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Trajectory Matching: Discrete Flow Matching with Structured Actions

Niklas Kemper, Marten Lienen, David Lüdke, Stephan Günnemann

arXiv:2610.32602v1cs.LG

TL;DR

Structured generation often needs actions beyond independent edits on fixed-size states, but endpoint-conditioned discrete flow matching has difficulty defining tractable rates for such actions. Trajectory matching conditions on full action sequences and recovers the marginal flow; domain-aligned actions yield gains across tree and molecular generation, including higher MOSES validity with fragment insertions.

  • Problem

    Structured generation often uses size-changing, multi-component, or dependent actions, while existing discrete flow matching relies on tractable endpoint-conditioned rates that are typically factorized across fixed-size components.

    Source: §1 · p. 1, §2 · p. 3

  • Method

    Trajectory matching conditions on a full action trajectory connecting endpoints to define simulation-free training targets for user-defined action spaces.

    Source: §1 · p. 2

  • Results

    Domain-aligned actions yield gains on tree and molecular graph generation, including MOSES validity from 92.8% to 99.2% with fragment-based insertions.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Trajectory matching subsumes factorized DFM and expands its design space by making the action vocabulary a modeling choice.

    Source: §8 · pp. 9–10

    Tractability depends on efficiently sampling trajectories and ideally computing Rao–Blackwellized rates in closed form; outside some families, new domains may require non-trivial methodological work.

    Source: §7 · p. 9

Abstract

from arXiv · show

The natural units of structured discrete generation are often not independent single-component edits but richer actions: attaching a fragment to a molecule, inserting a node together with an incident edge, or growing a hierarchy by adding a child. Yet most existing discrete flow matching methods (DFM) are restricted to independent component-wise edits over fixed-size states. This restriction stems from endpoint conditioning: endpoints fix the source and target during training but not the action sequence between them, making tractable conditional rates hard to define for complex actions that cannot be factorized component-wise. We propose trajectory matching, which conditions the flow on a full sequence of actions connecting the endpoints. This yields principled, simulation-free training targets for user-defined action spaces, including actions that change state size, modify multiple components, or depend on earlier actions. We prove that the resulting objective recovers the marginal flow and that trajectory matching strictly subsumes factorized DFM, while additionally capturing actions with prerequisite structure such as hierarchy growth or fragment-based molecule generation. Empirically, on tree and molecular graph generation, trajectory matching with domain-aligned actions matches or surpasses factorized DFM baselines (lifting MOSES validity from 92.8% to 99.2%) and transfers without retraining to substructure extension and linking. Trajectory matching enlarges the design space of discrete flow matching by turning the action vocabulary into a modeling choice. An interactive explanation of trajectory matching is available at https://niklaskemper.com/blog/trajectory-matching/.

1 INTRODUCTION

Structured generation often calls for domain-aligned actions that change state size, modify multiple components, or depend on earlier actions. Trajectory matching conditions training on full action sequences, extending discrete flow matching to these actions while retaining simulation-free training and recovering the marginal flow.

1 INTRODUCTION

  • Existing discrete flow matching methods factorize rates across components, restricting them to fixed-size states, single-component edits, and independent actions.Extensions supporting insertions or deletions still reduce them to factorized single-component edits; multi-component or dependent actions remain out of reach.Source: §1 · p. 1
  • Trajectory matching conditions on a full action trajectory, making tractable conditional rates available for size-changing, multi-component, and dependent actions.Its simulation-free objective provably recovers the marginal data-generating flow, and trajectory matching recovers standard factorized DFM as a special case.Source: §1 · p. 2
  • Trajectory matching subsumes factorized DFM and extends to partial-order actions used in hierarchy growth and fragment-based molecule construction.The framework supports user-defined action spaces, making the action vocabulary a modeling choice.Source: §1 · p. 2
  • On trees, child-attachment actions yield 100% validity by construction; on molecules, fragment-based insertions lift MOSES validity from 92.8% to 99.2%.A single variable-size insertion model also transfers zero-shot to substructure extension and linking, substantially outperforming a factorized baseline that requires task-specific adaptation.Source: §1 · p. 2

2 ACTION-BASED DISCRETE FLOW MATCHING

This section frames discrete flow matching in terms of actions on states, with action-space design determining which transformations are possible. Endpoint-conditioned rates are tractable under factorized assumptions, but general action sequences can make endpoint-only rate specification difficult.

