arXiv paper re-derives Flow Matching from a Lagrangian view, explains straight trajectories
burny_tech · x · 2026-09-03
Peyman Milanfar's arXiv paper (2609.00198) re-derives Flow Matching and Rectified Flow bottom-up from a Lagrangian (particle-centric) perspective, replacing the standard optimal-transport Eulerian derivation.
- A local Taylor expansion of the continuous denoiser motivates a strict invariance condition for optimal single-step generation: conservation of target identity.
- Enforcing it yields a quasi-linear advection PDE whose solution via the Method of Characteristics analytically recovers straight-line trajectories.
- The Jacobian of the denoiser is isolated as the primary source of trajectory curvature, explaining why straight-line flows permit massive step sizes and why distillation is needed to flatten intersecting characteristics.
Related event: Google researcher reframes Flow Matching through a Lagrangian lens(3 posts)→
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