PaperlayerView exact revision on arXiv

Source-linked AI summary

Unlocking Guidance for Discrete State-Space Diffusion and Flow Models

Hunter Nisonoff, Junhao Xiong, Stephan Allenspach, Jennifer Listgarten

arXiv:2406.01572v4cs.LG

TL;DR

Guidance for discrete diffusion and flow models is difficult because existing approaches lack a principled, tractable way to condition generation. The paper introduces Discrete Guidance using continuous-time Markov processes and demonstrates guided generation across small-molecules, DNA, and proteins.

  • Problem

    Discrete state-space diffusion and flow models lack a principled, tractable method for constructing conditional generative models through guidance.

  • Method

    Discrete Guidance uses continuous-time Markov chains, where only one state-space dimension changes at a time, to compute exact guided transition rates tractably.

  • Results

    Discrete Guidance effectively supports guided conditional generation across small-molecules, DNA sequences, and protein sequences, outperforming or matching alternatives on reported tasks.

  • Takeaways & Limitations

    Guided conditional generation in discrete state-spaces can be leveraged across natural-science applications.

  • Takeaways & Limitations

    Taylor-approximated guidance lacks exact guidance’s theoretical guarantees, and effective training strategies for predictors on noised samples remain unclear.

Abstract

from arXiv · show

Generative models on discrete state-spaces have a wide range of potential applications, particularly in the domain of natural sciences. In continuous state-spaces, controllable and flexible generation of samples with desired properties has been realized using guidance on diffusion and flow models. However, these guidance approaches are not readily amenable to discrete state-space models. Consequently, we introduce a general and principled method for applying guidance on such models. Our method depends on leveraging continuous-time Markov processes on discrete state-spaces, which unlocks computational tractability for sampling from a desired guided distribution. We demonstrate the utility of our approach, Discrete Guidance, on a range of applications including guided generation of small-molecules, DNA sequences and protein sequences.

1 DIFFUSION AND FLOW-BASED GENERATIVE MODELS FOR SCIENCE

Discrete diffusion and flow models address scientific data in discrete state-spaces, but lacked principled conditioning through guidance. The paper introduces Discrete Guidance, an exact and tractable CTMC-based framework demonstrated across molecular and biological sequence generation.

  • Discrete state-spaces encompass biological sequences and molecular graphs, but continuous-space score functions and linear flow interpolation behave poorly when applied directly to them.Existing CTMC formulations replace scores or vector fields with neural estimates of transition rates, yet previously lacked principled guidance.
  • The framework supports conditional generation by modulating an already-trained unconditional model, which is valuable when unlabelled scientific data greatly exceed labelled data.Scientific conditioning targets include protein backbone structures, binding affinity, and enzymatic activity.
  • Discrete Guidance enables exact, theoretically rigorous predictor and predictor-free guidance for CTMC-based generative models, including discrete diffusion and flow matching.Because only one state-space dimension can change at any instant in continuous time, the guidance remains computationally tractable.
  • The method’s utility is demonstrated across conditional generation tasks involving small molecules, DNA sequences, and protein sequences, using both diffusion and flow models.
  • An efficient approximation to Discrete Guidance preserves sample quality while reducing sampling cost.

2 CHALLENGES OF GUIDANCE IN DISCRETE STATE-SPACES

Guidance in discrete state-spaces cannot directly use continuous diffusion scores, while Bayes-based conditioning requires an generally intractable exponential normalizing constant. Continuous-time Markov formulations address this by constraining transitions to enable tractable computation without sacrificing model expressibility, also supporting flow-based guidance.

  • Bayes-based guidance in a D-dimensional discrete space with cardinality S requires a normalizing constant containing S^D terms, making it generally intractable.The exponential number of terms arises because unconstrained transitions allow any state to reach any other state.
  • Continuous-time discrete diffusion can impose transition constraints that make the normalizing constant tractable without losing model expressibility, while the same computations enable flow-based guidance.Prior work on guidance for flow models is limited, including exploration in the continuous probabilistic simplex for discrete sample generation.