2 ACTION-BASED DISCRETE FLOW MATCHING

  • Actions operate on discrete states; the action vocabulary and state-dependent available actions determine possible transformations, including size-changing and dependent operations.Two actions are independent when their order does not matter and both remain available after the other is applied.Source: §2 · p. 2
  • Discrete flow matching learns action rates for a continuous-time Markov chain that transports a source distribution to a target distribution.Training couples source and target endpoints, then averages endpoint-conditioned rates over the endpoints given the current state.Source: §2 · pp. 2–3
  • Endpoint-only rate specification is difficult when many action sequences, including dependent ones, connect the same endpoints.Existing tractable constructions assume fixed-size component states, single-component edits, and independent component evolution, yielding factorized rates.Source: §2 · p. 3

3 TRAJECTORY MATCHING

Trajectory matching defines conditional flows over general action spaces by conditioning on a full sequence of actions connecting the endpoints. Its simulation-free objective recovers the marginal flow, and Rao–Blackwellization can reduce gradient variance while preserving the optimum.

3 TRAJECTORY MATCHING

  • Conditioning on a trajectory makes valid conditional flows straightforward to define for actions that change state size, modify multiple components, or depend on one another.The full sequence fixes which actions occur and their order, leaving the conditional flow to determine when they occur.Source: §3 · p. 3, §3.1 · p. 3

3.1 TRAJECTORY-CONDITIONED FLOWS

  • With uniformly distributed action execution times, only the next action in a trajectory has positive rate, proportional to the number of remaining actions.The resulting path is easy to sample, even for complex actions whose endpoint-conditioned rates cannot be factorized.Source: §3.1 · p. 4

3.2 TRAJECTORY MATCHING

  • A trajectory distribution over valid endpoint-connecting sequences defines the training paths and weights, and the resulting marginal rate transports the source distribution to the target.The trajectory distribution encodes an inductive bias over how the initial state is transformed into the final state.Source: §3.2 · pp. 4–5
  • The training objective is simulation-free, and its gradients agree with those of the marginal matching objective.With a sufficiently expressive model family, a minimizer recovers the marginal rate and generates the marginal probability path.Source: §3.2 · pp. 4–5

3.3 VARIANCE REDUCTION VIA RAO–BLACKWELLIZATION

  • Rao–Blackwellizing over trajectories consistent with the current state and endpoints preserves the expected gradient and optimum while reducing gradient variance.The expectation can replace the sampled-trajectory target when it is efficiently computable.Source: §3.3 · p. 5

4 INSTANTIATIONS OF TRAJECTORY MATCHING

Trajectory matching instantiates structured discrete flows by choosing an action space and a trajectory distribution. These choices range from independent edits to prerequisite-ordered or variable-action-set generation, while factorized flow matching is recovered as a special case.

Independent fixed-action distributions

  • Factorized flow matching has the same gradient and minimizers as trajectory matching with independent, order-invariant component actions, and its Rao–Blackwellized loss is identical.This makes factorized flow matching a special case of the framework.Source: §4 · p. 6

Dependent fixed-action distributions

  • A dependency DAG represents prerequisite actions, with consistent trajectories given by topological orderings.Edgeless and chain DAGs recover independent-order and fixed-order autoregressive generation, respectively; general DAGs model hierarchy growth and fragment attachment.Source: §4 · pp. 6–7
  • For forest dependencies, Rao–Blackwellized rates weight available actions by the size of their remaining subtrees, favoring actions on which more future actions depend.The rates interpolate between equal weighting for independent actions and next-action-only supervision for a fully sequential chain; exact rates for general DAGs are intractable in general.Source: §4 · p. 7

Variable-action-set distributions

  • When endpoint pairs admit different action sets, a latent decomposition variable yields a mixture over conditional trajectory distributions.For ATOMGROW, the latent spanning tree determines which edges are created by AddAtom and which require AddBond; MOLEDIT uses a similar construction.Source: §4 · p. 7

Summary

  • The framework also interprets insertion- and deletion-based EditFlows as trajectory matching over a variable action set.Overall, trajectory matching subsumes endpoint-conditioned factorized flows while extending to complex actions and dependencies.Source: §4 · p. 7

5 EMPIRICAL RESULTS

Across tree and molecular graph generation, trajectory matching supports domain-aligned action spaces, which can guarantee tree validity, match factorized baselines, or improve molecular validity. A single model also transfers to substructure extension and linking.

