Flow Matching Along a Foliation

A flow-matching path is a curve in the space of distributions, and on a dual-flat statistical manifold the natural curve to transport along is a geodesic. Reading the manifold’s foliation by expectation-parameter level sets as exactly the structure a flow moves through, the path can be designed so the flow follows the likelihood-optimal -geodesic, assembled from iterated -projections onto neighbouring leaves.

The two geodesics

On an exponential-family with natural parameter and expectation parameter , the dual-flat structure gives two straight lines between and :

• the -geodesic, straight in , the mixture . Its flow-matching velocity field is time-independent and trivial to learn, but it is not the likelihood-optimal path;
• the -geodesic, straight in , the exponential interpolation . This is the maximum-likelihood, minimum-Kullback–Leibler path, and the one worth designing a flow to follow.

The foliation is what the flow moves through

Construction. Leaves transverse to the geodesic

Foliate the manifold by the level sets of the expectation parameter , equivalently the -flat leaves. A path that raises crosses these leaves transversally, so the leaves are exactly the fibres a flow’s expectation coordinate sweeps through. The moving quantity of the path is the coordinate that indexes the foliation. Transporting a distribution along the flow is then “step to the next leaf”, and the design question is which transverse curve to pick: the -geodesic, leaf-to-leaf by mixture, or the -geodesic, leaf-to-leaf by -projection.

Iterated e-projection assembles the e-geodesic

Does stepping leaf-to-leaf by -projection stay on the -geodesic?

Design the path by repeatedly -projecting onto the next leaf, the KL-nearest point of the target leaf along an -geodesic, which is a Pythagorean projection in the dual-flat geometry. The conjecture is that a composite of -projections onto a nested or transverse family of leaves is itself an -projection, so the discretised leaf-by-leaf path converges to the global -geodesic. If that holds, the result is a likelihood-optimal flow-matching path built incrementally, since the sampler only ever solves a local projection yet the whole trajectory is the -geodesic. The generalised Pythagorean theorem gives the two-step case. The open part is the coherence of the full iterate when leaves are not -parallel, or off the family.

This construction improves on the convex/α construction. There the family is closed by choosing the interpolation operation, whereas here the foliation supplies the path, and the choice of - versus -geodesic is the choice of which optimality the flow inherits.

The α-path generalisation

The - and -geodesics are the two endpoints of a one-parameter family, so the choice between them relaxes into a single continuous knob. On the exponential-family the dual-flat structure carries a family of affine connections , ,

with the -connection straight in , the -connection straight in , and the self-dual Fisher (Levi-Civita) connection; the pair is Fisher-dual.

Construction. The α-geodesic as FM path

Take the conditional path from to to be the -geodesic, which turns into a design knob for the flow-matching target. Its FM velocity is the constant-speed tangent in the -affine coordinate pushed to -space, and at it reduces to the exponential and mixture paths above. Intermediate therefore dials continuously between the likelihood-optimality of the -path and the mixture-simplicity of the -path, giving a one-scalar family of FM targets in place of the two discrete choices.

Near the velocity is the harder one to learn while near it is time-independent and trivial, so plausibly acts as a regularisation dial trading the optimality of the target against the learnability of the field, with whether an interior ever beats the endpoints the open empirical part. The same closure demand applies as for the geodesics themselves: an -geodesic between two family members stays in-family only on the -autoparallel (-flat) submanifolds, the condition convex-flow-matching makes precise, so choosing also chooses which family the path is allowed to close in.