3 BACKGROUND ON DISCRETE DIFFUSION AND FLOW MODELS

CTMCs provide the mathematical foundation for discrete diffusion and flow models by encoding holding and jumping behavior in time-dependent rate matrices. These models learn to reverse noise through denoising-induced rates, motivating guidance for conditional generation.

  • CTMCs define discrete generative processes through exponentially distributed holding times and jumps governed by time-dependent rate matrices R_t.The rates determine both jump frequencies and destinations, while off-diagonal entries are nonnegative and diagonal entries are nonpositive.
  • CTDD and DFM use CTMCs to progressively noise discrete data and learn denoising models that reverse the process (Campbell et al., 2022; Campbell et al., 2024).Their learned denoisers predict clean data from noisy states, with the training loss depending on whether the model is diffusion- or flow-based.
  • Sampling starts from an easily sampled noise distribution and integrates learned time-dependent rate matrices toward the target distribution.The rate matrices induced by denoising models can be simulated through time, including by Euler integration or more sophisticated methods.
  • Conditional generation could use rate matrices conditioned on a desired property, but direct conditional modeling has typically been outperformed by guidance-based approaches.This limitation motivates introducing Discrete Guidance for discrete diffusion and flow models.

4 UNLOCKING GUIDANCE IN DISCRETE STATE-SPACES

The method makes guidance tractable in discrete state-spaces by exploiting sparse transition rates and reformulating Bayes’ theorem in continuous-time Markov-process rates. It also introduces Taylor-approximated guidance to reduce predictor evaluations while usually matching exact-guidance sample quality.

  • 4 UNLOCKING GUIDANCE IN DISCRETE STATE-SPACES: Independent per-dimension noising leaves only D × (S −1) + 1 non-zero rate-matrix entries, preserving tractable unguided and guided sampling without constraining expressivity.The same sparsity reduces Bayes-normalizer computation from SD terms to D × (S −1) + 1 terms.
  • 4 UNLOCKING GUIDANCE IN DISCRETE STATE-SPACES: Discrete Guidance derives conditional rates from unconditional rates and a predictive distribution, modulating them with a likelihood ratio whose influence can be tuned by guidance strength γ ∈R+.The framework also derives predictor-free guidance by blending unconditional and conditional rate models, without relying on a predictive distribution.
  • 4 UNLOCKING GUIDANCE IN DISCRETE STATE-SPACES: The rate adjustments extend beyond discrete diffusion and flow models to a broad class of CTMC-based generative models on discrete state-spaces.The paper states this generalization holds in principle for models realized through continuous-time Markov chains.
  • 4.1 EFFICIENT APPROXIMATIONS FOR PREDICTOR GUIDANCE: Taylor-approximated guidance reduces predictor computation from D × (S −1) + 1 forward passes to one forward and one backward pass, making adjusted-rate estimation O(1) rather than O(D × S).TAG uses a first-order Taylor approximation of the predictor log-likelihood ratio evaluated at discrete states.
  • 4.1 EFFICIENT APPROXIMATIONS FOR PREDICTOR GUIDANCE: Across the paper’s empirical investigations, TAG usually matches exact guidance in sample quality and generally outperforms DiGress.The comparison is reported as systematic for TAG versus exact guidance and as generally superior for TAG versus DiGress.

5 RELATED WORK

Discrete Guidance extends continuous-time discrete-state diffusion and flow-model work by enabling guidance directly on native discrete spaces. It differs from embedding-based, Dirichlet-simplex, and DiGress approaches in training requirements, tractability, and generality.

