Direct Intermediate Initialization for Tilted Diffusion Samplers

arXiv:2610.06834v1 [stat.ML] 5 Oct 2026

Direct Intermediate Initialization for Tilted Diffusion Samplers

Gregory D. Bellchambers PhysicsX Ltd. London, EC2A 4DQ greg.bellchambers@physicsx.ai

Abstract Some diffusion posterior samplers construct Gaussian-tilted intermediate distributions along the reverse process. We observe that these targets can be pulled back to clean-space posteriors with weaker conditioning, with samples transported analytically to the corresponding noisy-space target through a Gaussian bridge. For the sequential Monte Carlo (SMC) sampler MCGDiff, the effective observation variance of this pulled-back problem is up to twice the diffusion-noise variance. We exploit this structure to initialize MCGDiff directly at an intermediate time: an approximate solver samples the softened clean-space posterior, the Gaussian bridge maps these samples to the tilted target, and only the remaining SMC suffix is run. This trades asymptotic consistency for finite-particle performance. With moment-matching posterior sampling (MMPS) as the solver, the hybrid improves sliced Wasserstein distance by roughly 2× at matched particle count on a structured Gaussian-mixture inverse problem, and by more than an order of magnitude when the posterior-relevant mode is rare under the prior. A prior-initialization control, which retains the bridge but drops the clean-space conditioning, shows that on MCGDiff’s standard Gaussian-mixture benchmark most of the improvement is insensitive to the conditioning. Conditioning the initialization gives a further consistent gain on the structured problem, and becomes decisive on a rare-mode problem, where resampling cannot repopulate a mode absent from the initial population.

1

Introduction

Diffusion generative models [Song and Ermon, 2019, Ho et al., 2020, Karras et al., 2022] provide expressive priors for Bayesian inverse problems [Daras et al., 2024]. Given a pretrained unconditional model, training-free posterior samplers typically incorporate observations either through approximations to the posterior score [Chung et al., 2023, Rozet et al., 2024] or through a sequence of auxiliary distributions that gradually introduce the likelihood [Cardoso et al., 2024, Janati et al., 2024, Wu et al., 2024]. The latter include particle and SMC [Doucet et al., 2001] methods such as MCGDiff [Cardoso et al., 2024]. Although SMC methods are asymptotically consistent under appropriate assumptions, finite particle populations can suffer from weight degeneracy, ancestral collapse, and loss of posterior modes. We consider linear inverse problems y = Ax0 + σy ϵ,

ϵ ∼ N (0, I),

(1)

with posterior p(x0 |y). Several posterior samplers [Cardoso et al., 2024, Janati et al., 2024, Wu et al., 2024, Zhao, 2026] move through Gaussian tilts of diffusion marginals pt , vt (xt ) ∝ pt (xt )N (bt ; Ct xt , Rt ), 40th Conference on Neural Information Processing Systems (NeurIPS 2026). Workshop: AI for Stochastic Dynamics.

(2)

which we call tilted intermediate targets. The pseudo-observation bt , linear operator Ct , and covariance Rt are designed so that the sequence terminates at the posterior of Eq 1, while the tilt constrains x0 more weakly with increasing t. Each vt induces a clean-space posterior with weaker conditioning (Section 2.1), and our key observation is that this pullback can be substantially easier to sample than the final posterior p(x0 |y). We therefore combine an approximate posterior sampler with SMC by initializing the particle population directly at an intermediate time t⋆ : we approximately sample the pullback, transport those samples to vt⋆ through an exact Gaussian bridge, and run only the remaining SMC steps from t⋆ to 0 (the suffix), skipping those from the terminal time to t⋆ (the prefix). We apply this strategy to MCGDiff [Cardoso et al., 2024] with MMPS [Rozet et al., 2024] as the clean-space sampler. At matched particle count, the hybrid halves the sliced Wasserstein distance (SWD) of ESS-gated MCGDiff on a structured Gaussian mixture model (GMM) benchmark and improves on it about 15× on a rare-mode problem. Our first benchmark is the GMM problem introduced by MCGDiff and since used to compare against it [Janati et al., 2024, Ekström Kelvinius et al., 2025, Boys et al., 2024]. Here, initializing from the prior instead of MMPS also beats MCGDiff given 16× the particles, and still matches it at 8× when MCGDiff starts from exact terminal samples (Appendix A.3): the benchmark rewards prior coverage rather than fidelity to the conditioning. We therefore add a structured GMM and a rare-mode problem, the latter isolating conditioning fidelity from prior coverage.