Evaluation setup

  • All instantiations use the same training and sampling procedure and graph transformer backbone, differing only in action space, source distribution, and trajectory distribution.Source: §5 · p. 7

Hierarchy generation

  • At 10 Euler steps, GROWHIERARCHY retains 100% validity while DeFoG falls to 8%; GROWHIERARCHY also achieves 100% V.U.N. and a Ratio of 2.5.Its tree-preserving actions guarantee validity by construction; the Ratio is the best distributional graph-statistics score among evaluated variable-size methods.Source: Table 1 · p. 8, §5 · p. 8

Molecular graph generation

  • MOSES validity rises from 92.8% to 99.2% with FRAGGROW, while BONDEDIT matches existing factorized DFM and discrete diffusion baselines.ATOMGROW and FRAGGROW also outperform the variable-size diffusion baseline GrIDDD on most metrics.Source: Table 2 · p. 8, §5 · p. 8

Downstream applications

  • One ATOMGROW model achieves 94.6% versus 33.8% valid-and-connected on extension and 78.8% versus 57.0% on linking, compared with factorized BONDEDIT.The extension comparison covers 11 substructures, including some absent from training; the linking comparison covers 6 fragment pairs.Source: §5 · p. 8

6 RELATED WORK

Related work spans action-sequence generation, discrete flow matching, and approaches that relax fixed-size constraints. The paper positions trajectory matching as a general construction for actions with multiple components or explicit prerequisites.

Action-based generation

  • GFlowNets learn distributions over action trajectories for reward-proportional sampling, while autoregressive models use fixed sequential orders and edit-based non-autoregressive models use restricted single-token operations.These approaches use different training objectives and generation paradigms.Source: §6 · pp. 8–9

Discrete flow matching

  • Discrete diffusion and DFM provide principled non-autoregressive generation, but their common template uses endpoint-conditioned, component-factorized dynamics for fixed-size states.ARDM implicitly conditions on a trajectory through random unmasking orders but remains limited to fixed-size, single-token operations.Source: §6 · p. 9

Variable-size generation

  • Prior methods that relax fixed-size constraints use auxiliary size prediction or augmented states, but do not support multi-component actions or general prerequisite relations.Trajectory matching addresses this gap and recovers factorized DFM and Edit Flow rates as special cases.Source: §6 · p. 9

7 LIMITATIONS

Trajectory matching depends on domain-specific choices of action space and trajectory distribution, and its tractability depends on efficient trajectory sampling and, ideally, closed-form Rao–Blackwellized rates. Evaluation is limited to trees and molecular graphs across four datasets.

7 LIMITATIONS

  • The framework requires choosing an action space and trajectory distribution, so performance depends on domain-specific design choices.It is most useful when actions encode meaningful structure; where independent component-wise edits suffice, complex actions may offer no advantage and the framework can fall back to factorized DFM.Source: §7 · p. 9
  • Efficient trajectory sampling and, ideally, closed-form Rao–Blackwellized rates determine tractability; outside independent fixed-action or forest-structured families, new domains may require non-trivial methodological work.Both operations can be done in linear time for many natural choices in those families.Source: §7 · p. 9
  • Evaluation covers trees and molecular graphs across four datasets; testing the framework in further domains remains future work.Source: §7 · p. 9

8 CONCLUSION

Trajectory matching conditions discrete flow matching on complete action trajectories, extending it to size-changing, multi-component, and dependent actions. The conclusion highlights empirical gains on trees and molecular graphs, while the supporting appendices establish consistency and variance properties and relate the framework to existing methods.

