Conditional Flow Matching

A categorical account of flow matching and its conditional training, spanning the flow primitive that transports distributions and the mixture construction that carries data outside a parametric family.

Flow matching carries a reference law (“noise”) to a data law along a path of distributions; conditional flow matching trains it by carrying each sample, inside a family , to the concentration boundary, then regressing on the resulting mixture. This note collects the flow primitive that does the transport and the conditional construction that reaches data.

The flow primitive

A flow on a state space (a smooth manifold) is a family with and each (an affine transformation of the connection; an isometry in the metric case), smooth in . Its lift to distributions is the pushforward

so a reference law gives both a point trajectory and a distributional trajectory , with . The path is in general not a convex combination of its endpoints. That is the separate interpolation. By naturality, with the Dirac unit of the probability monad, , so is the unique functorial lift agreeing with on Dirac masses.

The generating velocity

Differentiating and integrating by parts gives the continuity equation , which defines the velocity .

Proposition. Determinacy of the velocity

The continuity equation fixes only up to divergence-free fields (); a Riemannian metric selects the unique minimal-norm gradient representative (Otto); and on a finite-dimensional family the intrinsic tangent is unique, , the residual ambiguity being .

Reaching data outside

The closure results live inside a family ; a dataset lies outside it. Conditional flow matching bridges the gap. Each sample is carried inside to the concentration boundary, and the trained marginal is their mixture, outside .

The concentration boundary of is its set of Dirac limits, the lying in the weak closure of : as ; von Mises–Fisher laws concentrate as ; gamma laws of fixed mean as shape ; the categorical vertices are the one-hot laws (members already). The boundary lies at infinite Fisher–Rao distance from the interior, so a terminal floor (, finite ) keeps energy and Lipschitz integrals finite.

A conditional path for is a curve , on , with a common source and ; the marginal deforms from to and generically leaves .

The training identity

A path is trainable if a computable objective has the same minimiser as the continuity-equation velocity. Conditional flow matching supplies one:

the mixture is generated by the posterior mean , and for any strictly convex with Bregman divergence ,

so regressing on the conditional velocity has the marginal velocity as its minimiser.

The proof is by conditioning. With the cross term vanishes, leaving .

The per-sample paths rely on flow closure. The flow acts affinely on the sufficient statistic, so a transitive group (Gaussian/affine, Wishart/congruence) carries any parameter to any other, which makes the conditional velocity a flow generator in the explicit-target case.

Where the target is explicit. Two sources are pointwise evaluable: the flow generator when the structure-preserving group is transitive on the parameters (Gaussian, Wishart), and the one-dimensional quantile formula (gamma). Otherwise the target is fitted (two-stage), the velocity still existing as the gradient representative. The prevalence of the Gaussian case is structural, since its group is transitive over a vector-space sample space.

References

• Y. Lipman, R. T. Q. Chen, H. Ben-Hamu, M. Nickel, M. Le, Flow Matching for Generative Modeling (2023)