  • Prior continuous-time discrete-state diffusion and flow models define sampling through CTMCs but do not themselves propose guidance, making them compatible with Discrete Guidance.These include DFM, CTDD, alternative- objective CTMC methods, and generalized flow-matching probability paths, extending earlier discrete-time diffusion work.
  • Continuous-space guidance methods use classifiers or blend conditional and unconditional score estimates, providing precedents for controllable generation outside native discrete state-spaces.Sohl-Dickstein et al. (2015), Song et al. (2021), Dhariwal & Nichol (2021), and Ho & Salimans (2021) represent these conditioning strategies.
  • Embedding-based methods apply standard continuous-space guidance but require additional embedding training and hyperparameter selection, whereas Discrete Guidance reuses native-space predictor models.The paper cites predictor-model reuse as an easier way to repurpose existing models without learning a continuous embedding.
  • Unlike Dirichlet FM, which performs guidance on the probability simplex, Discrete Guidance operates directly on the native discrete state-space and achieves better predictor-guidance results on the same task.The comparison concerns the task presented by St¨ark et al. (2024), with empirical results reported in Section 6.2.
  • Compared with DiGress, Discrete Guidance offers tractable exact guidance and supports flow matching, while DiGress remains discrete-time, approximate, diffusion-specific, and requires further conditioning approximations.Both methods use Taylor-based predictor approximations for discrete inputs, but DiGress’s discrete-time formulation makes the conditioning normalizing constant intractable.

6 EMPIRICAL INVESTIGATIONS

Discrete Guidance was evaluated for conditional generation of small molecules, DNA enhancers, and protein sequences, generally matching or outperforming discrete diffusion and task-specific baselines. Across these applications, guidance shifted samples toward molecular targets, achieved competitive DNA conditioning, and improved protein stability while preserving folding.

  • 6 EMPIRICAL INVESTIGATIONS: The experiments used unconditional flow-matching models with predictor or predictor-free guidance across three domains, comparing Discrete Guidance with DiGress and additional task-specific baselines.The study aimed to test whether Discrete Guidance worked across multiple domains rather than claim optimal generation for any single problem.
  • 6.1 SMALL-MOLECULES GENERATION: Discrete Guidance significantly reduced molecular property MAE relative to DiGress, while exact guidance performed comparably to or better than Taylor-approximated guidance.Guidance also shifted generated histograms toward specified ring counts and lipophilicity targets.
  • 6.2 ENHANCER DNA DESIGN: Across all eight DNA cell-type classes, DG-Exact, DG-TAG, and DiGress performed comparably, while DirFM-CG performed more poorly.Evaluation used Class-Conditional FBD, where lower is better, and target-class probability, where higher is better; target-class probability better reflects guidance effectiveness.
  • 6.3 GUIDING INVERSE-FOLDING MODELS FOR STABILITY: FMIF-DG-Exact and FMIF-DG-TAG consistently achieved the highest stability-guided inverse-folding success rates, with FMIF-DG-Exact outperforming or matching all other methods on seven of eight proteins.Success required both RMSD ≤2 ˚A and predicted stability ∆∆G ≥0; FMIF-DG-TAG performed better on protein 6ACV.

7 DISCUSSION … C.8 SUMMARY OF GUIDANCE IN CTMCS

This section presents guidance methods for discrete CTMC diffusion and flow models, including predictor-guided and predictor-free sampling rules. It also characterizes the sparse transition structure of CTDD- and DFM-realized CTMCs and discusses theoretical, training, and safety considerations.

