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Flow Matching from Viewpoint of Proximal Operators

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

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Reformulates OT-CFM via an Aleksandrov–Brenier potential so the inverse map from x_t to x_1 is an exact Euclidean proximal operator (no density assumed), proves minibatch convergence, and shows terminal normal hyperbolicity on manifold-supported targets.

Problem

Analyzing the vector field and terminal dynamics of OT-based conditional flow matching when the target distribution has no density and concentrates on a low-dimensional manifold.

Approach

Express the OT-CFM interpolation inverse as the proximal operator of an Aleksandrov–Brenier convex potential, giving an exact proximal formula for the FM vector field; analyze terminal Lyapunov exponents via second epi-derivatives of the potential.

Key results

• On toy circle target, Jacobian eigenvalues of the trained OT-CFM vector field approach lambda_1=0 (tangential) and lambda_2=-gamma=-1.0 (normal) as t->1, matching the theory.
• On MNIST with minibatch size 256, about 100 eigenvalues of (1-t)D_x v are significantly larger than the rest, indicating a tangent/normal spectral gap.
• Theory: normal directions have Lyapunov exponent -gamma and tangent directions have zero exponent under the stated assumptions.

Surprising findings

• The Two Moons example, which is a manifold with boundary, falls outside the theoretical assumptions yet still shows the predicted spectrum.

Limitations

• Main analysis uses population OT, whereas practice uses minibatch OT; convergence is only along subsequences and needs additional assumptions for ODE/pushforward convergence.
• Lyapunov theorem requires strong assumptions (A1)-(A3) and twice epi-differentiability, and does not cover manifolds with boundary.
• Experiments are small-scale toy examples and MNIST only, with no quantitative generation metrics.

Experimental setup

Datasets: MNIST two moons (toy) circle (toy)

Benchmarks: MNIST two moons circle

Metrics: Jacobian eigenvalues visual inspection of generated samples

Baselines: Optimal Flow Matching

Builds on: conditional flow matching optimal transport conditional flow matching Brenier theorem Tweedie's formula

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