2

Method

Let p0 (x0 ) denote the data distribution represented by a pretrained diffusion model. In variancepreserving (VP) diffusion [Ho et al., 2020, Song et al., 2021b] with t ∈ [0, 1], the forward kernel is
√ (3) qt (xt |x0 ) = N xt ; ᾱt x0 , (1 − ᾱt )I , R where ᾱ0 = 1 and ᾱ1 ≈ 0. The corresponding marginal is pt (xt ) = qt (xt |x0 )p0 (x0 )dx0 . We consider initializing an existing diffusion posterior sampler at an intermediate time t⋆ < 1 by directly sampling the intermediate target vt⋆ as an auxiliary inverse problem. When the intermediate is a Gaussian-tilted diffusion marginal, this can be done by sampling a softened inverse problem at clean-data time, followed by an analytic tilt-conditioned Gaussian bridge. We discuss computational cost in A.4. 2.1

Direct sampling of a Gaussian-tilted diffusion marginal

Recall the tilted intermediate target vt of Eq. 2; it is the xt -marginal of the joint Πt (x0 , xt ) ∝ p0 (x0 )qt (xt |x0 )N (bt ; Ct xt , Rt ). Its x0 -marginal is a clean-space posterior with weaker conditioning than vt : √ π̃t (x0 ) ∝ p0 (x0 )N (bt ; ᾱt Ct x0 , Rt + (1 − ᾱt )Ct CtT ).

(4) (5)

We refer to π̃t as the pullback of the tilt. The conditional Πt (xt |x0 ) ∝ qt (xt |x0 ) N (bt ; Ct xt , Rt ) is Gaussian and available in closed form. Drawing x0 ∼ π̃t and then xt ∼ Πt (·|x0 ) gives (x0 , xt ) ∼ Πt , so xt ∼ vt exactly. 2.2

Application to MCGDiff

MCGDiff [Cardoso et al., 2024] constructs its intermediate Gaussian tilts in the singular-vector basis of the forward operator, with each measured direction becoming active once the diffusion noise exceeds a measurement-dependent threshold τi . We restrict our initialization time to t⋆ > maxi τi , so that all measured directions are active and the target is a single tilt of the form of Eq. 2. Here t⋆ is a fractional position on MCGDiff’s adapted grid (Appendix A.3); t increases with noise, so larger t⋆ gives a softer clean-space problem and a longer suffix. The τi lie on the same grid, and maxi τi < 0.25 in all benchmark runs; above this bound, t⋆ is a free hyperparameter. The corresponding bt , Ct , and Rt , together with the exact pullback and bridge, are given in Appendix A.2. An important consequence of the pullback is that the clean-space inverse problem can be substantially softer than the original one. MCGDiff itself introduces Gaussian noise at approximately the diffusion 2

Figure 1: Effect of initialization time t⋆ on the standard GMM benchmark. (a) Final posterior SWD for oracle and MMPS initialization. (b) MMPS approximation error before and after the exact bridge to vt⋆ . Generated with 1000 MMPS denoising steps and averaged over 80 seeds. noise level into its intermediate tilt; in the near-noiseless measurement limit, with unit singular values, ᾱt 2 the resulting clean-space likelihood therefore has effective variance σeff,t ≃ 2σt2 , where σt2 = 1− ᾱt . This makes the auxiliary inverse problem for MCGDiff particularly weakly conditioned. We sample it with MMPS [Rozet et al., 2024], which approximates the intractable likelihood score by replacing p(x0 |xt ) with a Gaussian matching its first two moments; the approximation improves as the effective observation noise grows, and MMPS needs no guidance scale or other problem-specific tuning. We transport the resulting samples through the exact Gaussian bridge to initialize vt⋆ , and then run the unchanged MCGDiff suffix. The choice of t⋆ therefore trades an increasingly easy MMPS problem against a longer remaining SMC trajectory. The hybrid does not inherit MCGDiff’s asymptotic consistency, since the suffix is initialized with equal-weight approximate samples, but this could be restored by replacing MMPS with another asymptotically consistent method.

3

Related work

Our approach is related to recent methods that improve diffusion posterior sampling through rewardor observation-informed initialization (Ψ-Sampler [Yoon et al., 2025], Chang et al. 2026) or easier auxiliary posterior distributions (DCPS [Janati et al., 2024], TGD [Makris et al., 2026], Zhao 2026). Unlike these, we directly initialize an intermediate target already defined by an existing sampler, using its induced softened clean-space posterior and an exact bridge back to the original diffusion-time target. See Appendix C for further discussion.

4

Experiments

We evaluate three inverse GMM problems for which both the exact posterior and MCGDiff intermediate targets vt can be sampled analytically. This allows us to isolate the quality of intermediate initialization from the behavior of the subsequent SMC suffix. Performance is measured by sliced Wasserstein distance (SWD) to exact posterior samples; see Appendix B for full benchmark definitions. As also observed by Zhao [2026], we find that MCGDiff improves when resampling is performed only when the particle ESS falls below 0.7N , rather than at every step. We therefore use this stronger ESS-gated variant for all full MCGDiff and initialized suffixes, while also reporting the original MCGDiff baseline. We use N = 1,000 particles for MCGDiff and intermediate initialization runs, plus baselines with larger N (up to 16,000) for full MCGDiff and, on the standard GMM, for oracle initialization at t⋆ = 1. The benchmarks separate two finite-particle failures. Loss of diversity can be repaired by a population with broad prior coverage; loss of support cannot, since resampling prunes particles but does not create them. We therefore include prior initialization at t⋆ as a control: it replaces the pullback posterior π̃t⋆ with the prior p0 in the clean-space step, and transports through the same exact bridge to vt⋆ . Both initializations bypass the prefix and enter the suffix with the observed directions conditioned 3