  • 7 DISCUSSION: The approach applies across CTMC-based discrete diffusion and flow models and was evaluated for guided conditional generation of small molecules, DNA sequences, and protein sequences.The discussion frames guided generation as broadly useful across natural-science applications.
  • 7 DISCUSSION: Taylor-approximated guidance lacks exact guidance’s theoretical guarantees, while predictor training strategies remain unclear and controllable text generation is an open direction.The paper also notes potential benefits for drug discovery and protein engineering alongside risks of bioweapon misuse and amplified training-data bias.
  • C.3 PREDICTOR GUIDANCE IN CTMCS: Discrete Guidance conditions CTMC transitions by combining unconditional rates with predictive likelihoods, yielding valid nonnegative guide-adjusted rates.The method provides a principled predictor-guidance construction for discrete state-space diffusion and flow models.
  • C.4 SIMPLIFYING PREDICTOR GUIDANCE IN CTMCS: Under guide-independent noising, predictor guidance simplifies to a likelihood ratio that favors states with higher predictive probability for the desired property.The likelihood ratio is raised to the guidance strength, making conditioning stronger as guidance increases.
  • C.5 TRAINING A NOISY PREDICTOR MODEL: A noisy predictor is trained by sampling labeled data, a time, and a noised state, then maximizing the label log-probability conditioned on that state and time.Alternative approximations that avoid training a noisy predictor are also discussed.
  • C.6 TEMPERATURE SAMPLING: Guidance temperature T, or inverse strength γ = 1/T, interpolates from unconditional sampling at γ = 0 to stronger-than-exact conditioning when γ > 1.The paper notes that true temperature-annealed sampling for diffusion models remains an open problem (Du et al., 2023).
  • C.7 PREDICTOR-FREE GUIDANCE IN CTMCS: Predictor-free guidance equivalently realizes temperature-modulated conditioning using unconditional and conditional CTMC rates, without requiring a predictor.In practice, both conditional and unconditional models are learned.
  • C.8 SUMMARY OF GUIDANCE IN CTMCS: In the limit Δt →0, CTMC sampling supports both predictor-guided and predictor-free rates at guidance strength γ = 1/T.Predictor guidance uses a learnable p(y|x*, t), whereas predictor-free guidance combines unconditional and conditional rates.
  • C.8 SUMMARY OF GUIDANCE IN CTMCS: CTDD (Campbell et al., 2022) and DFM (Campbell et al., 2024) realize CTMCs where only one state component can change in infinitesimal time.
  • C.8 SUMMARY OF GUIDANCE IN CTMCS: Only D × (S − 1)+1 transition rates R_t(x, x′) are non-zero in these CTMCs.
  • C.8 SUMMARY OF GUIDANCE IN CTMCS: The corresponding D × (S − 1)+1 guided rates R_t^(γ)(x, x′|y) are also non-zero, ensuring the identity rate R^(γ).

C.9 GENERALITY OF OUR RESULTS

The guidance results require tractable identity rates in practice, but the framework conceptually extends beyond the instantiated CTDD and DFM models to any suitable independent-dimension CTMC.

  • The framework’s derivation assumes only that the rates of a general unguided CTMC are known, while practical guidance additionally requires tractable identity-rate computation.The relevant identity rate is the sum in Equation 27.
  • Tractable unconditional or conditional identity rates in an unguided CTMC should guarantee tractability of the corresponding identity rate in the guided CTMC.This condition supports applying the results summarized in Equations 25–27 in practice.
  • Discrete Guidance conceptually applies to any generative model realized by a CTMC with transitions independent across dimensions, regardless of how its rate matrices are obtained.The authors instantiated it only for CTDD (Campbell et al., 2022) and DFM (Campbell et al., 2024), while noting possible future frameworks.

D IMPLEMENTATION DETAILS … E.1 ADDITIONAL DISCUSSION

This section details the implementation and sampling of Discrete Guidance, including masked flow-model training, guide-adjusted rates, predictor and predictor-free guidance, and handling of fixed components. It also discusses its relationship to alternative guidance approaches and the reuse of pretrained models without fine-tuning.

