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Tessellations of Semi-Discrete Flow Matching

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling · 0 citations (OpenAlex)

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Analyzes the exact closed-form velocity field of semi-discrete Flow Matching (Gaussian source, finitely many target atoms), proving that terminal assignment cells are open, simply connected (and homeomorphic to a ball under an assumption), and differ geometrically from semi-discrete OT Laguerre cells.

Problem

What geometry does the exact Flow Matching velocity field induce on the terminal assignment regions when transporting a Gaussian to a finitely supported target, and how does it compare to semi-discrete optimal transport Laguerre cells?

Approach

Uses the closed-form semi-discrete FM velocity (softmax-weighted drifts toward atoms) to study the preimages of atoms under the terminal flow map, proving topological properties via late-time capture and a planar four-point counterexample showing non-convex, curved, unbounded cells unlike Laguerre cells.

Key results

• Proves each terminal assignment cell Gamma_k is open and simply connected for the semi-discrete FM flow.
• Under the assumption that the limit of gamma_t^{-1}(a_k) lies in the interior of the cell, Gamma_k is contractible and homeomorphic to the unit ball.
• In a four-point planar example, the FM cell Gamma_4 is unbounded and non-convex while the corresponding Laguerre cell is bounded; the flow map is shown to be non-monotone.

Surprising findings

• The flow map is non-monotone in the four-point example, violating the monotonicity condition used as a sufficient criterion for straight second reflow.
• A point may fail to belong to its own assignment cell, as with Laguerre cells.

Limitations

• Results concern the exact semi-discrete FM field, not trained neural velocity fields.
• Homeomorphism to the ball requires an unproven assumption that the center lies in the cell interior (conjectured true numerically).
• Analysis restricted to equal-weight distinct atoms; general weights and mean-field limits left open.

Experimental setup

Datasets: MNIST

Metrics: topological properties of cells monotonicity (min eigenvalue of symmetrized Jacobian)

Baselines: semi-discrete optimal transport Laguerre cells

Builds on: Flow Matching rectified flow semi-discrete optimal transport reflow

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