on the measurement via the tilt; they differ only in whether the clean-space draw is conditioned. The standard and structured GMMs primarily probe lost diversity; the rare-mode benchmark isolates lost support, with ρ setting the prior probability of the posterior-relevant mode and hence how likely a finite prior-drawn population is to miss it entirely. Figure 1(a) compares oracle vt⋆ initialization with MMPS initialization across values of t⋆ on the standard GMM benchmark. As t⋆ increases, the induced clean-space posterior becomes progressively softer and is sampled more accurately by MMPS (Figure 1(b)). This comes at the cost of a longer remaining MCGDiff suffix, so the final posterior SWD exhibits a broad trade-off between initialization accuracy and suffix length rather than a sharply tuned optimum. In contrast, oracle initialization improves as t⋆ decreases, as expected when increasingly large portions of the SMC trajectory are bypassed using exact intermediate samples. At the terminal time (t⋆ = 1), where no trajectory is bypassed, there is still substantial improvement with exact initialization of vt⋆ over standard MCGDiff. This is not a consequence of bypassing the prefix, but of vt⋆ correcting the Gaussian approximation to the terminal marginal, which is inaccurate under this benchmark’s diffusion schedule (Appendix A.3). Across benchmarks, Table 1 compares MMPS initialization with MCGDiff, direct MMPS, oracle initialization, and prior-based controls using 100 MMPS denoising steps. Direct MMPS on the original posterior is consistently weaker than most baselines, showing that the gain is not a result of replacing MCGDiff with MMPS, but of the hybrid. On the standard GMM, both MMPS- and prior-initialization substantially improve on ESS-gated MCGDiff, with roughly 2× and 3× SWD improvement, respectively. Even with 16× the population size allocated to the MCGDiff baseline, there is a clear benefit of intermediate initialization. Much of this margin is the terminal correction discussed above: MCGDiff initialized with exact terminal samples (t⋆ = 1) needs about 2× the particles to match MMPS initialization and about 8× to match prior initialization. Since prior initialization discards the conditioning of unobserved directions entirely over [t⋆ , 1], and only weakly conditions observed directions via the bridge, this shows that MCGDiff’s prefix returns a worse finite-particle approximation than a single biased step. Beyond the terminal-marginal correction, the gain continues to grow as t⋆ decreases: over the bypassed interval the tilt is weakly informative, so resampling incurs ancestral collapse in exchange for almost no selection signal. ESS-gated resampling improves on MCGDiff for the same reason. This identifies finite-particle degradation in the high-noise prefix of MCGDiff, which is precisely the portion our construction bypasses. The structured GMM, inspired by Zhao [2026], introduces random means, component-dependent covariances, and a shifted measurement. Here prior initialization remains useful, but MMPS initialization is better at every t⋆ below the terminal time (SWD half the MCGDiff baseline). Only at 16× population size does MCGDiff outperform intermediate initialization. Finally, the rare-mode benchmark is deliberately constructed so that the observation strongly favors a component with very small prior probability. A finite prior-drawn population may therefore contain few or no particles from the posterior-relevant mode. In this setting, prior-based initialization and MCGDiff can fail because the relevant mode is never adequately represented, whereas MMPS initialization can populate that mode, despite direct MMPS performing poorly on this task. This benchmark gives the clearest advantage for MMPS initialization, which improves on MCGDiff by an order of magnitude. Figures 2 and 3 show representative samples for all methods on the standard GMM and rare-mode GMM problems, respectively.

5

Conclusion

Gaussian-tilted diffusion marginals can be sampled indirectly: pull the tilt back to a softer clean-space posterior, sample it, and transport forward through an exact Gaussian bridge. Applied to MCGDiff, this bypasses the high-noise prefix, improving SWD by 2× at matched particle count on a structured GMM and by more than an order of magnitude when the posterior-relevant mode is rare under the prior. Prior rather than posterior clean-space samples show that MCGDiff’s standard GMM benchmark is largely insensitive to conditioning fidelity. The evaluation is confined to GMMs with one-dimensional observations; imaging and scientific inverse problems are the natural next steps. Our construction requires only that the intermediate tilt is Gaussian and linear in xt ; the original measurement model does not enter the pullback. Extending to non-linear problems with intermediate targets built from a measurement linearization [Zheng et al., 2025, 2026] is a promising direction. 4

