.
29
/ 5TA6 70
20.2088/100 75128
77.03
24,22 9.633) 6.463 1458
19.0 2OW.//0/////111 1971
23 59 /0756P in 1913/
37.57 BASINO/ IL3945:
14944
129.4
17.40) OLSIKS F..0
12 TO 5731 3355
317.3
21.33 O 4L4D65 /8/041
20 I1 /0 70.005 178.1
4
181:
20.0
1L8643 (8:2255 69.08
33.00 (0
27 0.7907 8,3530 182.5
(0/001 1576
25.76
3601 BU8265/ IL26EN
/U/R01 IN 1295/
181.2
12888
18700 ELST92 (8.0991/ 239A
91
22.56 TO 0--21 158
23 (8.5097 (8.5322 156,2
7y--
5
(0,8809 08.110/
25.30 88831/ 8,3135/
18 8.837// (1.5000 128.1
19.60 BLEDST (L4501/
23.76 (L.962 4757
201 (1.611
3SM3/O8THH/ 50355/ 1179
FUST99
121.5
1O 4100 /1R.
2163/0 BETS /0-4228)
10
50 5o (18607 TO 7007 766
0.22050
64.26
31.36 (029989)
30 /0/855 21/66////71///000 DE 70
6,2184/ $7.44
20
19.36 8.3536 (L2107 98.45
/U8/// 11 T5A1 /1008
22.96 0,7572 (13200/ $21.8
31 55 1 TTMS ID SO01
/1417
75 35 /O GRAE TO 3836 TISS
(L7193 (9,775/ 110LS
15
121:
39.70
53.00 87221
(LR727
/8.2015
70.2052/
/ 62.40
33.81 (88217)
12A7/OM171 2111211 (181
8.1744 67.36
19-55 FEUSON
39,87
B69629/
11.110 197.8)
(L30733 199.1
24,8. (LS139 /1)
23. 15- 1957 (O 225221 1ORKS
(8.2025/ /9645
26.29 (0.7225 (8,3249 186.
26. 16 /O 11122111-0.000
-0.3000000000// 382.5
Table 36: Results of DAPS on CelebA-HQ Dataset.
Conditio
Stepe
Smriegy
PSNETSSIMI (FID)
Delia (ss5so)
Japanring
Delter (cHnlioca)
3
L11
3452/
24.09 (EAT20 /AA///////5/77 55.90
(L5005 A474 58394
19.54
21.6 /03345
FO.SOF/ 7 119.
8.5233 74.14
21.55/0444// FLSTBR 66:22
/A835/ NA113 $2.38
16BL ALEV2S FLAATE 2463
16 16.10:033148 IO FUIMRR 22553
22,00 FO.200 (8.9857 75876
22.5 (0.422900.555 /0.5133/ S0L50
4
L1E
35 105525 9,492/ / 50.32
24.58 (8.229)
8A359: 96.76
06465 (.3T1L 48.09
17.54: (L400 (0.5985
160.4
23 06 0.467440.4497
/ $4.56
TTy
3005
2577
(02314
15
FLL100 47.60
0L400
23.56 /84444444
7
(6.327/ (3981
25
26/70
6,3591
36.35 /6/999/ (0.377)
25 ON FOAZ11 IL 41774
43.55
0.530005100
(1 G017T $3.55
77.58
/
10
L1E
27.25 /0.805 70.314// 1502
27.43/ 01:846 (8,2999/
33.93
27
21.56
/OS457 033300 39.54
9.659 / 0,3350 46 16
25,53 7188400 1L31666 35.59
36.99 8.7220 (L3012) 51.97
31-16/05550 (O 01,3555 (6.31
22.0 0.5667 (0,3051/ 59..8
25 FE /O (U.54D -0.3533 742.01
25 (U.477 / -8,3565 /4.55
155
1.E
27.65
0.799 / / 36661
70.2856
28.74/ 9,2002 6,2999 35.08
21,47 (C (03055)
STAS 11728/6006//00000 (029561029111
37.33
6/714/ 07.53/07744// (028000333 33.33
23.5
24 BLEMS (8. (0.444
3500
46.05
26 30 O 5806 0.3336 37.59
57.574477 3548 755.3
/0.200/
15.13/ (0.2765 0199527 127.
5007 2218.
(00000 (0.3934
LiE
30.26
(1) /0:2010 / 10L13
23.42 FONSTS BAADNN 19.66
ZELL JUATSS 1L3321 41165
12-00
ELSAME (12 5054/
1996
27.45
FU CALL /O.5E11 S7,
1.E.
36.11
26.31
6,3699
56.77
38/71 /67744/10 (0.0044
18.00
23
5
LLE
32
50.00
008891 (8.2022 / 25.30
7927 70.248/31.8 /31.80
25.52
9.7479 / 8.3160 51.54 27
(1322200552222 03353) 56.46
21.22 /06550/ FL3000 13.35
20112 8.7147 113845 44.00
1915 UL5464 (0,427/
18.5/
(O 4807777
LS5.8
136.3
23 0080644 0.2727 22.51
20.30/0.33
7547 (0.3057 23.57
11B
33.55
103866 011844 23.35
29 98176 6.3666 42.1
39.47 (.799) 16447/67999 (13693 41. 96
31.56/ BL6A3 (8. 10.3075
(66.0
36 (8816 9,3387 24.93
/OL1935
31.20 (035547 (12466
FO0506/ 101526 28.0
22.60 FUTOON 3361 705.56
20
LiE
0.22344
15
11.
24.4
STAS
(6.163
30.37
17.89
31,65/ (33949)
33 AT (0.5788
/OTBS// 34.44
6.176) 70.70
39:35
1641 /6876) /03445//30 20
34.9
18.32 OUT810 (0.37N (6:30
25 11T0 TO
2754 45.85
23 09 TO 70.21 22. 15
23.73
49

---

Table 37: Results of DAPS on FFHQ Dataset.
Cadiiio
Teltee (aibuo)
Tegainiing
Ex. SR
CS 5095
Telter (renllenar)
18.1 /8,3777
0.4741 (9.4858/ 132.1
/LASN 126.3
16.54 0,367
/0.309// 337.4
19.0/0.200
1184
DUER86 DAATR 1218
5041/ 787.5
2185
/8.8884
3:05
5
22 5353 70 /
/1126
24,8 0.5513
64346/
120
26.24/05562/ F.4421
IL1133
1157
1194
17.50
64411
0.4299 (0 5053
/0.500
235.5
212.8
21,8 (0.397 129.5
/0.3991/9.22.1/
599 70 1994
10%.5
75 /0.4241
1019: 112.1
15.07/
1000
113.3
EL47H9 0018/117 /1723
22.9/
22.97/ /84549/ 121L5
123:
26.25 FUB16
1834
25.55 25.75/0.501/
0.5883 8.7861 1055
866648 1L3814
1146
10.S I 0-4700 (8.0005
16177
23 26 (0 (0/218/0-4007
/
/0.3613/5877
1111
26.79 (0.617 /8.2077 99.59
26.72/94194
(SA19) 033664 52.47
15.92 BAN9T/ FL3565
195.7
3144:005522
(0.4305
21.7 (045322 (8.463
70-1948 90.10
TA /UAE/T 15E7 101-3
TE 1O61/
21,1/0 /0-4133
78 5015 / 5005
18 ///5/////// 200
54 /US555 1055
199.2
16 0-131 (093
3756
70 11/0001126/0 5901 T66
13.
36.70 0077633 /6.2657
76.23
28.55 0.3887 6.5338 1667
37.00 BU6896 IL3851
125.
1633 0L4623 (154)
392.5
21.21 (0.4441 0.5000 1667
/0.290/133.1 73.34
23.70 /0.5144 DUABOTT
DUBCSS 1120
19.65
12.29/ BL4727 (8.5619
201.5
22,57 70-0577 131
183:
51.1 (1R248 10:34410:22344 86.55
/ BASES / 8-4220
127-2
/671/6 (LL300 111.5
12.52 12 9600 7O 5622/
201.1
23 S (USSII) (87757 12ML
/8.2261
99/034 8,3662 116.5
18.11
(L.3227
(8.514/
FUARRG 1V:1 83345
10
0.1867 54.45
20.91 0.822) 8.2629)
75
3OV 7A5T WIL17
78. /C7776 7801 77.51
012823 $7.14
7875/ BUSTT TO 3857 /141
(1300A
75 S6 GRBS TO 3151/ 95.53
15
33.75 FLSAA6
BU1TES 11007
747.55
11.39 TOOSTS DUZAOS MAD1
31.51 (8872))
8.2151 62:15 R7.15
2012N FUOISTT
29-41/ BUBB9O) DI (12606/172
FL2MES 70.31
22.91 1358
26.86/11 16352
(L.1AA (11-4
187.5
26 IA (OLTEA 19.122
26.27 /0.7566
50