  • D IMPLEMENTATION DETAILS: The implementation section provides algorithmic summaries and PyTorch code for training, guide-rate calculation, and integrating guidance into masked DFM sampling.It also includes an extension for discrete flow models with fixed components.
  • D.1 TRAINING DISCRETE STATE-SPACE FLOW MODELS: Masked discrete flow-model training samples a uniform time and masked state for each data batch, then uses cross-entropy loss for gradient updates.The training procedure is summarized in Algorithm 1 and follows Campbell et al. (2024).
  • D.2 CALCULATION OF GUIDE-ADJUSTED RATES: Guide-adjusted rates are computed either from exact log-probability ratios over all one-jump states or from a Taylor approximation using the predictor gradient.Both procedures adjust learned unconditional rates for guided sampling; the TAG implementation assumes categorical data.
  • D.2 CALCULATION OF GUIDE-ADJUSTED RATES: Exact guidance evaluates the predictor on the current state and every possible one-jump successor, while TAG computes the corresponding likelihood-ratio approximation from gradients.The implementations assume predictor log p(y | x_t, t), unconditional rates R_t, and a function enumerating all possible one-jump states.
  • D.3 DFM SAMPLING WITH GUIDANCE: Guided DFM sampling initializes masked states except fixed positions, obtains predictor or predictor-free logits, adjusts and normalizes rates, then samples transitions until t reaches 1.The sampling procedure returns the generated sequence and supports both predictor guidance and predictor-free guidance.
  • D.3 DFM SAMPLING WITH GUIDANCE: DFM sampling combines unconditional and conditional denoising rates, adds remasking rates, and applies predictor-free guidance through inverse-temperature rate adjustment before sampling the next state.The loop handles masked states through unmasking rates and non-masked states through remasking rates, then converts adjusted rates into transition probabilities.
  • D.4 DISCRETE FLOW MATCHING FOR STATES WITH FIXED COMPONENTS: For padded or otherwise fixed components, the augmented masking flow leaves the fixed component-state unchanged throughout noising and denoising by forbidding transitions to or from it.During generation, unmasking and remasking rates are set to zero for fixed component-states, making them isolated from both operations from t = 0 to t = 1.
  • E.1 ADDITIONAL DISCUSSION: Unlike score-matching alternatives, continuous latent-space methods, and autoregressive guidance, Discrete Guidance modulates discrete-state sampling with predictor likelihood ratios while reusing pretrained unconditional models without fine-tuning.Predictor guidance can also combine different predictors modularly, while latent-space approaches may lose discrete structure and some alternatives require specialized modeling, extra training, or additional hyperparameters.
  • E FURTHER DISCUSSION OF RELATED WORK: Discrete Guidance differs from DiGress by providing exact continuous-time rate adjustment, transition-specific Taylor approximation, inference-time step flexibility, and selectable sampling algorithms.DiGress applies gradient-based approximate guidance across possible transition states with a fixed training-time step count, whereas Discrete Guidance targets transitions changing one dimension.

E.2 COMPARISONS WITH OTHER GUIDANCE APPROACHES · F EXPERIMENTAL DETAILS · F.1 IMAGE MODELING

The paper unifies guidance methods across continuous and discrete state-spaces by identifying time-discretization and gradient approximations, then uses continuous-time Markov rates to make discrete guidance tractable and exact. Experiments apply Discrete Guidance to high-dimensional discrete-pixel images using a pretrained discrete diffusion model.

  • E.2 COMPARISONS WITH OTHER GUIDANCE APPROACHES: The comparison framework shows DiGress (Vignac et al., 2023) combines gradient and time-discretization approximations, whereas continuous-time limits make the latter exact.In continuous state-spaces, only time-discretization error remains, and it becomes exact in the continuous-time score-based framework (Song et al., 2021).
  • E.2 COMPARISONS WITH OTHER GUIDANCE APPROACHES: The underlying conditional transition is generally intractable because its normalizing denominator requires summing over discrete states or integrating over continuous states.Discrete-time approximations can therefore incur errors that depend on the number of training time steps, especially when the curvature assumption is poor.
  • E.2 COMPARISONS WITH OTHER GUIDANCE APPROACHES: In discrete state-spaces, the score function is not defined, so predictor guidance is expressed in log-space as an unconditional transition plus a likelihood-ratio correction.Specifically, log p(xt+∆t = ˜x|xt = x, y) = log p(xt+∆t = ˜x|xt = x) + [log p(y|xt+∆t = ˜x) − log p(y|xt = x)].
  • E.2 COMPARISONS WITH OTHER GUIDANCE APPROACHES: The rate-adjustment procedure is agnostic to how p(y|x_t) is estimated, but predictor quality determines how closely guided samples approach the true posterior p(x|y).Noisy predictors and clean-predictor estimates such as p(y|x̂_1) are both approximations to the true distribution; training modifications can improve the match.
  • E.2 COMPARISONS WITH OTHER GUIDANCE APPROACHES: Training-free guidance approaches face additional difficulties in discrete spaces because the posterior mean x̂_1 may lie inside the probabilistic simplex, outside the discrete predictor’s input domain.The paper notes that extending these approaches to discrete state-spaces is not immediate and leaves possible workarounds beyond scope.
  • F.1 IMAGE MODELING: The experimental details cover images, small molecules, enhancer DNA, and proteins, with image modeling using CIFAR-10 represented as discrete pixels.The image experiment tests whether Discrete Guidance can reuse a pretrained discrete diffusion model for high-dimensional, property-directed generation; architecture and results appear in Appendices F.1.1 and F.1.2.