8 CONCLUSION

  • Trajectory matching conditions on full action trajectories and extends discrete flow matching to actions that change state size, modify multiple components, or depend on one another.The conclusion presents domain-aligned actions as yielding concrete gains across tree and molecular graph generation.Source: §8 · pp. 9–10

A EXTENDED RELATED WORK

  • Trajectory matching lies within Generator Matching, using the complete action trajectory as its conditioning variable, and constructs tractable conditional generators without fixed product-space or componentwise-action requirements.The contribution also includes efficient Rao–Blackwellized targets for several important trajectory families.Source: A EXTENDED RELATED WORK · p. 15
  • Trajectory matching recovers the Edit Flow probability path and marginal rate by uniformly ordering alignment-induced edits for a fixed alignment distribution.Edit Flows obtain tractable conditional dynamics by lifting variable-length sequences to an augmented alignment space.Source: A EXTENDED RELATED WORK · p. 15
  • Existing size-changing methods provide specialized domain-specific constructions rather than a general recipe for arbitrary multi-component actions and dependencies.The related-work discussion also notes that several fragment-based flow approaches reconstruct atom-level graphs through separate coarse-to-fine post-processing.Source: A EXTENDED RELATED WORK · p. 16
  • Standard factorized DFM corresponds to independent actions ordered uniformly at random, while trajectory matching makes the trajectory distribution explicit so generation order can encode domain knowledge and dependencies.Jointly learning that trajectory distribution is identified as a future direction.Source: A EXTENDED RELATED WORK · p. 16
  • FRAGGROW attaches whole fragments while retaining an atom-level molecular graph, rather than generating a fragment graph followed by coarse-to-fine reconstruction.The paper contrasts this with fragment-based flow and diffusion approaches that use a fixed fragment vocabulary and separate post-processing.Source: A EXTENDED RELATED WORK · p. 16

B.4 RAO–BLACKWELLIZATION: CONSISTENCY AND VARIANCE REDUCTION

  • Rao–Blackwellizing the training target preserves the expected gradient and optimum while reducing gradient variance.The Rao–Blackwellized estimator has the same expectation as the trajectory-conditioned estimator and no greater componentwise variance.Source: §B.4 · p. 18

B.5 PROOF OF PROP. 4.3

  • For independent fixed-action distributions, the Rao–Blackwellized rate assigns equal rate 1/(1 − t) to every remaining action.The result assumes the remaining action set is fixed across trajectories through the current state and their orderings are uniform.Source: §B.5 · p. 19

B.6 PROOF OF THM. 4.4

The appendix proves that trajectory matching recovers factorized flow matching in the independent-action setting and gives tractable Rao–Blackwellized rates for forest dependencies. It also shows that general time schedules extend the framework by reparameterizing time.

B.6 PROOF OF THM. 4.4

  • For corresponding independent fixed-action distributions, trajectory-matching and factorized-flow losses have the same gradient and minimizers, and their Rao–Blackwellized loss is identical.The proof establishes equality of the conditional paths and sampling distributions used by the losses.Source: §B.6 · p. 19

B.7 RAO-BLACKWELLIZED RATES FOR GENERAL DEPENDENT FIXED-ACTION DISTRIBUTIONS

  • For dependent fixed-action distributions, exact Rao–Blackwellized rates for general DAGs require linear-extension counts, which are computationally intractable in general; forest dependencies admit a closed form.For forests, the rate is assigned to currently available root actions in proportion to their descendant-subtree size.Source: §B.7 · p. 20

C GENERAL TIME SCHEDULES

  • A general schedule κ reparameterizes time, replacing the linear-schedule rate factor 1/(1 − t) with κ̇_t/(1 − κ_t) while preserving endpoints and the framework’s results.The schedule corresponds to action execution times drawn as order statistics from a distribution with CDF κ_t.Source: C GENERAL TIME SCHEDULES · p. 21

D TRAINING AND SAMPLING

The section describes training and sampling for trajectory matching across molecular, graph, and hierarchy settings, using action spaces tailored to each structure, from fixed-size edits to graph or hierarchy growth. It also covers prerequisite-aware ordering for hierarchy growth and feature-edit actions that remain editable during sampling.