References Benjamin Boys, Mark Girolami, Jakiw Pidstrigach, Sebastian Reich, Alan Mosca, and Omer Deniz Akyildiz. Tweedie moment projected diffusions for inverse problems. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL https://openreview.net/forum?id= 4unJi0qrTE. Featured Certification. Joan Bruna and Jiequn Han. Provable posterior sampling with denoising oracles via tilted transport. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, editors, Advances in Neural Information Processing Systems, volume 37, pages 82863–82894. Curran Associates, Inc., 2024. doi: 10.52202/ 079017-2635. URL https://proceedings.neurips.cc/paper_files/paper/2024/ file/9705eec26e511d7844a591ce10c33a3f-Paper-Conference.pdf. Gabriel Cardoso, Yazid Janati el idrissi, Sylvain Le Corff, and Eric Moulines. Monte carlo guided denoising diffusion models for bayesian linear inverse problems. In B. Kim, Y. Yue, S. Chaudhuri, K. Fragkiadaki, M. Khan, and Y. Sun, editors, International Conference on Learning Representations, volume 2024, pages 44001– 44037, 2024. URL https://proceedings.iclr.cc/paper_files/paper/2024/file/ bfb3c1db0deda449ff4027f77dd48704-Paper-Conference.pdf. Jinyuan Chang, Chenguang Duan, Yuling Jiao, Ruoxuan Li, Jerry Zhijian Yang, and Cheng Yuan. Provable diffusion posterior sampling for bayesian inversion, 2026. URL https://arxiv.org/ abs/2512.08022. Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Marc Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id= OnD9zGAGT0k. Giannis Daras, Hyungjin Chung, Chieh-Hsin Lai, Yuki Mitsufuji, Jong Chul Ye, Peyman Milanfar, Alexandros G Dimakis, and Mauricio Delbracio. A survey on diffusion models for inverse problems. arXiv preprint arXiv:2410.00083, 2024. A. Doucet, N. de Freitas, and N. Gordon. Sequential Monte Carlo Methods in Practice. Springer, New York, 2001. Filip Ekström Kelvinius, Zheng Zhao, and Fredrik Lindsten. Solving linear-Gaussian Bayesian inverse problems with decoupled diffusion sequential Monte Carlo. In Aarti Singh, Maryam Fazel, Daniel Hsu, Simon Lacoste-Julien, Felix Berkenkamp, Tegan Maharaj, Kiri Wagstaff, and Jerry Zhu, editors, Proceedings of the 42nd International Conference on Machine Learning, volume 267 of Proceedings of Machine Learning Research, pages 15148–15181. PMLR, 13–19 Jul 2025. URL https://proceedings.mlr.press/v267/ekstrom-kelvinius25b.html. Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840–6851, 2020. Yazid Janati, Badr Moufad, Alain Durmus, Eric Moulines, and Jimmy Olsson. Divide-andconquer posterior sampling for denoising diffusion priors. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, editors, Advances in Neural Information Processing Systems, volume 37, pages 97408–97444. Curran Associates, Inc., 2024. doi: 10.52202/079017-3090. URL https://proceedings.neurips.cc/paper_files/paper/ 2024/file/b0ae046e198a5e43141519868a959c74-Paper-Conference.pdf. Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusionbased generative models. Advances in neural information processing systems, 35:26565–26577, 2022. Andreas Makris, Paul Fearnhead, and Chris Nemeth. Tempered guided diffusion, 2026. URL https://arxiv.org/abs/2605.03712. Kevin P. Murphy. Probabilistic Machine Learning: Advanced Topics. MIT Press, 2023. URL http://probml.github.io/book2. 5

François Rozet, Gérôme Andry, François Lanusse, and Gilles Louppe. Learning diffusion priors from observations by expectation maximization. In A. Globerson, L. Mackey, D. Belgrave, A. Fan, U. Paquet, J. Tomczak, and C. Zhang, editors, Advances in Neural Information Processing Systems, volume 37, pages 87647–87682. Curran Associates, Inc., 2024. doi: 10.52202/079017-2783. URL https://proceedings.neurips.cc/paper_files/paper/ 2024/file/9f94298bac4668db4dc77ddb0a244301-Paper-Conference.pdf. Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. In International Conference on Learning Representations, 2021a. URL https://openreview.net/ forum?id=St1giarCHLP. Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems, 32, 2019. Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021b. URL https://openreview.net/forum?id= PxTIG12RRHS. Zihui Wu, Yu Sun, Yifan Chen, Bingliang Zhang, Yisong Yue, and Katherine Bouman. Principled probabilistic imaging using diffusion models as plug-and-play priors. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL https://openreview.net/ forum?id=Xq9HQf7VNV. Taehoon Yoon, Yunhong Min, Kyeongmin Yeo, and Minhyuk Sung. \psi-sampler: Initial particle sampling for smc-based inference-time reward alignment in score models. In D. Belgrave, C. Zhang, H. Lin, R. Pascanu, P. Koniusz, M. Ghassemi, and N. Chen, editors, Advances in Neural Information Processing Systems, volume 38, Main Conference, pages 104745–104781. Curran Associates, Inc., 2025. doi: 10.52202/085713-3498. URL https://proceedings.neurips.cc/paper_files/ paper/2025/file/96f39c8de84678cb2a908cd52bfd7819-Paper-Conference.pdf. Bingliang Zhang, Wenda Chu, Julius Berner, Chenlin Meng, Anima Anandkumar, and Yang Song. Improving diffusion inverse problem solving with decoupled noise annealing, 2024. URL https: //arxiv.org/abs/2407.01521. Zheng Zhao. Generative diffusion posterior sampling for informative likelihoods. Communications in Information and Systems, 26(1):151–167, 2026. Special issue for celebrating Thomas Kailath’s 90th birthday. Hongkai Zheng, Wenda Chu, Austin Wang, Nikola Borislavov Kovachki, Ricardo Baptista, and Yisong Yue. Ensemble kalman diffusion guidance: A derivative-free method for inverse problems. Transactions on Machine Learning Research, 2025. ISSN 2835-8856. URL https://openreview.net/forum?id=XPEEsKneKs. Hongkai Zheng, Austin Wang, Zihui Wu, Zhengyu Huang, Ricardo Baptista, and Yisong Yue. Blade: A derivative-free bayesian inversion method using diffusion priors, 2026. URL https: //arxiv.org/abs/2510.10968.

