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
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 i denotes time in the Markov process, while t∈[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=10 (mean ± standard error over three seeds; lower is better). The ablation omits xi ; marginal flow omits both context states; the oracle uses true kernels. (c) Relative Top-5 retrieval accuracy lift over ENIGMA with only Ws adapted, versus EEG context length k ; both flow networks precede Ws 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
Adapter
Top-1
Top-5
Top-10
Ws only
17.78
43.87
57.40
CFM, width 64
20.55
49.05
62.40
CFM, width 14
18.60
46.73
59.82
Appendix
Table 1: Retrieval accuracy (%) at k=32 , averaging seeds within subjects and then ten subjects. Both CFM networks precede Ws .
Figure 3 : Additional EEG comparison with both flow networks after Ws . Relative Top-5 retrieval accuracy lift over the matched Ws -only baseline is plotted against context length k in transformed EEG coordinates. Accuracies average three seeds within each of ten subjects; error bars give 95% paired subject-bootstrap percentile intervals.
Adapter
Top-1
Top-5
Top-10
Ws only
17.78
43.87
57.40
CFM after Ws , width 64
20.65
48.97
62.23
CFM after Ws , width 14
19.38
47.05
60.32
Appendix
Table 2: Retrieval accuracy (%) at k=32 with CFM after Ws , averaging three seeds within subjects and then ten subjects.
Functional flow matching is posed on distributions of functions but implemented from finitely many coefficients or point values. Under scattered or adaptive refinement, the resulting conditioning sigma-algebras need not be nested, so martingale convergence does not justify the sensor limit. We prove strong L2 convergence of finite conditional velocity targets for every strongly consistent sequence of finite-rank reconstructions, with quantitative bounds for orthogonal projections and a point-sensor extension through a regularity space. For learned flows, coupling directly to a population superposition path yields an end-to-end Wasserstein bound without assuming uniqueness of the population finite-dimensional ODE. We verify sensor-independent constants for a normalized quadrature neural operator, including globally Lipschitz activations through an explicit magnitude recurrence. A noncommuting trace-class Gaussian example gives boundary multiplier 0 under projected restriction and 0.72 under exact conditioning. A spatial regularity--cubature certificate closes the operator-realization term, a Bernstein argument gives a O(n−1) excess-risk term for fixed model dimension and envelopes, and an exactly realizable clipped Gaussian scaling specialization yields an explicit end-to-end rate.
Lennon J. Shikhman
School of Computer Science, College of Computing Georgia Institute of Technology Atlanta, Georgia 30332, USA
We propose an instantiation of flow matching that relies on a time-independent velocity field (an \emph{autonomous flow}) to exactly map between two distributions, so long as the target is singular, i.e.\ supported on a lower-dimensional data manifold. We also show that the one-step generative map associated with this flow is the unique solution of a simple conservation equation, which can be used to learn the map directly from samples. These autonomous flows and maps give a dynamical meaning to the flux constraint of Beckmann's transportation problem. Their construction provides a unifying framework that recovers, for instance, the closed-form Poisson-flow generative model and equilibrium matching with a quadratic flow-matching regression loss. We illustrate how this theory corrects inconsistencies in existing methods and demonstrate the effectiveness of the autonomous flow and the one-step map on ImageNet 256x256.
Lee Cheuk-Kit, Florentin Coeurdoux, Yuyuan Chen +5
1Harvard University · 2Capital Fund Management · University of Pennsylvania +2
Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring the underlying trajectories, can be solved by multi-marginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by outperforming the existing baselines on spatial transcriptomics and cell tracking datasets, while performing on par with them on single-cell trajectory prediction. Code: https://github.com/mmacosha/adversarially-learned-interpolants.
Oskar Kviman, Kirill Tamogashev, Nicola Branchini +3