F.1.1 MODEL ARCHITECTURE AND TRAINING

The model uses a pre-trained unconditional U-net denoiser and a classifier formed from its downsampling trunk with global average pooling. The classifier is trained with the denoiser’s forward-noise process and specified optimization, augmentation, and early-stopping procedures.

  • F.1.1 MODEL ARCHITECTURE AND TRAINING: The denoiser is a standard U-net initialized from Campbell et al. (2022), while the classifier reuses its downsampling trunk with global average pooling.The denoiser weights come from the official repository.
  • F.1.1 MODEL ARCHITECTURE AND TRAINING: The noisy classifier was trained for 200 epochs on the denoiser’s training set using the same forward process, with validation accuracy for early stopping.Training used one RTX 6000A GPU, and 200 epochs required 2 hours.
  • F.1.1 MODEL ARCHITECTURE AND TRAINING: Optimization used Adam, a 0.0002 learning rate with 5,000 linear warmup steps, 0.1 dropout, and random horizontal flips.The horizontal flips were applied to input images during training.

F.1.2 RESULTS · F.2 MOLECULE GENERATION · F.2.1 DATASET CONSTRUCTION

Discrete Guidance produced visually plausible class-conditional images and maintained comparable unconditional sample quality while exposing a quality–diversity trade-off. The molecule-generation dataset was filtered from QMugs into 610,575 canonical SMILES molecules represented as padded discrete token sequences.

  • F.1.2 RESULTS: Predictor-guided sampling generated expected class-conditional images with reasonable visual quality at guidance strengths γ = 2 and γ = 3.Sampling used τ-leaping with τ = 0.001, without predictor-corrector steps, and Taylor-approximated predictor guidance.
  • F.1.2 RESULTS: CTDD-PG matched or sometimes exceeded CTDD on unconditional Inception Score and FID, indicating comparable quality while covering target-class distributions.The comparison was intended as a quantitative coverage check rather than evidence that guidance improves unconditional generation.
  • F.1.2 RESULTS: Increasing guidance strength from 2 to 3 improved Inception Score but worsened FID, revealing a trade-off between sample quality and diversity.The authors attribute the FID worsening to decreased sample diversity.
  • F.2 MOLECULE GENERATION: The molecule-generation study covers dataset construction, model training, SMILES generation and validity, property control, and comparison with DiGress (Vignac et al., 2023).These components define the scope of the molecule-generation evaluation.
  • F.2.1 DATASET CONSTRUCTION: Preprocessing QMugs molecules from ChEMBL yielded 630,508 unique canonical SMILES after standardization, stereochemistry removal, and simplified-string conversion.The preprocessing used RDKit and the ChEMBL Structure pipeline; the source datasets are attributed to Isert et al. (2021), Gaulton et al. (2011), and Weininger (1988).
  • F.2.1 DATASET CONSTRUCTION: Filtering by molecular weight, SMILES length, ring count, and LogP produced 610,575 molecules split 4:1 into train and holdout sets.The final representation padded SMILES to D = 100 tokens over S = 32 token states, including pad and mask tokens.