---

I More qualitative results
Here we present more visualization results. Anisotropic Deblurring results are shown Figure IT In- painting results are shown in Figure 12 Super-Resolution results are shown in Figure 13 Compressed
Sensing results are shown in Figure 14
Nonlinear Deblurring results are shown in Figure T5
DMPS Ong CelebA-1IQ
DMPS LLE CelebA-11Q
DMPS. Orig. ITRQQ
DMPS LLE
HIGDM Oris. CelchA-1IQ
HIDDM LLE.
HGDM Oriz.
HIGDM LLE FFJQ
Steps
Steps
Steps
Steps
10 Stops
15 Steps
y
33
Figure 11: Visualization of anisotropic Deblurring.
51

---

DAPS OOfig Celeba-1IQ
DAPS L.E. CelchA-IIQ
DAPS Onig FFIIQ
DAPS LLE FFIQ
DitPPR Onlg Ceaba
DHPPR LLE CelebA-IlQ
DIAPIR Ong FPIQ
DHAPIR LLE FFIQQ
Steps
4
Steps
10
Steps
Figure 12: Visualization of Inpainting.
52

---

DDNM Onli. CelebA-IIQ
DDNM LLE
CelebA-IIQ -IQ
DDNM Ong. FFIIQ
DDNM LLE FFIQ
ReSample- LLE ReSample. Ong. ReSumple LLB ReSample Ong. FFiIQ
Celbb HQQ
CelehA-IIQ
FiQQ
Steps
4 Steps
Steps
Steps
10 Steps
15 Steps
Figure 13: Visualization of Super Resoluteso
53

---

IGDDM Orig. CelebA-IIQ
IIDDM LLE
CelehA -IQ
HGGDM- Orig.
FFJIQ
IIDM LLB
FFRQQQ
DDRM Otig
DDRM LLE CelebA-IIQ
20
25
2
DDRM Ong FEJQ
DDRM LLE FFRQ
3
Steps
Steps
10 Steps
18 Steps
Figure 14: Visualization of Compressed Sensing.
54

---

DPS Orig Celeba-1IQ
DPS LLE. CelchA-IIQ
DPS .Ongg FTQIQ
DPS. LLE FFQQ
RED Origg
Celbb HHQ
RED RDD--iff LLE.
CelebA-IiQ
RED-did LLE RED diit Oni
FFIQ
FFIQQ
Steps
4
Steps
Steps
10
18 Steps
Figure 15: Visualization of nonlinear Deblurring.
55


