Robust Inference-Time Steering of Protein Diffusion Models via Embedding Optimization

Jiequn Han; Luhuan Wu; Minhuan Li; Pilar Cossio

arxiv: 2602.05285 · v2 · pith:ONIHY2PZnew · submitted 2026-02-05 · 💻 cs.LG

Robust Inference-Time Steering of Protein Diffusion Models via Embedding Optimization

Minhuan Li , Jiequn Han , Pilar Cossio , Luhuan Wu This is my paper

Pith reviewed 2026-05-16 07:32 UTC · model grok-4.3

classification 💻 cs.LG

keywords protein structure generationdiffusion modelsembedding optimizationinference-time steeringcryo-EM fittingexperimental constraintsposterior sampling

0 comments

The pith

Optimizing the conditional embedding steers protein diffusion models to fit experimental constraints more robustly than coordinate perturbation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Protein diffusion models generate plausible structures but require steering to satisfy experimental measurements such as distance restraints or density maps. Traditional posterior sampling perturbs atomic coordinates with likelihood gradients, which can destabilize when the target lies outside the prior's high-density region. EmbedOpt instead optimizes the conditional embedding that encodes coevolutionary signals, thereby shifting the structural prior itself to align with constraints. This yields performance that matches coordinate baselines on sparse distances and exceeds them on cryo-EM map fitting, including noisy experimental cases. The approach remains stable across hyperparameter ranges spanning two orders of magnitude and achieves similar quality with fewer diffusion steps.

Core claim

The paper establishes that updating the conditional embedding during inference shifts the diffusion model's structural prior to satisfy experimental likelihoods, producing conformations that match or exceed the fidelity of coordinate-based posterior sampling while avoiding its instability on difficult targets.

What carries the argument

The conditional embedding, which encodes coevolutionary information from the input sequence and is directly optimized at inference time to realign the generated distribution with experimental data.

If this is right

EmbedOpt matches coordinate-based posterior sampling on sparse distance constraints.
It outperforms coordinate methods when fitting to cryo-electron microscopy maps, including real noisy experimental data.
The optimization remains stable across hyperparameter values that span two orders of magnitude.
Comparable accuracy is reached with substantially fewer diffusion steps than standard sampling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same embedding-optimization axis could be used to combine multiple orthogonal experimental constraints in a single run.
Because the embedding carries sequence information, the method may allow rapid adaptation of a fixed diffusion model to new experimental modalities without retraining.
Scaling the approach to multi-domain or membrane proteins would test whether the embedding space remains sufficiently expressive for larger systems.

Load-bearing premise

That changes made to the conditional embedding can enforce experimental constraints without introducing non-physical artifacts or erasing the model's learned sequence-structure relationships.

What would settle it

If structures generated by EmbedOpt on the same experimental inputs exhibit systematically higher rates of steric clashes, bond-length violations, or Ramachandran outliers than those produced by coordinate-based posterior sampling.

Figures

Figures reproduced from arXiv: 2602.05285 by Jiequn Han, Luhuan Wu, Minhuan Li, Pilar Cossio.

**Figure 1.** Figure 1: Synthetic illustration comparing DPS and EmbedOpt under prior-likelihood mismatch. The diffusion prior has limited overlap with the measurement likelihood (top). With a likelihood weight = 1 (controlled by αt in Eq. (7)), the posterior is distant from the measurement (second). Upweigting the likelihood (third) moves the posterior toward the measurement but leads to an illconditioned sampling landscape. In… view at source ↗

**Figure 3.** Figure 3: Cryo-EM Map Fitting Benchmark. (a) Visualization of a challenging target: 8H1I requires significant inter-domain rearrangement of the prior structure (cc = 0.42) to fit the target density map (gray volume), DPS remains trapped in a local optimum (cc = 0.58), and EmbedOpt successfully reorients the domains (cc = 0.93). cc is tha map correlation coefficient. (b) Best-achieved Performance vs. Task Difficulty:… view at source ↗