6

Appendix A

Additional method details

A.1

Clean-space pullback and Gaussian bridge

The construction follows directly from standard linear-Gaussian marginalization and conditioning identities [Murphy, 2023]. Interpret the Gaussian tilt as an observation in reparameterized form, η ∼ N (0, Rt ).

bt = Ct xt + η,

Substitute in the VP forward process, Eq. 3, √ √ xt = ᾱt x0 + 1 − ᾱt ϵ, to give bt | x0 ∼ N

√

(6)

ϵ ∼ N (0, I),

(7)

ᾱt Ct x0 , Rt + (1 − ᾱt )Ct Ct⊤ .

(8)

Multiplying this likelihood by p0 (x0 ) gives π̃t in Eq 5. A.2

Application to MCGDiff

Let A = U SV ⊤ and work in the singular-vector coordinates used by MCGDiff [Cardoso et al., 2024]. For measured direction i, define (U ⊤ y)i ȳi = . (9) si The direction becomes active at the MCGDiff threshold τi , satisfying ᾱτi σy2 = (1 − ᾱτi )s2i . For t ≥ τi , MCGDiff uses the Gaussian factor
√ N ᾱt ȳi ; xt,i , γt,i ,

γt,i = 1 − (1 − κ)

(10) ᾱt , ᾱτi

(11)

where κ is MCGDiff’s noise floor parameter. Therefore, in the notation of Section 2.1, Ct selects the active singular-vector coordinates, √ bt,i = ᾱt ȳi , Rt,ii = γt,i . Applying Eq. (5), each active coordinate contributes the clean-space likelihood
γt,i + 1 − ᾱt 2 2 N ȳi ; x0,i , σeff,t,i , σeff,t,i = . ᾱt

(12)

(13)

This is the softened inverse problem sampled by MMPS. To transport a clean sample back to diffusion time t, note that for each active coordinate
√ qt (xt,i | x0,i ) = N xt,i ; ᾱt x0,i , 1 − ᾱt ,
√ gt (xt,i ) = N ᾱt ȳi ; xt,i , γt,i .

(14) (15)

Their product is Gaussian in xt,i , with −1 1 1 + , 1 − ᾱt γt,i √  √ ᾱt x0,i ᾱt ȳi 2 µbr,t,i = σbr,t,i + . 1 − ᾱt γt,i 2 σbr,t,i =



(16) (17)

Inactive directions (unmeasured directions, together with measured directions with t < τi ) are sampled from the ordinary forward kernel; in our experiments t⋆ > maxi τi , so only the former arise. Sampling these coordinate-wise Gaussian conditionals gives the exact bridge used to initialize the MCGDiff suffix. 7

A.3

Diffusion schedules

For MCGDiff we follow the schedule used by its GMM implementation. The underlying VP discretization contains 1,000 time points, with the linear β schedule stored in decreasing order from 0.02 to 10−4 . We note that, while the paper reports a maximum β value of βmax = 0.2, the accompanying code uses βmax = 0.02, as do Ekström Kelvinius et al. [2025] when replicating this benchmark. We adopt βmax = 0.02 to match the reference implementation. The same applies to MCGDiff’s noise-floor parameter: we use the repository default κ = 10−4 , whereas Ekström Kelvinius et al. [2025] use κ = 10−2 . MCGDiff does not uniformly subsample the raw diffusion indices, but uses a custom measurement-dependent timestep-selection procedure [Cardoso et al., 2024]. The resulting grid depends on the singular values of the forward operator and the observation-noise variance. Our reported t⋆ denotes a position in this adapted MCGDiff grid, rather than a raw diffusion index. We use 100 diffusion steps for MCGDiff throughout. √ Consequence for terminal initialization. At βmax = 0.02 we have ᾱ1 = 4.1 × 10−5 , so ᾱ1 ≈ 6.4 × 10−3 . In the standard GMM √ benchmark the outermost component means have ∥µ∥ = 452, giving a residual displacement ᾱ1 ∥µ∥ ≈ 2.9 against a noise standard deviation of 1 per coordinate. A Gaussian terminal distribution is therefore a poor approximation, and an N (0, I) initial population under-represents the outer components before any SMC step is taken. The issue is specific to this benchmark: N (0, I) is a good approximation to the terminal distribution of the structured and rare-mode priors, whose means are smaller and spread over more directions. This is reflected in Table 1, where oracle initialization at t⋆ = 1.0 improves on full MCGDiff by 1.7× on the standard GMM but only 1.1× on the structured GMM. For the standard GMM we therefore also report oracle initialization at t⋆ = 1 with larger N as a terminal-corrected baseline. We use η = 0.85 for the DDIM [Song et al., 2021a] steps of the MCGDiff suffix throughout. Repeating the standard GMM benchmark at η = 1 over the first 30 seeds and comparing to η = 0.85 results over the same seeds shows that the comparisons are insensitive to η (MCGDiff 1k: 2.383 → 2.380; with ESS gating: 1.958 → 1.804). For MMPS sampling of the clean-space posterior, we use η = 1 throughout. MMPS uses a separate auxiliary VP schedule with β increasing linearly from 10−4 to 0.2 over 1,000 time points. For the 100-step MMPS experiments, we truncate this auxiliary schedule at the first time for which ᾱt ≤ 10−8 and place 100 approximately uniformly spaced raw diffusion indices between zero and this terminal time. This avoids spending steps in the extreme high-noise tail, where the diffusion marginal is already effectively Gaussian. For the 1,000-step MMPS experiments we instead use the full auxiliary schedule. The MMPS schedule is used only to numerically sample the pulled-back clean-space posterior. The target vt⋆ and the subsequent Gaussian bridge are defined using the MCGDiff schedule. A.4