Reviews:
[{"review_id": "HfOu52eyfH", "reviewer": "Reviewer_hxDr", "summary": "The paper proposes Learnable Linear Extrapolation (LLE) which learns coefficients to combine previous estimates and the current prediction \nfor improved inference in solving inverse problems using diffusion models with few steps. The authors rewrite several diffusion-based algorithms for inverse problems \nin a canoncial form which can be combined with LLE. They show consistent improvements for the different algorithms on several standard reconstruction problems.", "questions": "- Using the history of estimates combined with learned coefficients has also been used in Bespoke Non-Stationary Solvers (https://arxiv.org/pdf/2403.01329).\n  Can the LLE method be seen as learning a type of non-stationary solver as defined in the paper?\n- How are the time steps chosen? Are they just uniformly spaced? Are there methods to choose them in a more optimal way?\n- What happens if we choose a large value for the number of steps $S$ and LLE? Do we still see improvements?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- LLE shows consistent improvements for a wide range of different algorithms \n- The canoncial form combines several existing algorithms in a unified framework, which is useful for practitioners\n- The paper is well written and easy to follow\n\nWeaknesses:\n\n- LLE requires additional finetuning and cannot be used out of the box\n- Corallary 2 that motivates LLE is based on ODEs, while the canonical form is non-deterministic due to the additional noiser step. I saw in the appendix that there is an extension to the more general SDE case; this should be highlighted more in the main paper\n- Improvements are only shown in the few-step regime (up to 14 steps); how large are the gaps to methods taking significantly more steps?\n- I would appreciate if the experiment section included problems where the diversity of solutions/full posterior is evaluated.", "was_revised": true, "final_justification": "I thank the authors for the thorough rebuttal. I will keep my positive rating.", "rebuttal": "We sincerely thank the reviewer for acknowledging the effectiveness and generality of our method. Below, we provide responses to the concerns and questions raised. **All additional experimental results are reported in terms of PSNR ↑ / SSIM ↑ / LPIPS ↓.**\n\n**1. \"LLE requires additional finetuning and cannot be used out of the box.\"**\n\nWe appreciate the reviewer's comment. We'd like to highlight that the training process of LLE is highly efficient, and LLE demonstrates cross-domain generalization ability. First of all, as shown in Table 6 on page 19, the training phase of LLE takes only a few minutes, making it practical and feasible in real-world scenarios. Moreover, as presented in Table 4 on page 18, LLE exhibits a certain degree of cross-domain generalization. We further validate this in a more challenging ImageNet-to-FFHQ cross-domain setting, where LLE still achieves good performance (see the table below, where we test DDNM with 5 steps). This indicates that domain shift does not significantly undermine the effectiveness of LLE. Lastly, we acknowledge that LLE currently requires retraining for different inverse problems. Given the minimal training cost, we consider this an acceptable trade-off. Designing more efficient ways to enhance LLE's performance on cross-task generalization is a promising direction for future work. \n\n| Testset  | Trainset | Deblur (noiseless)                | Inp (noiseless)                   | SR (noiseless)                | CS (noiseless)                    |\n| -------- | -------- | --------------------------------- | --------------------------------- | ----------------------------- | --------------------------------- |\n| FFHQ     | -        | 39.82 / 0.964 / 0.087             | 26.34 / 0.719 / 0.380             | **29.99** / **0.865** / 0.210 | 18.29 / 0.591 / 0.488                 |\n|  FFHQ        | FFHQ     | **40.08** / **0.966** / **0.084** | **28.71** / **0.838** / **0.292** | 29.94 / 0.863 / **0.206**     | **19.21** / 0.605 / 0.463             |\n|  FFHQ        | ImageNet | 39.94 / 0.965 / 0.086             | 28.30 / 0.818 / 0.317             | 29.94 / 0.863 / 0.208         | 19.01 / **0.608** / 0.470         |\n| ImageNet | -        | 36.48 / 0.949 / 0.099             | 23.35 / 0.628 / 0.416             | 25.74 / **0.754** / 0.302     | 17.86 / 0.535 / 0.486             |\n|  ImageNet        | ImageNet | 36.52 / 0.949 / **0.098**         | 24.87 / 0.726 / 0.364             | **25.76**  / 0.752 / 0.297    | **18.35** / **0.541** / **0.478** |\n|  ImageNet        | FFHQ     | **36.53** / 0.949 / 0.099         | **25.14** / **0.759** / **0.341** | 25.70 / 0.750 / **0.296**     | 18.20 / 0.527 / 0.485             |\n\n**2. \"The SDE version of the derivation should be highlighted more in the main paper.\"**\n\nWe appreciate the reviewer's valuable suggestion. As the reviewer pointed out, most diffusion-based inverse algorithms are formulated in the SDE framework, and the injected random noise often plays a significant role in improving solution quality. Thus, the SDE-based derivation provided in Appendix B.3 offers a more complete explanation of the motivation and formulation of LLE. We will highlight the SDE formulation in the main text and explicitly clarify its connection to the motivation behind LLE in the camera ready version.\n\n**3. \"How large are the gaps to methods taking significantly more steps?\"**\n\nRegarding the comparison between the few-step LLE-enhanced method and the original algorithm with significantly more steps, we kindly refer the reviewer to Table 8 in Appendix C.5 (page 20), where we compare the performance of DDNM with LLE within 15 steps and the default DDNM with 100 steps. The results show that in many cases, LLE-enhanced DDNM with  only 15 steps achieves performance comparable to the 100-step baseline.\n\nRegarding the comparison between the LLE-enhanced method and the original algorithm both under significantly more steps, we supplement an additional experiment comparing DDNM with and without LLE under a larger number of sampling steps (25, 50, and 100) on the FFHQ dataset, as presented in the table below. To avoid the need to store an excessive number of past results when the number of steps is large, we limit the lookback window to 10 steps. The results show that as the number of steps increases, the performance gap between methods with and without LLE becomes smaller. This is consistent with our expectation as LLE searches for an improved sampling trajectory within the linear subspace spanned by previous samples. When the original algorithm takes more steps, the discretization error is reduced, and its trajectory already closely approximates the optimal one. In such cases, LLE tends to follow the original trajectory more closely, resulting in smaller gaps.\n\nTherefore, LLE brings more significant improvements in few-step regimes, which aligns with our primary objective of enhancing the performance of inverse algorithms under a limited number of steps. We will include these results and the accompanying discussions in the appendix as an additional ablation study. We appreciate the reviewer's thoughtful question.\n\n| Steps |      | Deblur (noiseless)        | Inp (noiseless)           | SR (noiseless)            | CS (noiseless)            |\n| ----- | ---- | ------------------------- | ------------------------- | ------------------------- | ------------------------- |\n| 25    | -    | 41.14 / 0.972 / 0.064     | **34.33 / 0.941 / 0.109** | 30.92 / **0.885 / 0.146** | 25.01 / 0.837 / 0.220     |\n| 25    | LLE  | **41.35 / 0.974 / 0.061** | 34.23 / **0.941** / 0.110 | **30.93 / 0.885** / 0.147 | **25.61 / 0.841 / 0.215** |\n| 50    | -    | 41.57 / 0.974 / 0.056     | **35.48 / 0.953 / 0.077** | 31.14 / **0.889 / 0.142** | **28.94 / 0.897 / 0.158** |\n| 50    | LLE  | **41.71 / 0.976 / 0.055** | 35.33 / 0.952 / 0.079     | **31.15 / 0.889 / 0.142** | 28.81 / **0.897 / 0.158** |\n| 100   | -    | 41.70 / 0.974 / 0.056     | **36.29 / 0.960 / 0.058** | **31.41 / 0.883** / 0.140 | **32.15 / 0.927 / 0.122** |\n| 100   | LLE  | **41.86 / 0.976 / 0.054** | 36.18 / 0.959 / 0.060     | 31.37 / **0.883 / 0.139** | 31.56 / **0.927 / 0.122** |\n\n**4. \"Evaluating the diversity of solutions/full posterior.\"**\n\nRegarding the diversity on the full test set, we have included results in Appendix H that report FID scores, which serve as a quantitative measure of diversity across the dataset. Regarding the diversity of individual posteriors, we would like to clarify that the image restoration tasks considered in this paper typically have highly concentrated posteriors. For instance, in the 4$\\times$ super-resolution task on 256$\\times$256 images, the solution is generally close to deterministic and does not deviate significantly from the ground truth. As a result, a common practice in such settings is to evaluate performance using reconstruction metrics such as PSNR, SSIM, and LPIPS. We will include visualizations of solutions corresponding to different initial noises in the camery ready version to better illustrate the diversity of solutions. Sorry for the inconvenience that we are unable to include any visualizations during the rebuttal phase due to submission constraints. We thank the reviewer for this insightful suggestion.\n\n**5. \"Can the LLE method be seen as learning a type of non-stationary solver?\"**\n\nYes, LLE can be viewed as learning a broad class of non-stationary solvers as defined in the provided paper. Specifically, the non-stationary solver in the paper is formulated as a linear combination of all previous steps' estimates and velocity fields (or scores equivalently). Accordingly, LLE can be interpreted as a data-driven approach to learn the optimal coefficients of this linear combination. We will include this paper as a reference and discuss the connection between LLE and these non-stationary solvers in the camera ready version. We thank the reviewer for the valuable comment!