**Figure 4.** Figure 4: Distance Constraint Benchmark. (a) Optimization Trajectory: Representative surrogate reward traces for system 6V7W show EmbedOpt increases surrogate reward smoothly and monotonically to the optimum, while DPS has high-frequency volatility. (b) StepEfficiency Scaling: We substantially reduce # of diffusion steps from 200 down to 20 while keeping the base learning rate α×# of steps = const., where const. is… view at source ↗

**Figure 5.** Figure 5: Cryo-EM Map Fitting Benchmark: Sample Gallery of Representative Results across Test Systems (HyperparameterTuned). Structures display the best samples from each method following hyperparameter sweeping. Both methods perform robustly on targets where the unguided prior is already well-aligned with the target map (e.g., 8CAW). However, for targets requiring significant global conformational rearrangement (e… view at source ↗

**Figure 6.** Figure 6: Cryo-EM Map Fitting Benchmark: Sample Gallery of EmbedOpt and DPS Failure Modes under High Learning Rates. We visualize the impact of high learning rates (α=1.0) on generation quality for both methods. DPS (bottom) that directly steers noisy coordinates can push trajectories off the data manifold, resulting in unphysical, unraveled structures that defy energy relaxation. In contrast, EmbedOpt (middle) rema… view at source ↗

**Figure 7.** Figure 7: Cryo-EM Map Fitting Benchmark: Stereochemical Quality Analysis. Comparison of MolProbity scores (lower is better) across varying learning rates. EmbedOpt preserves structural validity even at high learning rates, whereas DPS suffers from severe geometric degradation when α > 0.1 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

**Figure 8.** Figure 8: Distance-Constrained Structure Determination Benchmark: Performance Analysis. (a) Best-Achieved Performance: Comparison of constraint satisfaction on the K = 20 constraint benchmark, where systems are ordered by task difficulty (initial deviation between prior and target distances per constraint). Unlike the unimodal likelihood in Cryo-EM task, sparse constraints allow both methods to achieve comparable pe… view at source ↗

read the original abstract

A core challenge in structural biophysics is generating biomolecular conformations that are both physically plausible and consistent with experimental measurements. While sequence-to-structure diffusion models provide powerful priors, posterior sampling methods steer generation by perturbing atomic coordinates with gradients from experimental likelihoods. However, when the target lies in a low-density region of the prior, these methods require aggressive upweighting of the likelihood that can destabilize sampling and be sensitive to hyperparameters. We propose EmbedOpt, an inference-time steering framework that introduces an orthogonal optimization axis: rather than performing posterior sampling under a fixed prior, EmbedOpt directly optimizes the prior by updating the model's conditional embedding. This embedding space encodes rich coevolutionary signals, so optimizing it shifts the structural prior to align with experimental constraints. Empirically, EmbedOpt matches coordinate-based posterior sampling baselines on sparse distance constraints and outperforms them on cryo-electron microscopy map fitting, including real, noisy experimental ones. Furthermore, EmbedOpt's smooth optimization behavior yields robustness to hyperparameters spanning two orders of magnitude and enables comparable performance with fewer diffusion steps. Code is available at https://github.com/rs-station/embedopt.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

EmbedOpt gives a cleaner inference-time knob for protein diffusion models by optimizing the conditional embedding instead of coordinates, with reported gains in cryo-EM fitting and hyperparameter robustness, though physical validity checks are missing.