Computational cost

Table 2 repeats the benchmark comparison with 1,000 rather than 100 MMPS steps and shows little change in performance, suggesting that 100 MMPS steps are sufficient for these experiments. In a practical application, where the score is represented by a neural network, using 100 steps for both MMPS and MCGDiff with t⋆ = 0.5 requires 100 diffusion steps for the initialization followed by 50 for the MCGDiff suffix, against 100 for standard MCGDiff. The number of diffusion steps therefore increases only moderately. Per-step cost depends on the choice of clean-space sampler. MMPS requires gradients through the score network, and is among the more expensive options; in our GMM benchmarks its cost was negligible, since the score is analytic and the observation is one-dimensional. It was chosen because it needs no guidance-scale or other problem-specific tuning, which keeps the ablation of the initialization mechanism clean. The pullback is a standard linear-Gaussian inverse problem, so any approximate posterior sampler can be substituted, including cheaper samplers such as DPS [Chung et al., 2023]. A natural alternative is simply to give MCGDiff more particles. Table 1 reports 2× and 16× populations. On the standard GMM, even 16× does not make MCGDiff with the N (0, I) terminal competitive with MMPS initialization, although an exact-terminal MCGDiff matches it at about 2×. On the structured benchmark, 16× with ESS-gated resampling does close the gap. Particles are 8

evaluated in parallel, so in these experiments a larger population costs memory rather than sequential depth. In high-dimensional problems this no longer holds: the population will not fit in memory at once, and the extra sequential batches of score-network evaluations erode much of the apparent advantage over the initialization prefix. A.5

Sliced Wasserstein evaluation

We compute SW1 between 1,000 samples from each approximate method and 1,000 samples from the exact posterior, using 1,000 shared random projections per inverse-problem instance. Methods run with larger particle populations are subsampled to 1,000 final particles only after inference so that all SWD comparisons use the same empirical sample size.

B

GMM benchmark definitions

We use three Gaussian mixture model (GMM) inverse problems. The first follows the benchmark construction used by MCGDiff, while the second is inspired by, but is not identical to, the structured GMM experiments of Zhao [2026]. The third is a deliberately constructed rare-mode problem designed to stress methods initialized from finite prior samples. Standard GMM. Following MCGDiff, we use a d = 800 dimensional, 25-component unitcovariance GMM. For i, j ∈ {−2, −1, 0, 1, 2}, the component means are µij = (8i, 8j, . . . , 8i, 8j) ∈ R800 ,

(18)

and each component has covariance I. The mixture weights are randomly drawn from a χ2 distribution once and shared by all 80 instances, which differ in A, σy and y. We only use one-dimensional linear observations. We first sample a Gaussian random matrix and retain its singular vectors, while replacing its singular values by sorted draws from U(0, 1). An observation is generated from a prior sample x0 according to ϵ ∼ N (0, I),

y = Ax0 + σy ϵ,

(19)

where σy is sampled uniformly between zero and the largest singular value of A. No additional observation offset is used. Structured GMM. Our structured benchmark is inspired by, but is not identical to, the GMM experiment of Zhao [2026]. We use d = 800 and K = 10. The unnormalised mixture weights are independently sampled as w̃k w̃k ∼ χ21 , wk = P . (20) j w̃j Independently for each component, µk ∼ Unif([−8, 8]d ), The observation operator A ∈ R of Zhao [2026], we choose

1×d

Λk = λk λ⊤ k + I,

λk ∼ Unif([0, 1]d ).

(21)

is a random unit row vector. Following the outlier construction

y = Ex∼p0 [Ax] + ω = AEp0 [x] + ω,

ω = 5,

(22)

and define the posterior using a Gaussian likelihood with observation standard deviation σy = 0.5 (i.e. R = 0.25). Unlike Zhao [2026], we therefore use a fixed observation covariance and a unit-norm one-dimensional forward operator. Since the prior is a GMM and the likelihood is linear Gaussian, the posterior remains analytically tractable. Rare-mode GMM. We construct a two-component GMM designed to test whether a finite particle population can recover a posterior-relevant mode that is very rare under the prior. We use d = 64, identity covariance for both components, and mixture weights (wcommon , wrare ) = (1 − ρ, ρ).

