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Conditional Flow Matching for Transport Between Markov Processes

Syamantak Kumar, Dheeraj Nagaraj, Saptarshi Roy, Purnamrita Sarkar

arXiv:2610.07229v1cs.LGstat.ML

TL;DR

With few source and target trajectories, the paper learns a conditional-flow map that transports source Markov trajectories while preserving target transition structure. It proves population correctness and mixing-dependent finite-sample bounds, and reports improved EEG adaptation between subjects.

  • Problem

    The paper asks how to learn from limited, temporally dependent trajectories a map that turns a fresh source trajectory into one following target Markov dynamics.

    Source: §1 · p. 3

  • Method

    Conditional flow matching learns a reusable transition map conditioned on current source and generated target states, then applies it recursively to a fresh source trajectory.

    Source: §1 · pp. 3–4

  • Results

    The method gives exact target trajectory generation at the population level, finite-sample bounds with unavoidable mixing-time dependence, and improved adaptation between subjects in EEG experiments.

    Source: §1.1 · p. 4, §6 · p. 13

  • Takeaways & Limitations

    Under suitable assumptions, recursively applying the population transition map from a stationary target initialization yields the exact target trajectory law.

    Source: §1.1 · p. 4

    Open directions include extending analysis beyond finite classes and longer memory; sequential ODE solves can also make generation slow.

    Source: §6 · p. 13

Abstract

from arXiv · show

Motivated by sequence-to-sequence transport in the context time-series domain adaptation, we study the problem of transportation between trajectories of Markov processes. Given a limited number of trajectories from source distribution and the target distribution, we formulate a flow matching based algorithm which learns a transport map from the source to target trajectory distribution, while preserving the Markov structure. We show that this is consistent in the population limit and derive finite-sample error bounds under mixing time assumptions, following the analysis of classical statistical problems including regression (Nagaraj et al., 2020), principal component analysis (Kumar and Sarkar, 2023), and matrix concentration (Neeman et al., 2024) in the Markov setting. We complement that with a lower-bound construction showing that a mixing-time dependent sample complexity is unavoidable even with regular Gaussian conditional transitions. We evaluate on synthetic and real-world data. For image retrieval from electroencephalography (EEG) on THINGS-EEG2 (Gifford et al., 2022), the task is to identify the viewed image from EEG signals captured from human subjects, which suffers from high inter subject variability. We augment the ENIGMA decoder (Kneeland et al., 2026) with a conditional flow before its subject-specific temporal map. This improves mean top-5 retrieval accuracy from 43.87% to 49.05%, an 11.82% relative improvement.

1 Introduction

The paper studies transport from a source trajectory to one that follows target Markov dynamics, using shared transition structure to learn from limited trajectories. It proposes conditional flow matching and analyzes population correctness, finite-sample error, and empirical performance.

1 Introduction

  • The problem is to transform a fresh source trajectory into one following target dynamics while preserving Markov structure, despite having few trajectories whose observations remain temporally dependent.Under stationarity and time homogeneity, successive observations share a transition-pair distribution that can support learning a common transition rule.Source: §1 · p. 3
  • The method learns a recursively reusable transition map with conditional flow matching, using current source and generated target states to transport each next source state toward the target transition.It addresses limited trajectory availability while preserving the target transition law, unlike marginal matching, which can leave transition behavior different despite matching state frequencies.Source: §1 · pp. 3–4

1.1 Our results

  • The population flow maps each source transition to the target transition and, from a stationary target initialization independent of the fresh source, recursively generates the exact target trajectory law.The proof relies on suitable assumptions.Source: §1.1 · p. 4
  • For a finite candidate class containing the population velocity, the squared-velocity-error bound depends on the effective sample size m/K, up to logarithmic factors, where K is a mixing-time-based observation spacing.The bound is for two independent stationary trajectories; C depends on assumption parameters.Source: §1.1 · p. 4
  • Even with two candidate velocities and Gaussian conditional transitions, achieving small squared velocity error with high probability requires a number of transitions proportional to mixing time.This lower bound makes the mixing-time dependence unavoidable in that setting.Source: §1.1 · p. 4
  • For contractive target dynamics with coverage uniform over the horizon, the state-marginal error bound has a decaying initialization term and an estimation term bounded independently of trajectory length.A separate bound controls trajectory Wasserstein distance using the sum of squared state distances as transport cost.Source: §1.1 · p. 5
  • 49.05% versus 43.87% mean top-5 retrieval accuracy, an 11.82% relative improvement, when a conditional flow is added to the ENIGMA decoder for EEG-to-image retrieval.The comparison uses 4,096 calibration images per subject and a width-64 flow conditioned on 32 previous EEG samples.Source: §1.1 · p. 5

1.2 Related work

  • The paper distinguishes its learned continuous transition map from prior transport work based on cost-minimizing transition or trajectory couplings.Its analysis addresses dependent transitions with finite-sample bounds in terms of mixing time and a necessary mixing-dependent sample threshold.Source: §1.2 · p. 5

2 Preliminaries and notation

The preliminaries establish notation for distributions, norms, and transport distances, then introduce flow matching and Markov mixing concepts used in the analysis.

