Flow Matching
Stub. Designing a generative model directly as a probability path: conditional and marginal probability paths, designing a target vector field, the continuity equation that ties a vector field to its induced density path, and conditional flow matching (CFM) as a tractable surrogate loss for learning it — including the special (Gaussian) case where the score comes for free.
Generalizes the diffusion picture: instead of deriving a generative model from a fixed noising process, directly design a probability path from the data distribution to a simple prior (or between any two distributions) and a vector field that transports it, connected by the continuity equation (the ODE analogue of the Fokker-Planck equation). Regressing onto the true marginal vector field is intractable, but conditional flow matching (CFM) regresses onto a per-sample conditional vector field instead, and is provably a valid surrogate — this is the “ODE to SDE to score matching” arc that shows CFM, score matching, and diffusion training are all facets of the same idea, with the Gaussian conditional path recovering diffusion models’ score for free.
Outline (TODO — flesh out each)
- Conditional and marginal probability paths
- Designing a target vector field
- The continuity equation
- Zoom out: visualizing flows
- Score functions, revisited from the flow perspective
- Conditional flow matching (CFM)
- From conditional to marginal
- CFM as a surrogate loss for flow matching (FM)
- Score matching as a special case; the Gaussian case (score for free)