read the letter

The main thing here is that EmbedOpt steers generation by directly updating the model's conditional embedding at inference time rather than adding gradients to atomic coordinates. This avoids the instability that comes from aggressive likelihood weighting when the target is outside the prior's high-density region. The abstract shows it matches coordinate-based posterior sampling on sparse distance constraints and beats it on cryo-EM map fitting, including real noisy experimental maps. It also claims smoother optimization that tolerates hyperparameters over two orders of magnitude and reaches comparable results with fewer diffusion steps. Code is released, which is helpful. The core move—treating the embedding as the thing to optimize because it carries coevolutionary signals—is presented as a distinct axis and looks like a genuine shift from prior posterior-sampling setups. That framing is the clearest novelty. The results on robustness and efficiency are the practical upside. The soft spot is exactly the one the stress-test flags: no reported checks on whether the embedding-optimized structures stay physically plausible. There is nothing in the abstract about steric clashes, bond geometry, secondary-structure fidelity, or energy-minimized baselines for the real cryo-EM cases. If the updates are just pulling the density fit without preserving the prior's learned constraints, the reported gains could be overfitting rather than a better prior shift. The abstract also skips details on statistical significance and exact protocols, so the strength of the outperformance is hard to judge from what's here. This is for computational structural biologists who already use diffusion models and need to fold in experimental data without constant hyperparameter tuning. A reader working on inference-time control of generative models would get the most from the robustness claims. I would send it to peer review. The idea is clean enough and the empirical pattern is worth closer examination, even if the physical-validity gap needs to be closed in revision.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces EmbedOpt, an inference-time steering method for protein diffusion models that optimizes the conditional embedding (rather than perturbing atomic coordinates) to align generated structures with experimental constraints such as sparse distance restraints and cryo-EM density maps. It claims to match coordinate-based posterior sampling baselines on distance constraints, outperform them on cryo-EM fitting (including real noisy data), exhibit robustness to hyperparameters over two orders of magnitude, and achieve comparable results with fewer diffusion steps.

Significance. If the central claims hold after addressing validation gaps, EmbedOpt would offer a valuable orthogonal axis for posterior sampling in generative protein models, potentially improving stability and accuracy when targets lie in low-density regions of the learned prior. The open-source code release supports reproducibility and extension.

major comments (2)

[Results (cryo-EM experiments)] The abstract and results claim outperformance on real, noisy experimental cryo-EM maps, but provide no quantitative checks (e.g., steric clash scores, bond-length/angle deviations, Ramachandran statistics, or secondary-structure fidelity) on the generated ensembles. This validation is load-bearing for the premise that embedding optimization shifts the prior without non-physical artifacts.
[Method (embedding optimization)] The motivation rests on the embedding space encoding coevolutionary signals, yet there is no ablation or metric (e.g., comparison of contact maps or evolutionary coupling recovery before/after optimization) demonstrating that these signals are preserved after embedding updates.

minor comments (1)

[Abstract] The abstract states 'including real, noisy experimental ones' without specifying the number of maps, resolution range, or exact fitting metric (e.g., cross-correlation coefficient); adding these details would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback. We address each major comment point by point below. Where the comments identify validation gaps, we have revised the manuscript to incorporate the suggested analyses.

read point-by-point responses

Referee: [Results (cryo-EM experiments)] The abstract and results claim outperformance on real, noisy experimental cryo-EM maps, but provide no quantitative checks (e.g., steric clash scores, bond-length/angle deviations, Ramachandran statistics, or secondary-structure fidelity) on the generated ensembles. This validation is load-bearing for the premise that embedding optimization shifts the prior without non-physical artifacts.

Authors: We agree that explicit quantification of structural quality is essential to substantiate that embedding optimization preserves physical plausibility. In the revised manuscript we have added MolProbity-derived steric clash scores, bond-length and angle RMSDs relative to ideal geometry, Ramachandran outlier percentages, and DSSP-based secondary-structure fidelity metrics for all ensembles generated on the real experimental maps. These metrics show that EmbedOpt structures remain within acceptable physical ranges and are comparable to (or modestly better than) the coordinate-based posterior sampling baselines, supporting the claim that the method shifts the prior without introducing non-physical artifacts. revision: yes
Referee: [Method (embedding optimization)] The motivation rests on the embedding space encoding coevolutionary signals, yet there is no ablation or metric (e.g., comparison of contact maps or evolutionary coupling recovery before/after optimization) demonstrating that these signals are preserved after embedding updates.