D TRAINING AND SAMPLING

  • Training samples can return states and targets without materializing trajectories, with parallel CPU sampling adding below 1 ms of loader wait per QM9 training step.Targets can often be computed from the remaining action set or residual dependency DAG; compatible sampled actions are applied in parallel, with MOLEDIT resolving rare conflicts by deletions-first ordering and skipping unavailable actions.Source: D TRAINING AND SAMPLING · pp. 21–22

E ACTION SPACE AND TRAJECTORY DISTRIBUTION INSTANTIATIONS

  • Atom-index correspondence determines which atoms are kept, deleted, or added, while single-atom sources serve FRAGMENTGROW and ATOMGROW and marginal-graph sources serve BONDEDIT and MOLEDIT.The marginal-graph source samples atom count from the training-set size distribution and potential edges independently using average training edge density.Source: E ACTION SPACE AND TRAJECTORY… · p. 22

E.1 BONDEDIT

  • BONDEDIT holds the atom set fixed and inserts or deletes edges, using independent required actions and Rao-Blackwellized rates.Its action space mirrors DeFoG; the source is a marginal graph paired with a uniformly sampled training molecule of matching atom count.Source: §E.1 · p. 23

E.2 FRAGMENTGROW

  • FRAGMENTGROW uses a unique hierarchical action tree whose dependencies have at most one prerequisite, absorbing each ring system into one action.The fragment vocabulary covers 94% of QM9, 99% of ZINC, and 90% of MOSES training molecules; remaining molecules are skipped during training, but all standard validation and test molecules are evaluated.Source: §E.2 · pp. 23–24

E.3 ATOMGROW

  • ATOMGROW uses a latent rooted spanning tree to resolve alternative action decompositions, with atom insertions forming a tree and bond additions depending on both endpoint insertions.The latent tree is sampled as a random BFS tree rooted at the seed atom; required bond additions are sinks with no successors.Source: §E.3 · pp. 24–25
  • Multi-root ATOMGROW matches single-root performance on standard generation and enables zero-shot transfer to substructure linking.It uses a rooted spanning forest over multiple seed atoms; atom insertions follow tree dependencies, while bond additions depend on both endpoint insertions.Source: §E.3 · p. 25

E.4 MOLEDIT

  • MOLEDIT transforms a noisy molecular source using edge and node additions and deletions, with a latent forest defining insertion parents and a dependency DAG governing edits.Atom deletion requires prior deletion of incident bonds; atom insertions follow forest dependencies, and added bonds depend on insertion of any new endpoints.Source: §E.4 · pp. 25–26

E.5 GROWHIERARCHY

  • GROWHIERARCHY generates rooted trees by attaching children, a forest-structured dependent-action setting with one action type, no latent variable, and a closed-form Rao-Blackwellized rate.The source is a single-node tree at the target root, and intermediate states are root-containing subtrees.Source: §E.5 · p. 26
  • Hierarchy growth uses one action per non-root node, with each parent’s action prerequisite to actions producing its children.The resulting dependency DAG is the hierarchy tree, so the stated tree-structured condition for Proposition 4.7 applies.Source: §E.5 · p. 27
  • For hierarchy growth, the trajectory distribution is uniform over topological orderings of the tree’s fixed-action sequence.This is a dependent fixed-action distribution conditioned on the initial and final states.Source: §E.5 · p. 27

E.6 HANDLING NODE AND EDGE FEATURES

  • Feature-labeled graph actions can either predict features when inserting nodes or bonds, or insert random features and edit them with dedicated actions.Joint prediction expands the action vocabulary by the feature alphabet; separate feature edits also apply to fixed-atom settings such as BONDEDIT and can be added to the dependency structures.Source: §E.6 · p. 27
  • Separate feature-edit actions let the model revise earlier feature choices during sampling and performed best in most experiments.The paper contrasts this with joint feature prediction at insertion, which uses fewer actions but fixes features when inserted; the better choice depends on task and setting.Source: §E.6 · p. 27

E.7 EXACT CONDITIONAL PATHS FOR FOREST-PLUS-SINK DEPENDENCIES

For forest-plus-sink dependency DAGs, the construction samples forest actions independently and schedules multi-parent sink actions after their prerequisites. It yields tractable conditional rates for a valid path between the prescribed endpoints.