F.2.2 MODEL ARCHITECTURE AND TRAINING · F.2.3 SMILES STRING GENERATION · F.2.4 SMILES STRING VALIDITY

The molecular generator combines an unconditional masked discrete flow model with separate predictors for ring count and lipophilicity, then generates padded SMILES strings by fixed-pad denoising. With stochasticity selected for validity, unconditional generation achieves (12±3)% validity, while conditional validity is lowest when targeting zero rings.

  • F.2.2 MODEL ARCHITECTURE AND TRAINING: The system uses three fully connected networks: an unconditional masked discrete flow model, a number-of-rings predictor, and a lipophilicity predictor.The predictors model pϕ1(Nr|x,t) and pϕ2(LogP|x,t).
  • F.2.2 MODEL ARCHITECTURE AND TRAINING: The unconditional model has two 20,000-unit ReLU hidden layers, while each predictor has one 1,000-unit ReLU hidden layer.The noisy input xt implicitly conveys time because its average mask-token count is proportional to t, so explicit time input is unnecessary.
  • F.2.2 MODEL ARCHITECTURE AND TRAINING: Predictors use continuous normal likelihoods for ring count and LogP; discretizing the ring-count distribution performed comparably, so the simpler continuous model was selected.The LogP predictor uses σ^y_0 = 1.66, while σ^y_0 is estimated from the training property distribution at the fully masked state.
  • F.2.3 SMILES STRING GENERATION: Generation samples the number of pad tokens from the training-set SMILES-length distribution, fixes those pads throughout denoising, and starts from a fully masked string.Euler integration uses step size 0.001 to time 0.98, followed by argmax; guided sampling uses γ ∈ {0.2, 1, 2} and η = 30.
  • F.2.4 SMILES STRING VALIDITY: 12±3% validity is obtained for unconditional generation when stochasticity is set to η = 30, selected because it maximizes average validity.Validity is assessed by constructing molecules from generated SMILES strings with RDKit.
  • F.2.4 SMILES STRING VALIDITY: 4±2% validity occurs for conditional generation targeting zero rings, as ring guidance can discourage or promote unclosed-ring errors depending on the target.Validity fractions and uncertainties are means and standard deviations over batches of 100 simultaneously generated SMILES strings.
  • F.2.4 SMILES STRING VALIDITY: The study samples SMILES strings until obtaining 1,000 valid molecules for each molecular-generation condition.The authors suggest graph-structure generation as a possible route to improving validity.

F.2.5 WIDE RANGE CONDITIONAL GENERATION

Discrete Guidance shifts generated molecule-property distributions across wide target ranges for ring count and lipophilicity. Guidance largely preserves diversity, though tail targets produce slightly less diverse molecules and occasional property impurity.

  • F.2.5 WIDE RANGE CONDITIONAL GENERATION: Guidance clearly shifts molecule distributions toward demanded ring-count and LogP target values across wide ranges.The evaluation generated 1,000 valid SMILES strings with guidance strength γ = 2 using Discrete Guidance.
  • F.2.5 WIDE RANGE CONDITIONAL GENERATION: Ring-count histograms are sharp for low target values, whereas LogP distributions broaden in the tails and sharpen near the unconditional mode.These patterns characterize how accurately distributions concentrate under different target conditions.
  • F.2.5 WIDE RANGE CONDITIONAL GENERATION: Most generated molecules contain the requested ring count, while a minority differs by one ring; increasing guidance strength could reduce this impurity.The impurity is already visible in the ring-count histograms.
  • F.2.5 WIDE RANGE CONDITIONAL GENERATION: Increasing LogP targets produces molecules transitioning from primarily polar bonds to primarily non-polar bonds, consistent with lipophilicity being inversely related to bond polarity.The observed transition matches the expected structural interpretation of higher lipophilicity.

F.2.6 COMPARING DISCRETE GUIDANCE TO DIGRESS … F.4.5 METRICS

Across molecule, enhancer-DNA, and protein-generation experiments, Discrete Guidance generally matches or outperforms discrete and continuous guidance baselines, while the appendices specify the training, sampling, guidance, and evaluation procedures. The comparisons also identify cases where exact guidance improves on Taylor approximation and where high guidance strengths degrade performance.