Authors: We appreciate the referee’s emphasis on directly verifying preservation of coevolutionary information. In the revised version we have added an ablation that extracts predicted contact maps from the original and optimized embeddings and compares them against both the input sequence’s evolutionary couplings (computed via CCMpred) and the final generated structures. The results indicate that the top-L contact precision remains largely unchanged after optimization, with only localized adjustments that are consistent with the experimental constraints; we also report the change in evolutionary coupling recovery scores before and after the embedding update. revision: yes

Circularity Check

0 steps flagged

No circularity: EmbedOpt derivation is independent optimization over pretrained embedding space

full rationale

The paper frames EmbedOpt as a distinct inference-time procedure that optimizes the conditional embedding of a pretrained diffusion model to shift its structural prior, rather than perturbing atomic coordinates under a fixed prior. No equations or performance claims in the provided text reduce the reported matching or outperformance on distance constraints and cryo-EM maps to quantities defined by the method's own fitted parameters, self-referential definitions, or load-bearing self-citations. The central distinction (embedding optimization vs. coordinate perturbation) is presented as an orthogonal axis whose validity rests on the external property that the embedding encodes coevolutionary signals from the pretrained model. Empirical results are described as direct comparisons to baselines without any renaming of known patterns or ansatzes imported via author self-citation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the conditional embedding encodes usable coevolutionary signals that can be directly optimized to realign the generative prior; no free parameters or invented entities are explicitly introduced in the abstract.

free parameters (1)

embedding optimization hyperparameters
The method performs gradient-based optimization in embedding space, which necessarily involves at least one step-size or regularization hyperparameter whose value is not derived from first principles.

axioms (1)

domain assumption The model's conditional embedding encodes rich coevolutionary signals that can be leveraged to shift the structural prior.
Explicitly stated in the abstract as the justification for why embedding optimization works.

pith-pipeline@v0.9.0 · 5496 in / 1235 out tokens · 30033 ms · 2026-05-16T07:32:26.540811+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

EmbedOpt directly optimizes the prior by updating the model's conditional embedding... greedy optimization of the conditional embedding via single-step gradient ascent at each diffusion step
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reward R(x0)∝logp(y|x0) ... R(x0)=−1/N ∑(V(x0)−Vobs)² or distance violation

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

CrystalBoltz: End-to-End Protein Structure Determination via Experiment-Guided Diffusion for X-Ray Crystallography
cs.LG 2026-05 unverdicted novelty 6.0

CrystalBoltz performs experiment-guided posterior sampling with diffusion models on structure-factor amplitudes for protein structure determination, reporting lower RMSD and R-factors than baselines with 33x faster runtime.
ConforNets: Latents-Based Conformational Control in OpenFold3
q-bio.BM 2026-04 unverdicted novelty 6.0

ConforNets use channel-wise affine transforms on pre-Pairformer pair latents in OpenFold3 to achieve state-of-the-art unsupervised generation of alternate protein states and supervised conformational transfer across families.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 2 Pith papers · 3 internal anchors

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work page internal anchor Pith review Pith/arXiv arXiv doi:10.1101/2025.01.08 2025
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Prompt-tuning latent diffusion models for inverse problems.arXiv preprint arXiv:2310.01110, 2023

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He, Y ., Murata, N., Lai, C.-H., Takida, Y ., Uesaka, T., Kim, D., Liao, W.-H., Mitsufuji, Y ., Kolter, J. Z., Salakhut- dinov, R., et al. Manifold preserving guided diffusion. arXiv preprint arXiv:2311.16424,

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work page internal anchor Pith review Pith/arXiv arXiv
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Maddipatla, A., Sellam, N. B., Bojan, M., Vedula, S., Schanda, P., Marx, A., and Bronstein, A. M. Inverse prob- lems with experiment-guided alphafold.arXiv preprint arXiv:2502.09372,

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Song, Y ., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456,

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Wohlwend, J., Corso, G., Passaro, S., Getz, N., Reveiz, M., Leidal, K., Swiderski, W., Atkinson, L., Portnoi, T., Chinn, I., et al. Boltz-1 democratizing biomolecular interaction modeling.BioRxiv, pp. 2024–11,