\n\n**6. \"How are the time steps chosen? Are they just uniformly spaced? Are there methods to choose them in a more optimal way?\"**\n\nYes, in this paper, the time steps are uniformly spaced, as stated in Appendix F.2 (line 716 on page 25). We will add this clarification in the main text to make this setup clearer.\n\nAdaptive selection of time steps may indeed further improve the performance of inverse algorithms. For example, by leveraging LD3 [R.1], the time steps of inverse algorithms could be parameterized and optimized in a data-driven manner to find more optimal schedules. Combining such adaptive time step schedules with LLE is a promising direction for future work. We appreciate the reviewer's suggestion and will include this discussion in the camera ready version.\n\n**7. \"What happens if we choose a large value for the number of steps for LLE?\"**\n\nSee our response to point 3 above.\n\nWe would like to express our gratitude once again for the reviewer's comments and suggestions for our work. Should there be any further questions, we welcome continued discussion.\n\n\n\n[R.1] Tong, Vinh, et al. \"Learning to Discretize Denoising Diffusion ODEs.\" *ICLR*, 2025."}, {"review_id": "4nZSYQ8vg9", "reviewer": "Reviewer_enUs", "summary": "This paper introduces \"Learnable Linear Extrapolation\", a method to improve the performance of diffusion-based inverse algorithms with few sampling steps by leveraging previous estimation.\nThe authors propose a canonical form that unifies several diffusion-based inverse algorithms by decomposing them into three modules: Sampler, Corrector, and Noiser.\nThe method  operates by learning optimal linear combination coefficients to refine current predictions using previous estimates predictions.\nThe coefficient are pre-learned my minimizing a ``reconstruction + w * perceptual`` loss with respect to a reference dataset.\nThe authors validate the method through extensive experiments across multiple algorithms and tasks.", "questions": "None", "limitations": "- Memory Footprint: The method necessitates storing previous estimate, which could lead to considerable memory overhead\n- Out-of-Distribution Performance: the method might suffer from out-of-distribution samples as the extrapolation coefficient are computed beforehand on a selected dataset.\n- Algorithm Dependence: the method is dependent on  the specific diffusion algorithm employed. changing the diffusion sampler requires re-training to get the extrapolation coeffcients.", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths:\n\n- A rich framework that encompasses various diffusion posterior samplers\n- A novel method that improves performance with reduced required number of sampling steps\n\n\n## Weaknesses:\n\n**Theoretical Justification:**\n- Ambiguity in the justification for $\\hat{x}_0$ being a linear combination of previous estimate, particularly in Equation (10), it is unclear why it holds and a reference would great.\nIn fact, empirical observations of the extrapolation coefficient in Figures 6-8 suggests minimal dependence of the current iterate on previous estimates, in fact the matrix of weights is dominated by the diagonal or first diagonal elements, while the rest are almost zero.\n- The choice and size of the ground-truth datasets are not adequately addressed. Also the robustness of the method to out-of-distrbution samples. \nWhile the authors assess the the robustness on datasets FFHQ and CelebA-HQ, it is of little relevance as both dataset represent human faces. A more relevant discussion would be on Imagenet dataset, for instance.\n\n**Practical Considerations:**\n- **Computational Overhead:** Essential computational aspects, including the training time required to obtain the extrapolation coefficients and the memory requirements of the method during inference (as it requires storing previous estimates), are not discussed.\n\n**Inconsistencies:**\n- **Error in equation (2):** There is a missing logarithm in Equation (2) \n- **Figure 4 Clarity:** Figure 4 is unclear and lacks essential information. Specifically, the values of $ w $ are missing, and while the axes are labeled (LPIPS and PSNR), the context or meaning of $ w$ in relation to these metrics is not provided.", "was_revised": true, "final_justification": "To sum up,\n\n- the proposed idea is original\n- my concerns were addressed during the rebuttal\n- the authors committed to fixing the highlighted inconsistencies and typos in the manuscript\n\nOn the other hand, the work has few limitations\n\n- the method is bound to a specific algorithm (requires re-training for each algorithm)\n- minor improvements in out-of-distribution setup (e.g. using the method trained on Imagenet for FFHQ) as shown in the authors response \n- low dependencies between iterates (the matrix of weights) that may compromise the effectiveness of the extrapolation\n\nAs conclusion, I retain my rating", "rebuttal": "We sincerely thank the reviewer for recognizing the novelty and generality of our proposed method. Below, we provide point-by-point responses to the raised concerns and questions. **All additional experimental results are reported using the metrics PSNR ↑ / SSIM ↑ / LPIPS ↓.**\n\n**1. \"Ambiguity in the justification for being a linear combination of previous estimate, particularly in Equation (10), it is unclear why it holds and a reference would great.\"**\n\nEquation (10) employs a forward difference approximation of high-order derivatives. The related derivation can be found in [R.1] and in book [R.2] (Chapter 6.5). We will add these citations in the revised version to make the presentation clearer. We sincerely thank the reviewer for the helpful suggestion.\n\n**2. \"The matrix of weights in Figure 6-8 is dominated by the diagonal or first diagonal elements, while the rest are almost zero.\"**\n\nYes, and this is consistent with our expectations. According to Equation (10), elements farther from the diagonal correspond to higher-order derivatives of the sampling trajectory. Since most sampling trajectories are expected to be smooth, the high-order derivatives decay rapidly to zero, which results in the weight matrix being dominated by the diagonal and first off-diagonal elements. In a few cases, we observed non-zero elements farther from the diagonal, indicating possible non-smooth regions in the trajectory with larger high-order derivatives. We will include this discussion in Appendix C.1. We thank the reviewer for this insightful comment.\n\n**3. \"The choice and size of the ground-truth datasets are not adequately addressed.\"**\n\n**Regarding the train set for LLE**, we used 999-step DDIM to sample 50 synthetic images from the diffusion model as the reference set, as stated in the paper (line 52 on page 2, line 223 on page 7, and line 722 on page 25). Using only synthetic images makes LLE more practical in real world applications, since diffusion-based inverse algorithms assume access to a pre-trained diffusion model, thus making it straightforward to construct the reference set required by LLE.\n\n**Regarding the test set in our experiments**, we follow common practice (e.g., references [5, 6, 12, 14] in the main paper) to sample test examples, as described in the experiment section. Specifically, we use 1000 test samples for CelebA-HQ and 100 samples for FFHQ (and ImageNet in the additional experiments).\n\nWe will further add a paragraph at the beginning of the Experiment section to clarify these settings. We sincerely thank the reviewer for helping us improve the clarity of our paper.\n\n**4. A more relevant discussion of cross-domain performance on Imagenet**\n\nHere we supplement cross-domain experiments of LLE between the FFHQ and ImageNet datasets. We test DDNM with 5 steps, and the results are shown in the table below (\"–\" denotes results without using LLE). LLE demonstrates generalization ability across FFHQ and ImageNet. The coefficients trained on one dataset can still improve performance on the other in most cases, and sometimes even outperform those trained in-domain. This further indicates the out-of-domain robustness of LLE, which is critical for ensuring its general applicability.\n\nMoreover, we have conducted broader evaluations of LLE with multiple inverse algorithms on the ImageNet test set; due to space limitations, we kindly refer the reviewer to our response to *Reviewer bt2H (Point 1)* for details.\n\nWe will include these new experimental results in the camera ready version. We sincerely thank the reviewer for the valuable suggestion.\n\n| Testset     | Trainset    | Deblur (noiseless)                | Inp (noiseless)                   | SR (noiseless)                | CS (noiseless)                    |\n| -------- | -------- | --------------------------------- | --------------------------------- | ----------------------------- | --------------------------------- |\n| FFHQ     | -        | 39.82 / 0.964 / 0.087             | 26.34 / 0.719 / 0.380             | **29.99** / **0.865** / 0.210 | 18.29 / 0.591 / 0.488                 |\n|  FFHQ        | FFHQ     | **40.08** / **0.966** / **0.084** | **28.71** / **0.838** / **0.292** | 29.94 / 0.863 / **0.206**     | **19.21** / 0.605 / 0.463             |\n|   FFHQ       | ImageNet | 39.94 / 0.965 / 0.086             | 28.30 / 0.818 / 0.317             | 29.94 / 0.863 / 0.208         | 19.01 / **0.608** / 0.470         |\n| ImageNet | -        | 36.48 / **0.949** / 0.099         | 23.35 / 0.628 / 0.416             | 25.74 / **0.754** / 0.302     | 17.86 / 0.535 / 0.486             |\n|   ImageNet       | ImageNet | 36.52 / **0.949** / **0.098**     | 24.87 / 0.726 / 0.364             | **25.76**  / 0.752 / 0.297    | **18.35** / **0.541** / **0.478** |\n| ImageNet         | FFHQ     | **36.53** / **0.949** / 0.099     | **25.14** / **0.759** / **0.341** | 25.70 / 0.750 / **0.296**     | 18.20 / 0.527 / 0.485             |\n\n**5. Training time required to obtain the extrapolation coefficients and the memory requirements during inference.