cs.LGOct 5, 2026

Conditional Flow Matching for Transport Between Markov Processes

Authors: Syamantak Kumar, Dheeraj Nagaraj, Saptarshi Roy, Purnamrita Sarkar

Organizations: The University of Texas at Austin, Department of Computer Science · Google Deepmind · Mohamed bin Zayed University of Artificial Intelligence, CMS Division · The University of Texas at Austin

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Abstract

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.

Figures & tables

Figure 1 : Each learned flow transports one source transition to one target transition. The index ii denotes time in the Markov process, while t∈[0,1]t\in[0,1] denotes flow time within a single transition.
Algorithm 1 Conditional flow matching for transport between Markov processes
Figure 2 : Synthetic transition recovery and EEG image retrieval. (a–b) Next-state distribution error for d=10d=10 (mean ±\pm standard error over three seeds; lower is better). The ablation omits xi{\mathbf{x}}_{i} ; marginal flow omits both context states; the oracle uses true kernels. (c) Relative Top-5 retrieval accuracy lift over ENIGMA with only Ws{\mathbf{W}}_{s} adapted, versus EEG context length kk ; both flow networks precede Ws{\mathbf{W}}_{s} and have two hidden layers: width 64 and width 14. Accuracies average three seeds per subject, then ten subjects; error bars show 95% subject-bootstrap percentile intervals.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

AdapterTop-1Top-5Top-10
Ws{\mathbf{W}}_{s} only17.7843.8757.40
CFM, width 6420.5549.0562.40
CFM, width 1418.6046.7359.82
Appendix
Table 1: Retrieval accuracy (%) at k=32k=32 , averaging seeds within subjects and then ten subjects. Both CFM networks precede Ws{\mathbf{W}}_{s} .
Figure 3 : Additional EEG comparison with both flow networks after Ws{\mathbf{W}}_{s} . Relative Top-5 retrieval accuracy lift over the matched Ws{\mathbf{W}}_{s} -only baseline is plotted against context length kk in transformed EEG coordinates. Accuracies average three seeds within each of ten subjects; error bars give 95% paired subject-bootstrap percentile intervals.
AdapterTop-1Top-5Top-10
Ws{\mathbf{W}}_{s} only17.7843.8757.40
CFM after Ws{\mathbf{W}}_{s} , width 6420.6548.9762.23
CFM after Ws{\mathbf{W}}_{s} , width 1419.3847.0560.32
Appendix
Table 2: Retrieval accuracy (%) at k=32k=32 with CFM after Ws{\mathbf{W}}_{s} , averaging three seeds within subjects and then ten subjects.

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