Notation

  • The notation specifies Euclidean and Frobenius norms, probability laws, pushforward distributions, and total-variation and quadratic Wasserstein distances.Time index i denotes Markov-process time, while t denotes flow time within one transition.Source: §2 · p. 5

Markov processes and mixing

  • A Markov kernel gives the next-state law conditional on the current state; stationarity means the marginal law is unchanged by a transition, while time homogeneity makes the kernel independent of time index.In a stationary process initialized from its stationary distribution, each state has that same marginal distribution.Source: §2 · p. 6
  • The mixing coefficient measures average total-variation distance from stationarity after a lag, and the mixing time is the first lag below a chosen tolerance.The average mixing time is no greater than the standard worst-case total-variation mixing time.Source: §2 · p. 6

Flow matching

  • Flow matching linearly interpolates independent endpoint samples and uses the conditional expected endpoint displacement as its velocity field.Under the stated regularity, its ODE map transports the initial distribution to the target distribution.Source: §2 · p. 6

3 Problem and model

The model observes one source and one target training trajectory, then learns maps to transform a fresh source trajectory recursively. Its objective is to match target transitions conditional on current states, under separate flow-regularity and finite-sample assumptions.

Model

  • The setup uses independent, time-homogeneous Markov processes, one observed trajectory from each, and a fresh source trajectory independent of the training data for generation.The fresh source trajectory may have an arbitrary initial distribution.Source: §3 · p. 6

Stationary training trajectories

  • Finite-sample analysis assumes independently initialized stationary training trajectories whose mixing profiles tend to zero; the source and target need not share a mixing profile or time.The generation objective specifies the conditional law of each output transition.Source: §3 · pp. 6–7

Transport problem

  • The objective is a single family of maps applied recursively so each generated next state has the target transition law conditional on the current source and target states.Given current states, the next source and target states have their respective transition-kernel laws and are conditionally independent.Source: §3 · p. 7

Flow regularity

  • The population and learned velocities must satisfy flow-regularity conditions that ensure unique ODE solutions; exact stationary generation initializes the target state from πY independently of the fresh source.Finite-sample bounds retain error from approximate initialization.Source: §3 · p. 7

4 Main results

The section establishes that population conditional flow maps source transitions to target transitions and can be applied recursively to generate target trajectories. It bounds velocity estimation from dependent transitions and shows that the sample requirement depends on mixing time.

4 Main results

  • Under suitable stability assumptions, the section also bounds error in generated states and trajectories.The main-results overview notes this generation-error analysis after the velocity-estimation bounds.Source: §4 · p. 8

4.1 From one transition to a trajectory

  • The source Markov property keeps the next source state distributed according to its transition kernel even after conditioning on the shared generation history.This allows the transition map to be applied recursively while preserving the target transition kernel conditional on the full history.Source: §4.1 · p. 8
  • With a stationary target initialization and valid transition maps at visited state pairs, recursive generation has the stationary target trajectory law for every horizon.The population ODE maps each source transition to the target transition; the source Markov property preserves this identity under conditioning on shared history.Source: §4.1 · pp. 8–9

4.2 Velocity estimation from two dependent trajectories

  • The velocity-estimation risk scales as K/m up to class-complexity and logarithmic factors, where K is set by the transition-observation mixing time.The guarantee assumes realizability and conditional noise and moment bounds; it also holds with a fixed number of independent flow-time samples per transition.Source: §4.2 · p. 10
  • A lower-bound construction shows slowly mixing chains can avoid regions where candidate velocities differ, making mixing-dependent sample complexity necessary.Under geometric mixing, the upper and lower bounds agree in their mixing-time dependence up to logarithmic factors.Source: §4.2 · p. 11

5 Experiments

The experiments evaluate transition recovery on Gaussian and nonlinear processes and EEG-to-image retrieval. The proposed flow improves over the ENIGMA baseline on EEG retrieval and outperforms the stated transition-recovery baselines at the largest reported training size.