**\n\n**Regarding the training time**, we reported the training time of LLE with different inverse algorithms in Table 6 of Appendix C.4 (page 19). Training LLE takes only 2 to 20 minutes, thanks to its lightweight design and the use of a small reference set.\n\n**Regarding the memory footprint**, here we further report the GPU memory usage (in MB) of DDNM and DPS with and without LLE in the table below. We measure the CUDA peak allocated memory, and we account only for the memory used by model parameters, data, and additional storage required by the algorithm. The results show that LLE introduces only a few to several dozen MB of additional memory overhead, even under the 15-step setting. This is consistent with our expectations: during inference, LLE only needs to store a few 256$\\times$256 RGB images from previous steps, typically around 0.7 MB per image, resulting in at most a few dozen MB of additional cost. This is negligible compared to model weights or other algorithm-specific overhead (e.g., back-propagation in DPS).\n\nIn summary, both the training and inference overhead of LLE are minimal, which further supports the lightweight advantage of LLE. We will include these additional results and discussions in the camera ready version. We sincerely thank the reviewer for the helpful suggestion.\n\n| Steps    | 3       | 4       | 5       | 7       | 10      | 15      |\n| -------- | ------- | ------- | ------- | ------- | ------- | ------- |\n| DDNM     | 590.25  | 590.25  | 590.25  | 590.25  | 590.25  | 590.25  |\n| DDNM-LLE | 597.50  | 599.37  | 601.50  | 608.25  | 612.75  | 626.25  |\n| DPS      | 2388.78 | 2388.78 | 2388.78 | 2388.78 | 2388.78 | 2388.78 |\n| DPS-LLE  | 2396.41 | 2398.41 | 2400.41 | 2407.29 | 2411.79 | 2429.04 |\n\n**6. Missing logarithm in Equation (2).**\n\nWe thank the reviewer for pointing out the error, and we will correct it accordingly. The correct form is: $\\epsilon_{\\theta}(\\mathbf{x}_t, t)=-\\sqrt{1-\\overline{\\alpha}_t} \\nabla\\_{\\mathbf{x}\\_t}\\log q_t(\\mathbf{x}_t)$.\n\n**7. The values and meaning of $\\omega$ in Figure 4.**\n\nIn Figure 4, the ablated parameter $\\omega$ refers to the weight of the LPIPS loss in Equation (20). We tested $\\omega = \\{0.0, 0.025, 0.05, 0.075, 0.1, 0.2, 0.3, 0.4, 0.5\\}$, which correspond in the exact order to the points from the top-right to the bottom-left in Figure 4. This result suggests that smaller values of $\\omega$ encourage LLE to achieve higher PSNR, while larger values lead to better LPIPS. This provides a direct and practical way to control the trade-off between PSNR and LPIPS, depending on application-specific preferences.\n\nWe thank the reviewer for the helpful comments. We will annotate the corresponding $\\omega$ values in Figure 4 and add this explanation and discussion to the figure caption in the camera ready version. Sorry for the inconvenience as we are unable to include any updated figures during the rebuttal phase.\n\n**8. Limitations-\"The method is dependent on the specific diffusion algorithm employed.\"**\n\nWe agree that LLE requires training for each specific inverse algorithm. This is because different algorithms vary in their design principles, formulations, and posterior score estimations, which lead to different sampling trajectories. Consequently, the high-order approximation coefficients of these trajectories differ, necessitating separate training for each algorithm. Designing a more general approach, including aligning sampling trajectories across algorithms or developing unified high-order approximation parameters, is a promising direction for future work. We thank the reviewer for this comment.\n\nWe would like to express our gratitude once again for the reviewer's comments and suggestions for our work. Should there be any further questions, we welcome continued discussion.\n\n[R.1] Fornberg, Bengt. \"Generation of finite difference formulas on arbitrarily spaced grids.\" *Mathematics of computation* 51.184 (1988): 699-706.\n\n[R.2] Süli, Endre, and David F. Mayers. \"An introduction to numerical analysis\". *Cambridge university press,* 2003."}, {"review_id": "sSXjbmdB0i", "reviewer": "Reviewer_bt2H", "summary": "This paper tackles the high computational cost of diffusion-based inverse problem solvers, particularly in few-step scenarios. First, it proposes a \"canonical form\" that deconstructs diffusion-based inverse algorithms into three distinct modules: a Sampler, a Corrector, and a Noiser. This framework is designed to unify a wide range of existing methods. Second, based on this canonical form and an analysis of ODE solvers, the paper introduces Learnable Linear Extrapolation (LLE), a lightweight method that improves few-step performance by learning to linearly combine the current estimate with previous ones. The authors demonstrate that LLE consistently improves results across nine reformulated diffusion algorithms on several inverse problem tasks, such as deblurring and inpainting on facial datasets.", "questions": "1. How is LLE expected to perform against the baselines mentioned in the weakness section?\n\n2. What are the specific technical challenges in applying the canonical form and LLE to Latent Diffusion Models? Does linear extrapolation hold in the latent space, and how does the VAE's compression affect the Corrector module?\n\n3. How can the generalization of results from facial datasets to more complex domains like ImageNet be justified? The distinct priors of faces versus general scenes could affect the validity of the learned extrapolations.\n\n4. Please elaborate on the conceptual differences and key trade-offs (e.g., performance vs. training cost) between the LLE method and consistency-based approaches like Consistency Posterior Sampling.", "limitations": "A more thorough discussion of the challenges involved with LDMs and a justification for not addressing them is warranted.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "#### **Strengths**\n\n1.  LLE method is a lightweight and intuitive approach, well-motivated by an analysis of high-order ODE solvers. \n2.  The proposed canonical form is an insightful contribution that provides a structured, unified view of diverse diffusion-based inverse algorithms.\n\n#### **Weaknesses**\n\n1. The claim of enhancing performance lacks validation against the true state-of-the-art. The baseline set, while broad, omits recent, competitive methods like  LGD (Song et al., 2023), DCDP (Li et al., 2024), MPGD (He et al., 2024), FPS-SMC (Dou and Song, 2024), and  SGS-EDM (Wu et al., 2024).\n2. The over-reliance on facial datasets (CelebA-HQ, FFHQ) is not representative of general-purpose restoration; validation on a diverse benchmark like ImageNet is needed to prove general applicability. \n3. The paper's complete focus on pixel-space models is a critical weakness. As LDMs are the de facto standard for high-resolution restoration, a method that does not demonstrate a clear path for applicability to them has severely limited practical significance.\n4. The paper fails to discuss or compare its approach with consistency models (e.g., Consistency Posterior Sampling). This is a major concurrent research direction for accelerating inverse problem solvers, and its omission makes the literature review and experimental comparison incomplete.", "was_revised": true, "final_justification": "I thank the authors for their thorough rebuttal. In light of their response, I have raised my score to Borderline Accept.", "rebuttal": "We sincerely thank the reviewer for acknowledging the motivation and general applicability of our work. Below, we provide detailed responses to the concerns and questions raised. For clarity, we organize our response as follows:\n\n (1) We first present additional experiments on ImageNet to further validate the generality of LLE in more challenging settings.\n\n (2) We then report the performance of additional baselines on the FFHQ dataset as suggested by the reviewer, including five pixel-space methods, three latent-space methods, and one method based on consistency model.\n\n**All results are reported using PSNR ↑ / SSIM ↑ / LPIPS ↓ for consistency.**\n\n**1. \"Validation on a diverse benchmark like ImageNet is needed to prove general applicability.\"**\n\nWe supplement the results of $\\Pi$GDM, RED-diff, DPS, and ReSample on ImageNet with 4 steps and 10 steps in the following table. Following [12, 14] in the main paper, we sample 100 images from the ImageNet test set for evaluation. For the noisy tasks, we add Gaussian noise with $\\sigma = 0.1$ with respect to data range [-1, 1] (same in the following experiments). We use the ADM pre-trained checkpoint ([4] in the main paper).\n\nThe results demonstrate that LLE consistently improves performance across all methods in the few-step setting on ImageNet. This confirms that LLE is effective and broadly applicable on more complex datasets.\n\nIn addition, we conduct a cross-domain experiment between FFHQ and ImageNet. Due to space limitations, we kindly refer the reviewer to our response to *Reviewer enUs (Point 4)* for detailed results. These results further supports LLE's out-of-domain robustness.