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Zach, M., Haouchat, Y ., and Unser, M. A statistical bench- mark for diffusion posterior sampling algorithms.arXiv preprint arXiv:2509.12821,

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Method Details A.1

2 −1σ t ϵt, ϵ t ∼ N(0,I) 8:Amplify noise levelσ t ←(γ+ 1)σ t 9:end if 10:Make denoised predictionˆx 0 ←ˆxθ(xt, ct = (st, zt), σt) 11:Compute gradientg ct ←[g st , gzt]whereg st ← ∇ st R(ˆx0)andg zt ← ∇ zt R(ˆx0) 12:Normalize gradient by RMS¯g ct ←[¯gst ,¯gzt]where¯gst ← gstr 1 ds Pds i=1 g(i) st 2 and¯gzt ← gztq 1 dz Pdz i=1(gzt(i)) 2 13:Update embeddingc...

work page 2022
[13]

DPS Algorithm Adapted to AlphaFold 3 Sampling Scheme We follow the gradient normalization strategy in Maddipatla et al

B.2. DPS Algorithm Adapted to AlphaFold 3 Sampling Scheme We follow the gradient normalization strategy in Maddipatla et al. (2025) (official implementation can be found in https://github.com/sai-advaith/guided_alphafold, which is also built on the Protenix model), and summarize the DPS algorithm adapted for AlphaFold 3 sampling scheme in Algorithm

work page 2025
[14]

We set the diffusion noise schedule σ(t) =t for t∈[0,1]

conditioned on the location parameter c= 5 . We set the diffusion noise schedule σ(t) =t for t∈[0,1] . Since p(x0 |c)is Gaussian, we can access the conditional expectationE[x 0 |x t]∀(x t, t)without training a denoiser network. The measurement likelihood is given by N(y|x 0,1) with the measurement y= 20 . This setting simulates the case where the prior mo...

work page 2025
[15]

is implemented with SFC Torch (Li et al., 2025).We first compute the structure factors F(⃗h) in the frequency domain by summing the scattering contributions of individual atoms: F(⃗h) = X j Oj ·f ⃗h,j ·DWF( ⃗h)·exp h 2πi⃗h·⃗ xj i (44) where j indexes the atoms, Oj denotes occupancy (fixed at 1.0), and xj represents the fractional coordinates. The term fh,...

work page 2025

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doi: 10.1101/2025.01.08. 631967. Chung, H., Kim, J., Mccann, M. T., Klasky, M. L., and Ye, J. C. Diffusion posterior sampling for general noisy in- verse problems.arXiv preprint arXiv:2209.14687,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1101/2025.01.08 2025

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Prompt-tuning latent diffusion models for inverse problems.arXiv preprint arXiv:2310.01110, 2023

Chung, H., Ye, J. C., Milanfar, P., and Delbracio, M. Prompt- tuning latent diffusion models for inverse problems.arXiv preprint arXiv:2310.01110,

work page arXiv

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J., Terwilliger, T

Fadini, A., Li, M., McCoy, A. J., Terwilliger, T. C., Read, R. J., Hekstra, D., and AlQuraishi, M. Alphafold as a prior: Experimental structure determination conditioned on a pretrained neural network.bioRxiv, pp. 2025–02,

work page 2025

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Z., Salakhut- dinov, R., et al

He, Y ., Murata, N., Lai, C.-H., Takida, Y ., Uesaka, T., Kim, D., Liao, W.-H., Mitsufuji, Y ., Kolter, J. Z., Salakhut- dinov, R., et al. Manifold preserving guided diffusion. arXiv preprint arXiv:2311.16424,

work page arXiv

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Classifier-Free Diffusion Guidance

Ho, J. and Salimans, T. Classifier-free diffusion guidance. arXiv preprint arXiv:2207.12598,

work page internal anchor Pith review Pith/arXiv arXiv

[6] [6]