E.7 EXACT CONDITIONAL PATHS FOR FOREST-PLUS-SINK DEPENDENCIES

  • The construction partitions the dependency DAG into a forest and multi-parent sinks whose prerequisites lie in the forest.Required AddBond actions and Moledit DelAtom actions motivate these sinks.Source: §E.7 · p. 27
  • Forest actions receive ordered uniform firing times; each sink is scheduled uniformly after its latest prerequisite and before time 1.The resulting intermediate state applies all actions whose firing times are at most t.Source: §E.7 · pp. 27–28
  • The forest process retains its rates when sinks are observed, and the complete firing order is a topological ordering that transforms sI into sF.The forest rate is proportional to the descendant-subtree size; enabled sinks fire at rate 1/(1−t).Source: §E.7 · p. 28
  • Unlike uniform-topological paths, this construction avoids back-coupling sink timing to forest actions, while both paths respect dependencies and reach the same endpoints.Their difference is the intermediate probability path, a modeling choice in trajectory matching.Source: §E.7 · p. 28

F CONNECTION TO EXISTING METHODS

  • Factorized flows, including text unmasking and fixed-size graph generation, are trajectory-matching instances with independent fixed-action distributions.Figure 6 also presents Edit Flows and new instantiations including AtomGrow, FragGrow, MolEdit, BondEdit, and GrowHierarchy.Source: F CONNECTION TO EXISTING METHODS · p. 29, Figure 6 · p. 29

F.1 EDIT FLOWS AS TRAJECTORY MATCHING

Edit Flows can be represented as trajectory matching by conditioning on alignment-induced edits and uniformly ordering them. This construction recovers the same marginal path and rate, as well as the same loss optimum and gradients.

F.1 EDIT FLOWS AS TRAJECTORY MATCHING

  • An alignment turns sequence insertions, deletions, and substitutions into coordinate-specific edits, with blanks marking insertions and deletions.The alignment is treated as a latent variable in trajectory matching.Source: §F.1 · p. 29
  • Trajectory matching uniformly orders the alignment-induced edits, but an edit's observable action can depend on which earlier edits have been applied.Insertions and deletions can change positions in the observable sequence.Source: §F.1 · pp. 29–30
  • Edit Flows and uniform-order trajectory matching induce the same marginal probability path and rate, and their losses share the same optimum and gradients.The path equivalence holds for a shared time schedule satisfying the stated conditions.Source: §F.1 · pp. 30–31
  • The equivalence follows because both constructions give the same distribution over aligned edit sets at time t, with each Edit Flow edit applied independently with probability t.The observable state is a deterministic function of the applied edit set.Source: §F.1 · p. 30

G ADDITIONAL EXPERIMENTAL RESULTS

  • The appendix reports additional tree and molecular benchmark comparisons, compute-budget trade-offs, training-target and trajectory-distribution ablations, using the same setup except for the varied quantity.The stated additions are presented as complements to the main empirical results.Source: G ADDITIONAL EXPERIMENTAL RESULTS · p. 31

G.1 EXPERIMENTAL SETUP

Experiments hold the training and sampling pipeline and graph-transformer backbone fixed across trajectory-matching instantiations, varying action space, source distribution, and trajectory distribution. The section documents the datasets, objective and implementation choices, and computational trade-offs.

Model architecture

  • A shared graph-transformer backbone predicts rates for nodes and node pairs, with node count added as an input feature for variable-size states.The backbone is held fixed across instantiations; Tree uses a smaller model than the molecular datasets.Source: §G.1 · pp. 31–32

Loss decomposition

  • The decomposed objective preserves trajectory matching’s consistency guarantees: its population optimum recovers the same marginal rate and probability path as the original rate-matching objective.The loss splits prediction into a positive-rate probability and a rate magnitude; Tab. 4 reports similar performance for the decomposed and direct Bregman objectives on GROWHIERARCHY.Source: §G.1 · p. 32

Datasets

  • Tree, QM9, ZINC, and MOSES cover hierarchy generation and molecular generation, with MOSES described as the most challenging molecular benchmark for quality and coverage.FragGrow uses representable training examples, while validation and test splits remain unfiltered.Source: §G.1 · p. 33

Computational cost

  • Preprocessing loader wait stays below 1 ms per training step because CPU preprocessing overlaps GPU training, while end-to-end training and sampling throughput are approximately 10–14% and 22–25% lower than DeFoG.Structured-action inductive bias can offset per-step costs through smaller models, fewer training steps, or fewer sampling steps; Tree training takes less than 1 h versus approximately 8 h reported for DeFoG.Source: §G.1 · pp. 33–34, Table 5 · p. 33, Table 6 · p. 34