\n\n| Steps|Algo|  Deblur (noiseless)|  Inp (noisy) |SR (noiseless)| CS (noisy) |\n|:-|:-| :--:|:--:|:--:|:--: |\n| 4|$\\Pi$GDM |23.56/0.727/0.294|18.75/0.408/0.641|17.41/0.373/0.557|14.60/0.217/0.746|\n||$\\Pi$GDM-LLE|**30.14/0.875/0.196**|**19.41/0.435/0.619**|**20.53/0.524/0.531**|**16.73/0.359/0.650** |\n||RED-diff |22.57/0.589/0.490|18.00/0.357/0.592|24.81/0.719/0.419|17.10/0.409/0.574|\n||RED-diff-LLE|**22.7/0.602/0.486** |**21.69/0.524/0.492**|**25.19/0.726/0.359**|**17.77/0.468/0.531** |\n||DPS |15.90/0.268/0.681|22.92/0.672/0.421|17.10/0.358/0.567|15.30/0.300/0.633|\n||DPS-LLE |**18.42/0.388/0.622**|**24.41/0.690/0.410**|**20.33/0.517/0.533**|**18.03/0.474/0.556** |\n||ReSample |25.89/0.746/0.355|20.49/0.440/0.521|23.18/0.651/0.417|17.50/0.439/0.540|\n||ReSample-LLE|**26.18/0.757/0.340**|**23.28/0.561/0.443**|**24.98/0.712/0.369**|**17.80/0.454/0.529** |\n| 10|$\\Pi$GDM |26.22/0.789/0.289|21.99/0.609/0.487|21.49/0.532/0.503|16.21/0.360/0.681|\n||$\\Pi$GDM-LLE|**29.92/0.856/0.216**|**23.69/0.611/0.471**|**22.76/0.596/0.450**|**18.61/0.509/0.540** |\n||RED-diff|23.41/0.620/**0.458** |22.95/0.539/0.449|25.75/0.749/0.349|18.49/0.466/0.517|\n||RED-diff-LLE|**23.42/0.623**/0.462|**25.07/0.669/0.386**|**25.92/0.754/0.309**|**19.22/0.513/0.487** |\n||DPS |19.55/0.426/0.608|24.65/0.711/0.364|21.28/0.520/0.512|16.61/0.413/0.612|\n||DPS-LLE |**20.33/0.463/0.555**|**27.17/0.780/0.317**|**22.42/0.585/0.460**|**19.52/0.590/0.463** |\n||ReSample |26.49/0.765/0.331|25.26/0.625/0.367|25.27/0.721/0.353|19.60/0.519/0.449|\n||ReSample-LLE|**26.72/0.772/0.317**|**25.78/0.651/0.349**|**25.61/0.736/0.329**|**20.02/0.541/0.434** |\n\n**2. Including more baseline methods like LGD, DCDP, MPGD, FPS-SMC, and SGS-EDM.**\n\nWe supplement experiments with LGD, DCDP, MPGD, FPS-SMC, and SGS-EDM on noisy tasks on FFHQ with 4 steps. The results are reported in the **Pixel space** part of the table below. The results show that LLE consistently improves the performance of all five baselines in the few-step regime. This further validates the generality and broad applicability of LLE.\n\n**3. \"What are the specific technical challenges in applying the canonical form and LLE to Latent Diffusion Models?\"**\n\nThere are no difficulties in extending the canonical form and LLE from pixel space to latent space. The core theoretical analysis of linear extrapolation still holds. The observation function with LDMs is $\\mathbf{y} = \\mathcal{A}(\\mathcal{D}(\\mathbf{z}_0)) + \\sigma \\mathbf{n}$, where $\\mathcal{D}(\\cdot)$ denotes the decoder of the VAE, $\\mathcal{A}(\\cdot)$ is the observation operator, and $\\mathbf{z}_0$ is the latent. This formulation corresponds to a nonlinear inverse problem with respect to $\\mathbf{z}_0$, since the composed operator $\\mathcal{A} \\circ \\mathcal{D}$ is highly nonlinear. Our theoretical motivation remains valid, as it does not rely on any specific assumptions about the form of the observation function. \n\nTo solve inverse problems with LDMs, the Corrector (or the inverse algorithm) must be capable of handling nonlinear observation functions. Moreover, as long as the inverse algorithm is applicable with LDM, LLE can still serve as an effective enhancement strategy.\n\nTo validate LLE with LDMs, we include results of Latent-DPS (introduced in [12] of the main paper), PSLD [R.1], and STSL [R.2] on the FFHQ dataset. The results are reported in the **Latent space** part of the table below. LLE consistently improves performance for these LDM baselines. The only adjustment required for training LLE in the latent space is a minor change to the loss in Equation (20), which we modify to $\\mathcal{L}(\\mathcal{D}(\\tilde{\\mathbf{z}}_{0, t_i}^{(n)}), \\mathcal{D}(\\mathbf{z}_0^{(n)}))$, since LPIPS is computed in the pixel space. These results further confirm the generality and effectiveness of LLE when applied to LDMs.\n\n**4. The conceptual differences and key trade-offs between the LLE method and consistency-based approaches like Consistency Posterior Sampling.**\n\nWe would like to clarify that LLE and Consistency Posterior Sampling (referred to CPS below) [R.3] aim to improve performance from different but complementary perspectives. Specifically, LLE enhances the efficiency by learning a better solver over the original trajectory, while CPS leverages the strong single-/few-step performance of consistency models to improve reconstruction quality.\n\nMoreover, we note that CPS also fits into our proposed canonical form, and LLE can be applied to further enhance CPS's performance in the few-step regime. More specifically, CPS uses a consistency model as the Sampler, rather than a traditional diffusion model, but it still generates a sequence of $\\mathbf{x}_0$ estimates, forming a valid trajectory. This trajectory can be directly used with LLE's linear extrapolation framework.\n\nWe supplement experiments with CPS on the ImageNet-64 dataset. The results are shown in the **CM** part of the table below. We use the `openai/diffusers-cd_imagenet64_l2` checkpoint available on Hugging Face, and test on 100 randomly sampled images from the ImageNet test set.\n\nLLE consistently improves CPS in the few-step setting, demonstrating that LLE and CPS are not exclusive, but rather complementary approaches. This highlights LLE's broad applicability and its ability to boost performance when applied to advanced sampling frameworks with consistency models.\n\n| Steps|Type|Algo|Deblur (noisy) |Inp (noisy)|SR (noisy)|CS (noisy)\n|:-| - |:-|:-| :-- |:-|:-|\n| 4|**Pixel space** |DCDP|26.00/0.619/0.428| 24.75/0.558/0.436|24.16/0.585/0.578| 17.70/0.461/0.535|\n||| DCDP-LLE|**26.12/0.681/0.378**|**25.54/0.596/0.411**|**25.24/0.707/0.350**|**18.16/0.471/0.524** |\n||| LGD |22.81/0.645/0.433| 21.77/0.548/0.513|24.95/0.680/0.439| 15.84/0.451/0.513|\n||| LGD-LLE |**24.59/0.695/0.376**|**28.25/0.768/0.339**|**25.55/0.705/0.402**|**16.52/0.503/0.499** |\n||| MPGD|25.29/**0.705**/0.380|27.33/0.753/0.309|24.93/0.562/0.579| 17.36/0.437/0.565|\n||| MPGD-LLE|**25.43**/0.704/**0.366**|**27.76/0.768/0.298**|**25.42/0.703/0.360**|**18.02/0.446/0.543** |\n||| SGS-EDM |**27.72**/0.776/0.308|19.71/0.416/0.538|26.53/0.738/0.337| 17.72/0.482/0.530|\n||| SGS-EDM-LLE|27.67/**0.782/0.307**|**24.44/0.511/0.453**|**26.95/0.762/0.309**|**18.40/0.516/0.486** |\n||| FPS-SMC |25.11/0.517/0.475| 17.52/0.188/0.686|24.62/0.524/0.474| 16.58/0.224/0.666|\n||| FPS-SMC-LLE|**26.26/0.617/0.428**|**18.63/0.229/0.643**|**26.14/0.672/0.400**|**17.66/0.384/0.569** |\n| 4|**Latent space**|Latent-DPS|16.42/0.428/0.636| 15.43/0.426/**0.628**|15.30/0.369/0.682| 14.51/0.364/0.665|\n||| Latent-DPS-LLE|**17.08/0.441/0.609**|**17.83/0.492**/0.652|**15.49/0.399/0.648**|**14.70/0.366/0.646** |\n||| PSLD|17.55/0.468/0.630| 17.71/0.474/0.627|17.72/0.471/0.630| 15.06/**0.438**/0.647 |\n||| PSLD-LLE|**18.10/0.477/0.607**|**18.55/0.490/0.599**|**18.57/0.489/0.597**|**15.24**/0.433/**0.644** |\n||| STSL|16.08/0.378/0.655| 16.87/0.404/0.639|15.70/**0.368**/0.663|15.31/0.383/0.629|\n||| STSL-LLE|**16.97/0.392/0.616**|**17.50/0.407/0.598** |**16.27**/0.360/**0.628**|**15.72/0.384/0.621** |\n| 4|**CM**|CPS |12.69/0.204/0.613| 12.71/0.269/0.649|14.65/0.317/0.559| 14.80/0.242/0.619|\n||| CPS-LLE |**16.50/0.283/0.522**|**17.64/0.383/0.566**|**18.57/0.410/0.471**|**16.58/0.268/0.580** |\n\n**5. Ethical Concern**\n\nIt was unexpected for us to receive an *Ethical Concern* flag, as all datasets and algorithms used in our work are publicly available and widely adopted. We'd appreciate it if the reviewer could elaborate on the specific ethical concerns in detail.\n\nWe would like to express our gratitude once again for the reviewer's comments and suggestions for our work. All additional experiments and comparisons, including those on ImageNet and nine baseline methods on FFHQ, will be included in the camera ready version to further support the generality and validity of LLE. Should there be any further questions, we welcome continued discussion.\n\n[R.1] Rout, Litu, et al. \"Solving linear inverse problems provably via posterior sampling with latent diffusion models.\" *NeurIPS*, 2023.\n\n[R.2] Rout, Litu, et al. \"Beyond first-order tweedie: Solving inverse problems using latent diffusion.\" *CVPR*, 2024.\n\n[R.3] Xu, Tongda, et al. \"Consistency model is an effective posterior sample approximation for diffusion inverse solvers.\" *ICLR*, 2025."}, {"review_id": "EZi9KjDC70", "reviewer": "Reviewer_7BP1", "summary": "This paper proposes learnable linear extrapolation (LLE), designed to improve the performance of any diffusion-based inverse algorithm that follows the steps: (i) predict the clean signal, (ii) enforce measurement consistency, and (iii) map back to the noisy manifold to continue the reverse sampling process. The main insight of this paper, as far as I can tell, comes from Corollary 2, where they show that the estimate of the clean sample conditioned on the measurements, denoted $\\mathbf{x}_{0}(t, \\mathbf{y})$, is a linear combination of the estimate at the current timestep along with all previous timesteps. Motivated by this, they propose solving an optimization problem to learn such linear coefficients in order to obtain an accurate estimate of the clean signal, provided that a reference dataset exists.", "questions": "- Why are the results for DMPS in Table 1 so poor? I suggest the authors to look into this to provide a more fair comparison.\n- How precisely do Equations (19–20) need to be optimized? Did the authors find that the loss associated with this problem is easy to optimize? In cases where the optimization falls into a local minimum, did the authors find that LLE still outperforms the original method? I believe these questions are fundamental to understanding the effectiveness of LLE, if they have not already been addressed.", "limitations": "I have stated the limitations under weaknesses above. I would be willing to raise my score if my questions can be effectively addressed.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "I would say that the main strength of this paper is that they can improve upon existing diffusion-based inverse problem solvers in a lightweight fashion. Table 7 shows that LLE does not induce a large overhead, but can improve upon the existing methods up to some extent.\n\nFor weaknesses, I am not sure how effective this method would actually be considering that the improvements are not great. I would suggest the author to put more qualitative results such that we can look at the performance differences a bit better. On top of that, this method assumes that there exists a reference dataset to perform the optimization, which I am not sure holds in practice. How were these reference points chosen for experiments? I perhaps missed this in the main text.