Gaussian transitions

  • At 32,768 transitions, Gaussian next-state error is 0.133 for the proposed flow, versus 0.725 for the ablation and 0.827 for marginal flow.The experiment uses d = 10 Gaussian transitions and reports lower error as better.Source: §5 · p. 11, Figure 2 · p. 12

Nonlinear mixture transitions

  • At 32,768 transitions, nonlinear next-state error is 0.312 for the proposed flow, versus 0.675 for the ablation, 0.852 for marginal flow, and 0.131 for the oracle.The ablation performs better at the smallest sample sizes, while the proposed flow’s gains taper at larger sizes under the fixed training budget.Source: §5 · p. 12, Figure 2 · p. 12

Supervised EEG adaptation

  • For THINGS-EEG2’s 200-image decoding task, the flow transforms EEG recordings using preceding samples before decoding, and inference requires only the source recording.The evaluation freezes the encoder and projection network pretrained on other subjects and uses 4,096 calibration images per subject.Source: §5 · pp. 12–13
  • 49.05% versus 43.87% mean Top-5 retrieval accuracy: the width-64 flow gives an 11.82% relative lift over ENIGMA at 32-sample context.Figure 2c reports the comparison; both widths improve on the baseline for all ten subjects after averaging seeds.Source: Figure 2 · p. 12, §5 · p. 13

6 Conclusion

Conditional flow matching yields exact population-level transport between Markov trajectory distributions, with finite-sample guarantees tied to mixing time. The paper also identifies open theoretical and computational directions.

6 Conclusion

  • Conditional flow matching transforms a fresh source trajectory into the exact target trajectory distribution at the population level.For finite candidate classes, the sample-complexity upper bound scales with mixing time up to logarithmic factors, and a lower bound shows this dependence is unavoidable.Source: §6 · p. 13
  • Synthetic experiments show the value of conditioning on the previous source state, while the EEG study shows improved adaptation between subjects.Source: §6 · p. 13
  • Extending the analysis beyond finite classes, linking observed-transition accuracy to generated-trajectory accuracy, handling longer memory, and reducing sequential ODE-solve latency remain open directions.Source: §6 · p. 13

Ethics statement

The EEG experiments reuse THINGS-EEG2 and evaluate retrieval in a controlled setting with a fixed candidate-image set. Use beyond that setting requires separate validation and safeguards.

Ethics statement

  • The study reuses publicly released THINGS-EEG2 data and collects no new human-subject data.Source: Ethics statement · p. 14
  • The EEG retrieval evaluation ranks a fixed set of candidate images in a controlled experimental setting.Source: Ethics statement · p. 14
  • Use beyond the controlled evaluation setting requires separate validation and safeguards for participant consent and neural-data privacy.Source: Ethics statement · p. 14

AI use statement

The appendix develops the population transport and dependent-transition estimation arguments, and records the authors’ use and review of generative AI tools. Its finite-class analysis relies on mixing and loss assumptions, while optimization error remains uncertified for AdamW.

AI use statement

  • The authors used generative AI for proof exploration, writing, experiments, literature searches, code, and figures, then reviewed its contributions and retained responsibility for the final work.Source: AI use statement · p. 14

Appendix contents

  • The appendices cover transport proofs, dependent velocity estimation, lower bounds, Gaussian transitions, experiments, EEG study details, and prior-work comparisons.Source: Appendix contents · p. 19

A.1 Conditional transport and trajectory generation

  • The population flow transports a source transition to the target transition, and recursive generation from the target stationary law yields the stationary target trajectory distribution.Source: §A.1 · p. 20
  • Conditioning on generated history retains the source transition law; applying the transport identity then gives the target transition.Source: §A.1 · p. 20

A.2 Dependent velocity estimation

  • The paired transition observations form a stationary, time-homogeneous Markov process, whose mixing is bounded using the source and target chains’ mixing behavior.Source: §A.2 · pp. 21–22
  • Residue-class coupling produces independent observations within each class while preserving the joint law of the original data; different classes may remain dependent.Source: §A.2 · p. 22
  • For finite candidate classes, empirical excess-loss minimization controls velocity risk with high probability under the stated loss and mixing conditions.Source: §A.2 · pp. 24–26
  • Randomly sampled flow times retain the integrated-risk guarantee for each fixed sample count, but the analysis does not certify AdamW’s optimization error.Source: §A.2 · pp. 25–26

A.3 The sample-size lower bound

The lower-bound construction shows that learning transition maps from slowly mixing trajectories requires mixing-time-dependent data, even for regular Gaussian transitions. The section also compares mixing-time notions and develops conditions and bounds for translating learned velocity accuracy into rollout accuracy.