B., Bojan, M., Vedula, S., Schanda, P., Marx, A., and Bronstein, A

Maddipatla, A., Sellam, N. B., Bojan, M., Vedula, S., Schanda, P., Marx, A., and Bronstein, A. M. Inverse prob- lems with experiment-guided alphafold.arXiv preprint arXiv:2502.09372,

work page arXiv

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Driftlite: Lightweight drift control for inference-time scaling of diffusion models.arXiv preprint arXiv:2509.21655, 2025

Ren, Y ., Gao, W., Ying, L., Rotskoff, G. M., and Han, J. Driftlite: Lightweight drift control for inference-time scal- ing of diffusion models.arXiv preprint arXiv:2509.21655,

work page arXiv

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D., Karaguesian, J., Suomivuori, C.-M., and Dror, R

Richman, D. D., Karaguesian, J., Suomivuori, C.-M., and Dror, R. O. Unlocking hidden biomolecular conforma- tional landscapes in diffusion models at inference time. arXiv preprint arXiv:2512.03312,

work page arXiv

[9] [9]

Score-Based Generative Modeling through Stochastic Differential Equations

Song, Y ., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Er- mon, S., and Poole, B. Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456,

work page internal anchor Pith review Pith/arXiv arXiv 2011

[10] [10]

Boltz-1 democratizing biomolecular interaction modeling.BioRxiv, pp

Wohlwend, J., Corso, G., Passaro, S., Getz, N., Reveiz, M., Leidal, K., Swiderski, W., Atkinson, L., Portnoi, T., Chinn, I., et al. Boltz-1 democratizing biomolecular interaction modeling.BioRxiv, pp. 2024–11,

work page 2024

[11] [11]

A statistical bench- mark for diffusion posterior sampling algorithms.arXiv preprint arXiv:2509.12821,

Zach, M., Haouchat, Y ., and Unser, M. A statistical bench- mark for diffusion posterior sampling algorithms.arXiv preprint arXiv:2509.12821,

work page arXiv

[12] [12]

Method Details A.1

2 −1σ t ϵt, ϵ t ∼ N(0,I) 8:Amplify noise levelσ t ←(γ+ 1)σ t 9:end if 10:Make denoised predictionˆx 0 ←ˆxθ(xt, ct = (st, zt), σt) 11:Compute gradientg ct ←[g st , gzt]whereg st ← ∇ st R(ˆx0)andg zt ← ∇ zt R(ˆx0) 12:Normalize gradient by RMS¯g ct ←[¯gst ,¯gzt]where¯gst ← gstr 1 ds Pds i=1 g(i) st 2 and¯gzt ← gztq 1 dz Pdz i=1(gzt(i)) 2 13:Update embeddingc...

work page 2022

[13] [13]

DPS Algorithm Adapted to AlphaFold 3 Sampling Scheme We follow the gradient normalization strategy in Maddipatla et al

B.2. DPS Algorithm Adapted to AlphaFold 3 Sampling Scheme We follow the gradient normalization strategy in Maddipatla et al. (2025) (official implementation can be found in https://github.com/sai-advaith/guided_alphafold, which is also built on the Protenix model), and summarize the DPS algorithm adapted for AlphaFold 3 sampling scheme in Algorithm

work page 2025

[14] [14]

We set the diffusion noise schedule σ(t) =t for t∈[0,1]

conditioned on the location parameter c= 5 . We set the diffusion noise schedule σ(t) =t for t∈[0,1] . Since p(x0 |c)is Gaussian, we can access the conditional expectationE[x 0 |x t]∀(x t, t)without training a denoiser network. The measurement likelihood is given by N(y|x 0,1) with the measurement y= 20 . This setting simulates the case where the prior mo...

work page 2025

[15] [15]

is implemented with SFC Torch (Li et al., 2025).We first compute the structure factors F(⃗h) in the frequency domain by summing the scattering contributions of individual atoms: F(⃗h) = X j Oj ·f ⃗h,j ·DWF( ⃗h)·exp h 2πi⃗h·⃗ xj i (44) where j indexes the atoms, Oj denotes occupancy (fixed at 1.0), and xj represents the fractional coordinates. The term fh,...

work page 2025