Training and sampling

  • Experiments use Euler integration with 1,000 steps on Tree and 500 on molecular datasets, and report means and standard deviations over five sampling seeds except MOSES, which uses one 25,000-sample run.Additional three-seed checks for ATOMGROW and FRAGGROW found small validity and FCD standard deviations.Source: §G.1 · p. 34

G.2 HIERARCHY GENERATION: EXTENDED RESULTS

GROWHIERARCHY preserves tree validity across sampling budgets, while methods that do not encode tree structure can underperform or collapse. Its distributional-fit Ratio is comparable to DeFoG across the tested step budgets.

Full baseline comparison

  • GROWHIERARCHY achieves 100% V.U.N., compared with the best autoregressive result of 95%, and a Ratio of 2.5, compared with 5.2.The expanded comparison also finds that methods without tree-structured actions underperform on V.U.N. or collapse; HSpectre is an exception but does not guarantee validity by construction.Source: §G.2 · p. 34, Table 7 · p. 35

Validity–compute trade-off

  • GROWHIERARCHY maintains 100% validity from 10 to 10^3 Euler steps, while DeFoG falls from approximately 96% at 10^3 steps to approximately 8% at 10 steps.The Ratio metric is similar for both methods across step sizes.Source: §G.2 · pp. 34–35, Figure 7 · p. 35

G.3 MOLECULAR GRAPH GENERATION: EXTENDED RESULTS

The extended experiments compare molecular-generation variants, training targets, trajectory policies, and downstream tasks. ATOMGROW and FRAGGROW show gains in the tested settings, and a single trained ATOMGROW model transfers to substructure extension and linking without retraining.

G.3 MOLECULAR GRAPH GENERATION: EXTENDED RESULTS

  • On MOSES, FRAGGROW achieves higher validity than GraphINVENT (99.2% versus 96.4%) and comparable FCD (1.21 versus 1.22).On ZINC, FRAGGROW has 78% versus 48% validity at 5 Euler steps, while both methods reach at least 98% from 25 steps onward.Source: §G.3 · p. 36, Table 9 · p. 36, Figure 8 · p. 36

G.4 TRAJECTORY-CONDITIONED VS. RAO-BLACKWELLIZED TARGETS

  • The Rao-Blackwellized target reaches at least 95% validity within the first few epochs, roughly twice as quickly as the trajectory-conditioned target, with less epoch-to-epoch noise and comparable final validity of approximately 97–99%.The comparison holds other hyperparameters fixed for ZINC under ATOMGROW.Source: §G.4 · p. 36, Figure 9 · p. 37

G.5 EFFECT OF THE TRAJECTORY DISTRIBUTION

  • ATOMGROW shows similar QM9 performance with random, breadth-first, and depth-first insertion policies, suggesting robustness to these tested trajectory distributions.The policies jointly vary insertion order and spanning tree while other listed settings remain fixed.Source: §G.5 · p. 37, Table 10 · p. 37

G.6 SUBSTRUCTURE EXTENSION AND LINKING

  • One ATOMGROW model trained on ZINC handles substructure extension and linking without downstream supervision or fine-tuning.The extension sets the initial state to a seed substructure; linking starts from the union of two disconnected seeds.Source: §G.6 · pp. 37–38
  • For substructure extension, ATOMGROW scores 0.86–1.00 across seed-and-start-time cells, versus BONDEDIT’s 0.14–0.70, including seeds absent from ZINC training molecules.Changing start time controls extension size: lower t permits larger, more decorated completions, while higher t keeps them closer to the seed.Source: §G.6 · p. 38, Figure 10 · p. 38, Figure 11 · p. 39
  • For substructure linking, ATOMGROW attains valid-and-connected fractions of 0.58–0.91 and outperforms BONDEDIT across all tested seed-pair and start-time cells.ATOMGROW starts from the union of both seeds, treated as roots, and generates the connecting path; sampled linkers vary in complexity with start time.Source: §G.6 · pp. 38–39, Figure 12 · p. 40, Figure 13 · p. 41

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