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Metric Flow Matching for Smooth Interpolations on the Data Manifold
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Proposes Metric Flow Matching (MFM), a simulation-free conditional flow matching framework whose interpolants are learned approximate geodesics of a data-dependent Riemannian metric (LAND/RBF), and applies it to trajectory inference, LiDAR navigation, unpaired image translation, and single-cell dynamics.
Problem
Conditional flow matching uses straight-line interpolants under Euclidean geometry, which can leave the data manifold and fail to model nonlinear dynamics in trajectory inference (e.g., single-cell data).
Approach
Learn interpolants x_{t,eta} = (1-t)x0 + t x1 + t(1-t)phi(x0,x1) by minimizing the kinetic energy under a task-agnostic, data-dependent diagonal Riemannian metric (LAND or RBF), so that the approximate geodesics stay near the data; then regress a vector field with the standard CFM objective, without simulating geodesics.
Key results
- On 5-dim PCA single-cell data, OT-LAND reaches W1 0.724 on Cite, 0.713 on EB, and 0.890 on Multi, beating OT-CFM (0.882, 0.790, 0.937) and the other listed baselines.
- On 100-dim PCA single-cell data, OT-RBF reaches 41.784 on Cite versus 45.393 for OT-CFM and 43.731 for WLF-UOT, and 50.906 on Multi versus 54.814 for OT-CFM.
- MFM improves over the Euclidean OT-CFM on synthetic Arch and sphere experiments and on LiDAR navigation, where straight paths leave the surface.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use OT-LAND instead of OT-CFM | Single-cell trajectory inference · Cite (5-dim PCA) | Wasserstein-1 distance: 0.882 ± 0.058 -> 0.724 ± 0.070 | Helped |
| Use OT-LAND instead of OT-CFM | Single-cell trajectory inference · EB (5-dim PCA) | Wasserstein-1 distance: 0.790 ± 0.068 -> 0.713 ± 0.039 | Helped |
| Use OT-LAND instead of OT-CFM | Single-cell trajectory inference · Multi (5-dim PCA) | Wasserstein-1 distance: 0.937 ± 0.054 -> 0.890 ± 0.123 | Helped |
| Use OT-RBF instead of OT-CFM | Single-cell trajectory inference · Cite (100D) | Wasserstein-1 distance: 45.393 ± 0.416 -> 41.784 ± 1.020 | Helped |
| Use OT-RBF instead of OT-CFM | Single-cell trajectory inference · Multi (100D) | Wasserstein-1 distance: 54.814 ± 5.858 -> 50.906 ± 4.627 | Helped |
Limitations
- Task-specific constraints are not encoded into the metric; this is left for future work.
- Relies on Optimal Transport couplings, which carry standard scalability limits for high-dimensional or large datasets.
- Requires the data to be embedded in Euclidean space for interpolants to be defined.
Experimental setup
Datasets: Arch Sphere LiDAR Mt. Rainier AFHQ Embryoid body CITE-seq Multiome
Benchmarks: Arch Sphere LiDAR AFHQ EB CITE-seq Multiome
Metrics: Wasserstein-1 distance EMD FID LPIPS
Baselines: OT-CFM I-CFM SF2M-Geo SF2M-Exact WLF-UOT DSB TrajectoryNet Reg. CNF
Builds on: Conditional Flow Matching Riemannian Flow Matching LAND metric GSBM OT-CFM
Closest papers by meaning
- Compatible Likelihoods for Flow Matching on Manifolds (NeurIPS 2026 · similarity 0.90)
- Categorical Flow Matching on Statistical Manifolds (NeurIPS 2024 · similarity 0.90)
- Beckmann Transport Models: From Autonomous Flows to One-Step Maps (NeurIPS 2026 · similarity 0.89)
- Flow Matching on General Geometries (ICLR 2024 · similarity 0.89)
- Optimal Flow Matching: Learning Straight Trajectories in Just One Step (NeurIPS 2024 · similarity 0.88)
- Rectifying Categorical Flows on Statistical Manifolds for One-Step Generation (NeurIPS 2026 · similarity 0.88)
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