A.3 The sample-size lower bound

  • The Gaussian lower-bound construction makes the models observationally indistinguishable on an event where a slowly mixing coordinate avoids the region separating their velocities.The candidates satisfy the flow and moment assumptions, and their observation mixing time is proportional to τ.Source: §A.3 · pp. 26–27
  • For either θ, the stated trajectory-length requirement is m ≥ τ log(1/(2πδ)) ≥ (1/3)τmix,θ(1/4) log(1/(2πδ)), with the mixing-time term evaluated at 1/4.The candidates satisfy the flow and moment assumptions and have risk separation ϵ.Source: §A.3 · pp. 26–27

A.4 Comparison of mixing tolerances

  • The Gaussian lower-bound models have infinite worst-case mixing time despite their exponential stationary-average mixing profile.Under the stronger finite worst-case mixing assumption, the estimation theorem can instead use a worst-case mixing-time spacing.Source: §A.4 · p. 28

A.5 Accuracy of generated states and trajectories

  • The finite-sample rollout guarantee requires concentrability to transfer training accuracy to generated state pairs and target contraction to limit accumulated error.The bound separates decaying initialization error from accumulated estimation error, amplified by at most 1/(1 − ρ); verifying concentrability for broader model classes remains open.Source: §A.5 · pp. 28, 29, 31
  • A learned transition map’s one-step Wasserstein error is controlled by conditional velocity risk, and target-kernel contraction propagates that error across rollout steps.The one-step result uses flow regularity and a learned-velocity Lipschitz bound; the recursion also requires a target-kernel Lipschitz bound.Source: §A.5 · pp. 29–31

B Gaussian transitions, estimation, and rollout

This section specializes conditional flow matching to Gaussian transition models. It gives a Gaussian flow formula, applies the estimation theorem to a finite model class, and bounds rollout error by coupling trajectories.

B Gaussian transitions, estimation, and rollout

  • For Gaussian transitions with identity covariance and means a and b, the conditional flow lemma provides an explicit velocity and ODE flow.The interpolation variance is st = (1 − t)^2 + t^2.Source: B Gaussian transitions, estimation,… · p. 31
  • In the scalar Gaussian model, the increasing population flow transports the source law to the target law and therefore agrees with the quantile map.The example applies the estimation theorem to independent stationary Gaussian processes with parameters in a known finite set.Source: B Gaussian transitions, estimation,… · pp. 32–33
  • The Gaussian rollout bound follows by coupling fitted and target recursions with shared noise, which cancels when their updates are subtracted.The comparison trajectory has the stationary target law, and the initial discrepancy contributes through the initial coupling cost.Source: B Gaussian transitions, estimation,… · pp. 33–34

C Examples for the conditional noise assumption

The section gives examples satisfying the conditional noise assumption and describes synthetic transition benchmarks used to evaluate conditional flow matching. These include Gaussian transitions and a nonlinear mixture benchmark.

C Examples for the conditional noise assumption

  • Bounded displacements satisfy the conditional noise assumption with ν = B, while bounded additive noise can suffice even when states and transition means are unbounded.If both processes are supported in a ball of radius B, ν = 2B suffices; bounded additive-noise differences likewise give ν = 2B.Source: C Examples for the conditional noise… · pp. 34–35
  • For Gaussian transition kernels with covariance bounded above by σ²I, conditioning on the full input yields a centered Gaussian residual with a variance bound.This supplies a conditional noise bound even when the conditional means are arbitrary.Source: C Examples for the conditional noise… · p. 35

D.1 Gaussian transition experiments

  • The Gaussian benchmark uses linear Gaussian source and target transitions with independently randomized eigenbases and noise covariances.The main panels use dimension 10, while dimension 100 is used as a stress test.Source: D Experimental details · p. 35, §D.1 · p. 36

D Experimental details

  • The experiments compare conditional-transition recovery at supplied current-state pairs against an oracle and learned baselines using fresh evaluation samples.The reported metric uses projected quantiles and does not require an independently sampled target reference.Source: D Experimental details · pp. 35–36

D.2 Nonlinear mixture benchmark

  • The nonlinear benchmark uses rotated coordinatewise nonlinear transition means and a two-mode target mixture, with an oracle quantile map along a designated direction.The orthogonal component supplies independent Gaussian noise, yielding the target transition distribution.Source: §D.2 · pp. 36–37

D.3 Results at dimension 100

At dimension 100, the full conditional flow outperforms both learned baselines on both benchmarks at the reported sample size, though errors remain above the oracle. The EEG protocol evaluates subject adaptation, with a flow placed before ENIGMA’s subject-specific temporal map.