(23)

We draw a random unit vector a, which defines √the observed direction, and a second random vector q perpendicular to a, normalized so that ∥q∥ = d. Since q is orthogonal to a, displacement along q is 9

completely invisible to the measurement. A prior sample transported through the bridge is therefore corrected along a while its position along q is unchanged, so it can satisfy the measurement while remaining in the wrong component. The component means are µcommon = −4a − 4q,

µrare = +4a + 4q.

(24)

We use

A = a⊤ , y = 4, σy = 0.5. (25) Thus the measurement strongly favours the rare component, since Aµcommon = −4 while Aµrare = 4 = y, even though that component has very small prior probability. The additional separation along q makes the two components well separated in the full space while remaining invisible to the observation. We vary ρ to control how poorly the rare component is represented by a finite prior sample.

C

Related work (extended)

Several recent methods improve posterior sampling by replacing an unconditional high-noise initialization with one informed by the observation or reward. Ψ-Sampler [Yoon et al., 2025] samples a reward-aware terminal distribution before running SMC, while Chang et al. 2026 warm-start the conditional reverse diffusion with a Monte Carlo posterior-score estimator. These methods are close in spirit to our direct initialization, but initialize at the terminal time rather than at an intermediate target. A second line of work exploits easier intermediate posterior problems. DCPS [Janati et al., 2024] recursively decomposes posterior sampling into shorter diffusion-time subproblems with annealed likelihoods, while TGD [Makris et al., 2026] defines an explicit annealing path of clean-space tempered posteriors and uses diffusion states to move between them. Zhao [2026] instead proposes alternative Gaussian tilts to improve SMC under informative likelihoods. Our method differs in that we do not design a new bridging family: we directly initialize an existing SMC pathway at one of its own intermediate targets, thereby skipping the prefix. We derive the clean-space posterior implied by that target and the exact bridge back to it; because this pullback is more weakly conditioned, the tilted target can be populated with approximately independent MMPS samples before running only the remaining SMC suffix. Other methods condition in clean space and transport forward. DAPS [Zhang et al., 2024] reconstructs a clean sample from the current state, conditions it on the observation, and renoises through the forward kernel. DDSMC [Ekström Kelvinius et al., 2025] places this step inside an SMC algorithm. Both approximate the intractable noisy likelihood with a point estimate of the clean sample, so their intermediate targets approximate the noisy-space posterior rather than being auxiliary tilts, and there is no equivalent exact pullback posterior. Tilted transport [Bruna and Han, 2024] maps a linear-Gaussian posterior to a different, more strongly tilted “boosted” posterior that can be easier to sample, before transporting samples back to the target. Our construction moves in the opposite direction: an existing noisy-space tilt is pulled back to a more weakly conditioned clean-space posterior. MCGDiff [Cardoso et al., 2024] itself uses a related transport identity, converting noisy observations into noiseless constraints at intermediate diffusion times, but does not exploit the resulting clean-space softening considered here.

D

Additional figures and tables

10

Figure 2: Posterior samples on the standard GMM benchmark for MCGDiff, direct MMPS, MMPS initialization, and prior initialization. MMPS and Prior initialization give qualitatively comparable samples on this relatively simple benchmark. Teal: exact posterior samples; Red: approximate method samples.

Figure 3: Posterior samples on the rare-mode GMM benchmark for MCGDiff, direct MMPS, MMPS initialization, and two separate prior initialization examples. Unless marked otherwise, initialized methods use t⋆ = 0.3 and ESS-thresholded resampling at 0.7N . Prior initialization⋆ instead uses t⋆ = 0.4 with resampling at every step. This variant illustrates a failure mode of prior-based initialization: it collapses to the wrong posterior mode, whereas the ESS-thresholded prior initialization recovers the correct mode but remains less accurate than MMPS initialization. Teal: exact posterior samples; Red: approximate method samples.

11

Table 1: Sliced Wasserstein distance (lower is better) for the GMM benchmarks using MMPS initialization with 100 denoising steps. Values are mean ± 95% confidence interval over 80 random seeds. Initialization methods use the ESS< 0.7 MCGDiff suffix. Column N gives the number of MCGDiff particles used. Among approximate initialization methods, the best result is bold and the second best is underlined. “Exact posterior floor” is the SWD calculated from comparing two batches of 1000 samples of exact posterior samples. Oracle init values for t⋆ = 1 are included for higher particles N for the standard GMM benchmark to provide a fairer baseline, removing the bias due to the benchmark’s inaccurate terminal initialization (see Appendix A.3). Method

N

t⋆

GMM

Baselines MCGDiff MCGDiff (ESS 0.7) MCGDiff MCGDiff (ESS 0.7) MCGDiff MCGDiff (ESS 0.7) Oracle init

