Convex Flow Matching

Convex flow matching draws both the transport and the interpolation of a flow-matching path from the intrinsic geometry of the state space, so the intermediate laws stay inside a fixed exponential family exactly when that geometry is compatible with its sufficient statistic.

Convex flow matching is the coordinate-free construction that keeps the intermediate distributions of a flow-matching path inside a fixed exponential family by taking both the transport and the interpolation from the intrinsic geometry of the state space rather than from an ambient linear structure. Ordinary flow matching transports a reference law to a data law along a path of intermediate distributions, with targets built from an interpolation of the endpoints. On a structured state space one wants those intermediates to stay inside .

The single design datum is the convex space operation (a geodesic or Fréchet barycentre). From it: the transport is a flow of isometries acting by pushforward; the interpolation is the binary lift pushing an independent product through the same ; and reaching data is conditional-flow-matching. Because the operations are intrinsic they respect the geometry, and closes precisely when the geometry is compatible with its sufficient statistic. The rest of this note records the theorems and properties that make this precise.

Closure under transport (unary)

The transport is a flow of isometries . Writing for the base-measure distortion:

The flow acts affinely on the sufficient statistic, and the natural parameter evolves affinely, . Instances: Gaussian/affine group, Wishart/congruence, von Mises–Fisher/rotation. When the group is transitive on the parameters this gives the per-sample flows that carry each datum in conditional-flow-matching.

Closure under interpolation (binary)

The interpolation is the binary lift with and :

Solvability for all admissible parameters forces into the stable class (-homogeneous, closed when ; Gaussian at ) or the reproducible class (, convolution, index adds , the dispersion axis treated below). Unary transport and binary interpolation are two faces of one affineness, read on and on respectively.

Corollary. The sample–parameter homomorphism forces Gaussianity

The strongest form asks . On a flat this holds in both the - and -chart iff is affine, i.e. quadratic, i.e. the family is Gaussian (Morris’s constant variance function). In one chart alone it forces the stability condition.

Example. Deformed interpolation

Interpolating in a deformed coordinate preserves the $\chi$-deformed exponential families ; recover the exponential and mixture families, and Tsallis/Kaniadakis reach the heavy-tailed -/-exponential (-Gaussian, Student-) families.

The dispersion axis (reproduction)

The reproducible class is a genuine second axis, carried by the dispersion index :

Among natural exponential families the closed ones are exactly the power-variance Tweedie models , (, Bar-Lev–Enis); the discrete version is the reproductive property (precisions add).

Proposition. The index velocity is nonlocal unless Gaussian

The generator of has symbol , which is a genuine transport velocity iff the Lévy measure vanishes (Gaussian ) and otherwise a jump process (Poisson , gamma , inverse Gaussian , …; empty).

Bigraded synthesis

The two axes commute because the exponent is jointly affine in . Exponential interpolation with affine flow in and reproductive convolution in therefore generate a joint affine action on . The three admissible structures are vector space, convex cone, and convex space, which are the coordinate systems of , of , and of the normalised index of one graded object.

The design principle

The construction runs in reverse from its intended output. One does not first pick a family and then a transport. One picks the operation under which the path is to close, and the family is its image under the selection map , the family for which sample interpolation equals parameter interpolation. The obstruction is curvature, the flatness criterion. A curved is not a convex-space object, so the homomorphism has no domain.

operation	parameter axis
Euclidean barycentre	and	Gaussian
Aitchison barycentre (isometric log-ratio, ilr)	both, ilr chart	logistic-normal
geometric / log barycentre	both, log chart	log-normal
convolution, fixed	index	infinitely divisible / Tweedie
weighted stable sum	scale	-stable
spherical geodesic barycentre	n/a	none (no convex-space object)

Gaussianity is not a modelling assumption but the value of at the flat, self-dual operation ( affine, ); it loses its privilege under convolution ( infinitely divisible) or curvature ( none).

The flat case is worked out below on the probability simplex, whereas the / optimality trade-off it leaves open is dialled by the foliation construction and its -path generalisation.

On the probability simplex

The flat case is concrete on the interior of the probability simplex carried by the Fisher metric, the self-dual member of the dual-flat family. The Fisher–Rao geodesic is not a straight line in the raw coordinate , so linear FM coincides with the intrinsic path only after a chart that flattens the metric, and two charts do this in complementary ways.

The chart sends , a diffeomorphism onto the positive octant of the radius- sphere in under which the Fisher metric pulls back to the round metric. Fisher–Rao geodesics are therefore great-circle arcs, a slerp between endpoints is exactly the geodesic, and a linear interpolation in this chart renormalised through is geodesic transport whose FM velocity is the constant-speed tangent to the arc. The Aitchison isometric-log-ratio chart is the multiplicative counterpart: it sends to with the Aitchison inner product, making the simplex a vector space whose straight lines are the geodesics that linear FM lands on for free, the logistic-normal row of read as a chart rather than a barycentre.

Both charts give a closed-form velocity because the chart is flat: flatness makes the interpolation coefficient affine in , so its time-derivative is constant along the path, and pushing that constant tangent back through the nonlinear chart map yields the velocity in -coordinates. No learned target is needed for the per-sample conditional path, only for the marginal mixture, as in conditional-flow-matching.

Example. Dirichlet flow matching

Dirichlet FM instantiates the construction with the Dirichlet family as the path law. Samples on the simplex are transported to the categorical vertices, the one-hot laws, along simplex geodesics, and the conditional velocity is read off in closed form. It is the simplex analogue of Gaussian FM on , the natural instance for discrete data and sequence generation.

References

• Lipman, Chen, Ben-Hamu, Nickel, Le, “Flow Matching for Generative Modeling” (2023), arXiv:2210.02747
• Stark, Jing, Barzilay, Jaakkola, “Dirichlet Flow Matching with Applications to DNA Sequence Design” (2024)
• Amari & Nagaoka, “Methods of Information Geometry” (2000)
• Morris, “Natural Exponential Families with Quadratic Variance Functions,” Annals of Statistics (1982)
• Jorgensen, “The Theory of Dispersion Models” (1997)
• Bar-Lev & Enis, “Reproducibility and Natural Exponential Families with Power Variance Functions,” Annals of Statistics (1986)
• Aitchison, “The Statistical Analysis of Compositional Data” (1986)