  • F.2.6 COMPARING DISCRETE GUIDANCE TO DIGRESS: Discrete Guidance significantly outperformed DiGress across six molecular targets and three guidance strengths, while DG-exact was at least as good as DG-TAG.The comparison used MAE over 1,000 generated molecules and Mann-Whitney U tests; Figure 10 reports the evaluated conditions.
  • F.3 ENHANCER DNA DESIGN; F.3.1 DIRICHLET FM: The enhancer-DNA experiments reproduced Dirichlet FM using official checkpoints, matched reported results, and found similar metrics when sampling 1,000 to 10,000 sequences.Differences from St¨ark et al. (2024) may reflect random seeds, evaluation-sample counts, and reference-set construction.
  • F.3.2 MODEL ARCHITECTURE AND TRAINING; F.3.3 SAMPLING: Enhancer models used matched convolutional architectures and comparable training setups, while DiGress used 100 discrete time steps and separate noisy classifiers trained for each native-space method.Discrete Guidance used 20 denoising layers and five classifier layers; classifier training used unconditional-model samples labeled by the clean classifier.
  • F.3.4 PREDICTOR GUIDANCE: On eight enhancer cell types, DG-Exact, DG-TAG, and DiGress performed comparably and each slightly led on some classes or metrics, while all outperformed DirFM-CG.The same three discrete-space methods were also comparable under target-class success and diversity metrics.
  • F.3.5 PREDICTOR-FREE GUIDANCE: DG-PFG and DirFM-CFG performed comparably across eight enhancer cell types, with method-specific advantages and occasional metric deterioration at high guidance strengths.The proposed explanation is concentration on regions deviating from the data distribution or difficulty sampling the correct temperature-annealed distributions.
  • F.4.1 FMIF TRAINING; F.4.4 BASELINES: FMIF was trained as a non-autoregressive ProteinMPNN-based flow model, and its sequence recovery on the PDB test set improved over retrained ProteinMPNN.FMIF used four encoder and four decoder layers, backbone noise 0.1, 30-neighbor graphs, and Euler sampling with step size 0.01 and stochasticity η = 1.
  • F.4.2 STABILITY PREDICTION; F.4.3 STABILITY GUIDANCE: The stability predictor generalized to novel structures and sequences with average validation-cluster Spearman correlation 0.80, then supported protein-specific noisy-classifier guidance.The noisy classifiers were initialized from the regression model, refined with 1,000 guided samples labeled by the clean model, and used approximate guide-adjusted rates.
  • F.4.4 BASELINES; F.4.5 METRICS: Protein-generation evaluation defined success as generated sequences with RMSD ≤2 Å and ΔΔG ≥0, and computed diversity as average pairwise Hamming distance after sequence-only ColabFold folding.ProteinMPNN used pre-trained weights trained with backbone noise 0.1, the same noise used in FMIF.

F.4.6 DETAILED RESULTS

The section reports detailed inverse-folding results for eight proteins in Tables 4–11. Across these evaluations, methods are assessed by stability, correct folding, and sequence diversity.

  • Detailed results: Tables 4–9 report inverse-folding results for six proteins: 5KPH, 1F0M, 6M3N, 1O6X, 2JT1, and 2JN4.The section states that detailed results are provided for eight evaluated proteins.
  • Evaluation framework: Each table evaluates protein stability, correct folding with RMSD ≤2 Å and ΔΔG ≥0, and diversity measured by average pairwise Hamming distance.The tables compare temperature-controlled structure-conditioned generation, ProteinMPNN, unguided FMIF, DiGress, and TAG-approximated and exact FMIF discrete guidance.
  • Detailed results: Tables 10–11 extend the same inverse-folding evaluation to proteins 6ACV and 2K5N.Both tables use the same stability, folding, and diversity evaluation framework.
Loading 2406.01572v4…