D.3 Results at dimension 100

  • At m = 8192, the full conditional flow has lower mean error than both learned baselines in both dimension-100 benchmarks, but remains above the oracle.Its improvement over the source-state ablation is smaller for nonlinear transitions, and that ablation performs better at some smaller sample sizes under the fixed training budget.Source: §D.3 · p. 37

E.1 Protocol

  • The EEG protocol evaluates all ten THINGS-EEG2 subjects using matched splits and seeds, with retrieval averaged over 80 repetitions per image in a 200-image gallery.The adaptation uses 63-channel recordings and differs from ENIGMA’s published 32-channel joint-training experiment.Source: §E.1 · p. 37

E.2 Adapter configurations

  • The EEG flow precedes the subject-specific temporal map, and sequential generation uses only previously generated target samples rather than reference recordings.The paper does not assume the EEG data satisfy its stationary Markov or mixing hypotheses.Source: §E.2 · pp. 38–39
  • Past-state conditioning uses k ∈ {0, 1, 4, 16, 32} previous samples, with a fixed input dimension and one shared velocity network across EEG samples.The source and reference histories are inputs; the current source endpoint forms the interpolation but is not separately supplied to the velocity.Source: §E.2 · p. 38

E.3 Additional results

The additional EEG results show retrieval gains from conditional flow matching across placements and context lengths. The discussion distinguishes recursive transport of a prescribed source trajectory from approaches that learn couplings or whole-sequence dynamics.

E.3 Additional results

  • At k = 32, the width-64 model reaches 49.05% mean Top-5 accuracy versus 43.87% for baseline, an 11.82% relative lift.Both widths improve over baseline for all ten subjects, and longer history improves retrieval without increasing model size within either sweep.Source: §E.3 · p. 39, Table 1 · p. 39

E.4 Flow after the temporal map

  • After Ws, the flow acts on temporally transformed EEG endpoints, and context length counts preceding samples in that transformed sequence.Because the temporal maps mix all 250 raw time samples, this history need not correspond to preceding measurements in raw EEG time.Source: §E.4 · pp. 39–40
  • At k = 32, width 64 after Ws achieves 48.97% Top-5 accuracy, a 11.63% relative lift over the 43.87% baseline.Relative lift rises from 3.91% at k = 0 to 11.63% at k = 32 for width 64, and from 1.44% to 7.26% for width 14.Source: §E.4 · p. 40, Figure 3 · p. 40, Table 2 · p. 40

F.1 Sequential transport and generation

  • Unlike methods that generate whole functions or learn endpoint-law dynamics, this approach recursively transforms a prescribed fresh source trajectory with a conditional transition map.The discussion contrasts it with causal transport used as a training objective and with cost-based couplings for Markov chains.Source: §F.1 · p. 41
  • The statistical bound controls squared prediction error of a continuous conditional velocity, which is converted into generated-state accuracy.The discussion distinguishes this target from cost bounds for estimated stationary couplings of finite-valued processes.Source: §F.1 · p. 41

F.2 Statistical error and dependence

The analysis combines estimation from dependent transitions with conditional Wasserstein bounds and recursive generation. Its rollout guarantees rely on a concentrability condition and a conditional coupling that retains prescribed source transitions.

F.2 Statistical error and dependence

  • The estimation proof combines dependent-transition velocity estimation with conditional Wasserstein bounds and recursive generation.It uses a coupling and fourth-moment approach for dependent square-loss regression.Source: §F.2 · p. 41
  • The lower bound establishes a mixing-dependent minimum sample threshold, even though some predictor-class bounds need not place mixing time in their leading square-loss term.Dependence can still enter the minimum sample size and higher-order terms.Source: §F.2 · p. 41

F.3 Distribution mismatch and rollout stability

  • The concentrability condition compares conditional velocity error under source-generated state pairs with training-law error, with conditional risk playing the squared-residual role.This relates the rollout error to a mismatch between the generated-pair law and the training law.Source: §F.3 · p. 41
  • Proposition 5 uses a conditional W2 coupling that retains prescribed source transitions while controlling both state and joint-trajectory error.This extends error control beyond one-transition discrepancies propagated through contracting dynamics.Source: §F.3 · p. 42
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