1k 1k 2k 2k 16k 16k

– – – – – –

2.307 ± 0.45 1.856 ± 0.40 1.897 ± 0.31 1.519 ± 0.31 1.354 ± 0.23 1.326 ± 0.26

1.271 ± 0.20 0.762 ± 0.12 0.838 ± 0.12 0.548 ± 0.09 0.382 ± 0.05 0.310 ± 0.05

5.818 ± 0.40 2.836 ± 0.66 – – – –

3.052 ± 0.67 0.280 ± 0.14 – – – –

1k 1k 1k 1k 1k 2k 4k 8k 12k 16k

0.25 0.30 0.40 0.60 1.00 1.00 1.00 1.00 1.00 1.00

0.395 ± 0.06 0.446 ± 0.08 0.488 ± 0.07 0.777 ± 0.19 1.090 ± 0.24 0.877 ± 0.15 0.765 ± 0.15 0.641 ± 0.13 0.571 ± 0.14 0.570 ± 0.14

0.279 ± 0.03 0.337 ± 0.05 0.404 ± 0.07 0.532 ± 0.09 0.693 ± 0.11 – – – – –

0.080 ± 0.00 0.086 ± 0.00 0.125 ± 0.00 – – – – – – –

0.078 ± 0.00 0.079 ± 0.00 0.093 ± 0.00 – – – – – – –

1k 1k 1k 1k 1k

0.25 0.30 0.40 0.60 1.00

1.059 ± 0.17 0.417 ± 0.05 0.920 ± 0.16 0.382 ± 0.05 0.899 ± 0.17 0.412 ± 0.06 0.963 ± 0.18 0.507 ± 0.07 0.959 ± 0.17 0.735 ± 0.13

0.180 ± 0.00 0.191 ± 0.00 0.225 ± 0.01 – –

0.111 ± 0.00 0.115 ± 0.00 0.125 ± 0.00 – –

1k 1k 1k 1k 1k

0.25 0.670 ± 0.10 0.30 0.675 ± 0.10 0.40 0.610 ± 0.11 0.60 0.849 ± 0.16 1.00 1.155 ± 0.20

0.461 ± 0.06 0.461 ± 0.07 0.515 ± 0.08 0.546 ± 0.10 0.719 ± 0.13

1.256 ± 0.43 1.231 ± 0.43 1.180 ± 0.43 – –

0.260 ± 0.01 0.244 ± 0.01 0.232 ± 0.01 – –

3.216 ± 0.41 7.339 ± 0.51 0.240 ± 0.03

1.668 ± 0.19 3.037 ± 0.16 0.135 ± 0.01

5.879 ± 0.03 6.445 ± 0.04 0.057 ± 0.00

4.261 ± 0.03 6.394 ± 0.03 0.057 ± 0.00

Structured Rare (ρ = 0.002) Rare (ρ = 0.01)

MMPS init

Prior init (exact)

Additional references Direct MMPS 1k Prior 1k Exact posterior floor 1k

– – –

12

Table 2: Sliced Wasserstein distance (lower is better) for the GMM benchmarks using MMPS with 1000 denoising steps. Values are mean ± 95% confidence interval over 80 random seeds. Methods use the ESS< 0.7 MCGDiff suffix unless marked otherwise. The number of MCGDiff particles used for all relevant methods is 1,000. Among approximate methods, the best result is bold and the second best is underlined. Data in this table generated with different random seeds to corresponding data in Figure 1. Method

t⋆

GMM

Structured

Baselines MCGDiff (1k) – 2.615 ± 0.46 1.271 ± 0.20 MCGDiff (1k, ESS 0.7) – 1.784 ± 0.29 0.762 ± 0.12 Oracle init 0.25 0.444 ± 0.07 0.320 ± 0.06 0.30 0.466 ± 0.08 0.322 ± 0.04 0.40 0.556 ± 0.10 0.377 ± 0.06 0.60 0.665 ± 0.11 0.485 ± 0.07 1.00 1.206 ± 0.23 0.679 ± 0.10 MMPS init 0.25 1.063 ± 0.19 0.385 ± 0.05 0.30 1.013 ± 0.19 0.426 ± 0.06 0.40 0.867 ± 0.16 0.391 ± 0.06 0.60 0.782 ± 0.17 0.527 ± 0.09 1.00 1.164 ± 0.22 0.695 ± 0.12 Prior init (exact) 0.25 0.820 ± 0.15 0.456 ± 0.07 0.30 0.644 ± 0.10 0.485 ± 0.08 0.40 0.615 ± 0.10 0.487 ± 0.07 0.60 0.833 ± 0.21 0.506 ± 0.08 1.00 1.174 ± 0.22 0.634 ± 0.10 Additional references Direct MMPS – 3.267 ± 0.42 1.668 ± 0.19 Prior – 7.293 ± 0.52 3.037 ± 0.16 Exact posterior floor – 0.240 ± 0.03 0.135 ± 0.01

13

Related documents

• Direct Intermediate Initialization for Tilted Diffusion Samplers#1215914
• Fast Convergence for High-Order ODE Solvers in Diffusion Probabilistic Models#999662
• Kinetic Langevin Meets Split Gibbs: Accelerated Posterior Sampling for Imaging Inverse Problems with Diffusion Priors#1234554
• WEIRDO: WEak resIdual Regularized DOob's h-transform diffusion alignment#1144498
• Non-Gaussian Parameter Bias: Formalism and Validation#1157452