\n\nI also think that the presentation of this paper could be improved; I found the presentation of the methodology a bit convoluted. Perhaps there is a way to condense the main text to bring the algorithm box up to the main part.", "was_revised": true, "final_justification": "The authors have addressed my points and hence I have raised my score. I do still have some uncertainty about the actual practicality of the method, so I am on the fence.", "rebuttal": "We thank the reviewer for recognizing the lightweight advantage and general applicability of our method. Below, we provide point-by-point responses to the concerns and questions raised. **All additional experimental results are reported in terms of PSNR ↑ / SSIM ↑ / LPIPS ↓.**\n\n**1.  \"I am not sure how effective this method would actually be considering that the improvements are not great. I would suggest the author to put more qualitative results.\"**\n\nWe sincerely thank the reviewer for the valuable comments and helpful suggestion. We would like to clarify that in many cases, LLE improves the PSNR of the original algorithm by close to or more than 1dB, which is generally regarded a considerable improvement in the context of image restoration. Moreover, we consistently observe improvements across all three metrics—PSNR, SSIM, and LPIPS—indicating that LLE not only reduces pixel-wise reconstruction error but also enhances perceptual quality.\n\nRegarding the qualitative results, we appreciate the suggestion and will include more visualizations in the camera ready version. In the meantime, we refer the reviewer to Appendix I (pages 35–39) for additional qualitative examples beyond those presented in the main paper. We apologize for the inconvenience that, per this year's policy, visual results cannot be included during the rebuttal phase. Thank you again for this thoughtful feedback.\n\n**2. \"This method assumes that there exists a reference dataset to perform the optimization, which I am not sure holds in practice. How were these reference points chosen for experiments?\"**\n\nIn all of our experiments, we use 999-step DDIM to sample 50 synthetic images from the diffusion model as the reference dataset. This design ensures that LLE remains practical in real-world scenarios: since diffusion-based inverse algorithms assume access to a pre-trained diffusion model, it is straightforward to generate the reference dataset using the same model without requiring any real images.\n\nWhile this setup is mentioned at several points in the paper (e.g., line 52 on page 2, line 223 on page 7, and line 722 on page 25), we acknowledge that the current presentation may not be sufficiently clear or direct, and could lead to misunderstanding. We will revise the Experimental Setup section to explicitly describe the construction of the reference dataset used by LLE. We thank the reviewer again for this helpful question.\n\n**3. \"I found the presentation of the methodology a bit convoluted. Perhaps there is a way to condense the main text to bring the algorithm box up to the main part.\"**\n\nWe appreciate the reviewer's suggestion. We will move the algorithm box into Section 4 of the main text in the camera ready version to improve the overall flow and coherence. We will also refine the presentation of the methodology section to make the description clearer and more concise.\n\n**4. \"Why are the results for DMPS in Table 1 so poor? I suggest the authors to look into this to provide a more fair comparison.\"**\n\nWe thank the reviewer for pointing out this issue. We have double-checked our implementation of DMPS and confirmed that it is consistent with the original paper. We also verified our implementation under 100 and 1000 sampling steps, and the results closely match those reported in the original work. In fact, we observe that our implementation achieves comparable performance with the original paper using as few as 15 steps (see Table 18 and Table 19 on page 30), which supports the correctness of our implementation.\n\nUpon further investigation into the hyperparameter settings of DMPS, we find the primary issue lies in the scaling parameter $\\lambda$. While the original paper suggests setting $\\lambda \\geq 1$, we find this choice to be suboptimal in low-step regimes, especially when the number of steps is below 10. To address this, we conduct a step-by-step and task-by-task hyperparameter search for DMPS on the CelebA-HQ dataset with 3, 4, 5, and 7 sampling steps. The table below shows the optimal values of $\\lambda$ we identified. \n\n| Steps | Deblur | Inp  | SR   | CS   |\n| ----- | ------ | ---- | ---- | ---- |\n| 3     | 0.2    | 0.2  | 0.3  | 0.2  |\n| 4     | 0.3    | 0.3  | 0.4  | 0.3  |\n| 5     | 0.4    | 0.4  | 0.5  | 0.4  |\n| 7     | 0.5    | 0.6  | 0.6  | 0.5  |\n\nUsing these revised hyperparameters, we re-evaluate both the original DMPS and the LLE-enhanced version on noisy tasks on CelebA-HQ. The updated results are shown in the table below. We believe these results provide a more fair and accurate comparison for DMPS. We will incorporate them into the camera ready version, along with revised results on noiseless tasks on CelebA-HQ and on FFHQ. We sincerely thank the reviewer for helping us improve the fairness and credibility of our experimental evaluation.\n\n| Steps | Algo     | Deblur (noisy)            | Inp (noisy)               | SR (noisy)                | CS (noisy)                |\n| ----- | -------- | ------------------------- | ------------------------- | ------------------------- | ------------------------- |\n| 3     | DMPS     | 14.02 / 0.174 / 0.747     | 11.91 / 0.100 / 0.802     | 15.53 / 0.235 / **0.763** | 12.45 / 0.102 / 0.791     |\n|       | DMPS-LLE | **14.10 / 0.177 / 0.744** | **11.97 / 0.104 / 0.800** | **18.76 / 0.253** / 0.776 | **12.52 / 0.105 / 0.789** |\n| 4     | DMPS     | 17.95 / 0.322 / 0.619     | 13.41 / 0.144 / 0.762     | 19.33 / 0.365 / 0.645     | 14.09 / 0.151 / 0.741     |\n|       | DMPS-LLE | **18.23 / 0.325 / 0.617** | **13.55 / 0.149 / 0.759** | **23.07 / 0.511 / 0.544** | **14.19 / 0.155 / 0.738** |\n| 5     | DMPS     | 21.20 / 0.404 / 0.562     | 14.71 / 0.164 / 0.738     | 22.95 / 0.498 / 0.541     | 14.85 / 0.165 / 0.719     |\n|       | DMPS-LLE | **21.63 / 0.406 / 0.558** | **16.45 / 0.186 / 0.690** | **24.66 / 0.625 / 0.437** | **15.57 / 0.180 / 0.699** |\n| 7     | DMPS     | 25.12 / 0.559 / 0.446     | 15.87 / 0.187 / 0.716     | **25.96** / 0.669 / 0.396 | 16.04 / 0.213 / 0.679     |\n|       | DMPS-LLE | **25.84 / 0.576 / 0.436** | **20.12 / 0.337 / 0.562** | 25.82 / **0.707 / 0.355** | **17.36 / 0.303 / 0.594** |\n\n**5. \"How precisely do Equations (19–20) need to be optimized? Did the authors find that the loss associated with this problem is easy to optimize? In cases where the optimization falls into a local minimum, did the authors find that LLE still outperforms the original method?\"**\n\nWe thank the reviewer for the insightful question. We find that the optimization of the objective function (19–20) requires relatively few steps, is easy to optimize, and is not prone to getting stuck in local minima.\n\nTo better understand the impact of optimization iterations on LLE's performance, we conduct an additional experiment evaluating LLE trained for {25, 50, 100, 150, 200} iterations. The experiment is performed using 5-step DDNM on the FFHQ noiseless tasks. The results are summarized in the table below. We observe that LLE achieves comparable and strong performance when trained for iterations between 50 and 200. This suggests that LLE converges quickly with relatively few iterations and does not show obvious signs of overfitting. \n\nTheoretically, this is reasonable since LLE learns interpolation coefficients by minimizing a weighted sum of MSE loss and LPIPS loss. When the LPIPS weight $\\omega$ is zero, the problem reduces to a least squares problem, which is convex and free of local minima. Although the LPIPS loss is non-convex, our choice of a relatively small weight ($\\omega = 0.1$) ensures that the overall loss landscape does not contain many undesirable local minima, making optimization easier in practice. We sincerely thank the reviewer for this insightful question and will include these additional experiments and discussions in the final version of the paper.\n\n| Train steps | Deblur (noiseless)                | Inp (noiseless)                   | SR (noiseless)               | CS (noiseless)                    |\n| ----------- | --------------------------------- | --------------------------------- | ---------------------------- | --------------------------------- |\n| -           | 39.82 / 0.964 / 0.087             | 26.34 / 0.719 / 0.380             | **29.99** / **0.865** / 0.211 | 18.29 / 0.591 / 0.487             |\n| 25          | 39.93 / 0.965 / 0.086             | 28.21 / 0.821 / 0.316             | 29.93 / 0.863 / 0.207        | 18.87 / 0.597 / 0.476             |\n| 50          | 40.12 / **0.967** / 0.085         | 28.61 / 0.834 / 0.303             | 29.95 / 0.863 / 0.208        | 19.13 / 0.602 / 0.468             |\n| **100**     | 40.08 / 0.966 / **0.084**             | 28.71 / 0.838 / 0.293             | 29.94 / 0.863 / 0.206        | 19.21 / **0.605** / 0.463             |\n| 150         | 40.10 / 0.966 / **0.084**             | 28.77 / 0.840 / 0.290             | 29.91 / 0.862 / **0.204**    | 19.23 / 0.603 / **0.461**             |\n| 200         | **40.14** / **0.967** / **0.084** | **28.82** / **0.842** / **0.288** | 29.93 / 0.863 / **0.204**    | **19.25** / 0.604 / 0.462 |\n\nWe would like to express our gratitude once again for the reviewer's comments and suggestions for our work. Should there be any further questions, we welcome continued discussion."}]

Author rebuttal:

Meta-review: The paper examines the computational challenges of diffusion-based inverse algorithms, which achieve strong performance but typically require many denoising steps. The authors analyze ODE solvers in the inverse setting and identify a linear combination structure underlying the approximations used to trace inverse trajectories. From this perspective, they propose a canonical form that unifies various diffusion-based inverse methods. Building on this framework, they introduce Learnable Linear Extrapolation (LLE), a lightweight technique that refines predictions by optimizing linear combination coefficients of current and past estimates, thereby mitigating the instability of analytical solvers. The authors also carry out experiments for various algorithms and tasks to show that LLE enhances efficiency and performance.

Reviewers highlighted multiple strengths: the framework provides a clear and structured perspective on diverse diffusion methods, the idea of learning linear combination coefficients is interesting, and the method consistently yields performance improvements across a broad set of inverse problems. The paper is also praised for its minimal training overhead, and thorough ablations. Some weaknesses were raised, including overlap with prior work on non-stationary solvers, the reliance on facial datasets in the initial submission, and missing comparisons to certain SOTA baselines and consistency models. However, the rebuttal effectively addressed many of these issues by adding diverse experiments, clarifying theoretical connections, and discussing extensions to broader settings. Ethical concerns regarding dataset bias and potential misuse were acknowledged and will be addressed in the final version. While one reviewer was not fully satisfied with the paper and the response of the authors, overall I think the paper merits acceptance.
Decision: Accept (poster)