Preserving Exponential Families along Generative Paths

How to choose the family for a structured generative model, by not choosing it.

Flow matching and diffusion models carry noise to data along a path of intermediate distributions. In the vanilla recipe the noise is Gaussian, the interpolation is the straight line , and the intermediate marginals are Gaussian too. That closure is exactly why the training target has a closed form and the sampler is cheap. The Gaussian is doing real work, because it is the family that stays itself under the operations the method uses.

Now step off . The data may be unit vectors, covariance matrices, points on a simplex, or nonnegative counts. The straight line then leaves the manifold, since the chord between two rotations is not a rotation, and the Gaussian is no longer natural noise. Which family should the intermediates live in, von Mises–Fisher, Wishart, Dirichlet, or Poisson? The usual answer is a catalogue, one design per data type.

This article offers a single principle instead. Don’t choose the family. Choose the operation the path should close under, and the family is determined.

The selection question. Fix a way to interpolate on the state space, a binary operation . Is there a family for which interpolating the samples equals interpolating the parameters? Is it unique, and what is it? Write the answer .

A single geometric invariant sorts the answers, the curvature of the interpolation. Gaussianity turns out to be one value of , not a privileged assumption.

Pick the operation, not the family

Take the state space to be a smooth manifold. Its measurable structure comes for free, since it is Polish and hence standard Borel, so distributions, pushforwards and independent products are all available. The only real modelling choice is a connection or metric, which turns into a convex space with an intrinsic way to interpolate points by a geodesic or Fréchet barycentre . Fixing that one operation determines the rest. Transport is a flow of isometries such as rotations, congruences, and affine maps, and interpolation pushes an independent pair through the same . This is convex flow matching, where both the transport and the training-target interpolation are read off the geometry rather than from an ambient straight line that would fall off the manifold.

Recovering the velocity field from a path is the continuity-equation business every flow-matching method does. It is upstream background, taken as given here and kept on the flow note.

Why closure keeps the family tractable

The method is usable only if its intermediate laws stay inside a family that can be sampled and scored. That is a closure demand with two faces, worked out with all the boxed conditions in convex-flow-matching:

• Transport stays in the family iff the flow acts affinely on the family’s sufficient statistic, giving rotations for von Mises–Fisher, congruences for Wishart, and affine maps for the Gaussian.
• Interpolation stays in the family only for the special stable and reproducible families. Its strongest version demands that interpolating samples literally equals interpolating parameters, the sample–parameter homomorphism, which is exactly what the selection question asks to solve.

These are two faces of one requirement (affineness, read on the statistic vs. on the log-partition). When they hold, the inverse question becomes answerable.

Curvature sorts the families

Fix the interpolation and ask which family makes it a homomorphism. The obstruction is curvature. On a curved space the geodesic barycentre is not even a convex-space operation, because geodesic parallelograms do not close, so the question has no answer at all and the family is void before it is chosen. Flat spaces are the good case, and there Gaussianity is forced.

interpolation	curvature	family it selects
straight line / affine barycentre (flat)		Gaussian, and its isometric twins (logistic-normal, log-normal)
convolution (additive group)	n/a	infinitely divisible / Tweedie, along a dispersion axis
geodesic barycentre (curved)		none; the homomorphism breaks with the curvature

A structured generative model is therefore the choice of an operation, not of a family, and the family is that operation’s image under . Gaussianity is not a modelling assumption. It is the value at the flat, self-dual operation, the one point where interpolating in the mean coordinate and in the natural coordinate agree. Ask the path to close under convolution instead and infinitely divisible families take its place, namely the Tweedie spectrum of Poisson, gamma, and inverse Gaussian. Ask it to close along a curved geodesic and nothing qualifies. The finer table (Aitchison → logistic-normal, log → log-normal, weighted stable sums → -stable) and the dispersion axis are in convex-flow-matching and the dispersion extension.

What this buys in practice

The families with a closed-form training target are exactly the ones whose structure-preserving group is transitive on the parameters, such as the Gaussian on a vector space and the Wishart on the cone, the picture in which the family is a single group orbit. There a per-sample flow carries each datum and yields the target velocity for free. This is the structural reason the Gaussian recipe is everywhere, not an a-priori virtue but the flat, self-dual sweet spot. Moving off it has a cost, either a fitted, two-stage target on a curved space, or a jump process rather than a smooth transport once the dispersion axis is engaged on a non-Gaussian model. The bridge from these in-family paths to a real dataset is conditional flow matching, carrying each sample to the concentration boundary and training the mixture by a posterior-mean/Bregman objective.

Instances

state space	convex slot	family	structure-preserving flow	index
	vector space	Gaussian	affine group	precision
	manifold	von Mises–Fisher	rotations	n/a
(PD cone)	convex cone	Wishart	congruences	d.o.f.
(simplex)	convex space	categorical / multinomial	Markov semigroups	trials
	cone / discrete	Tweedie	scaling / drift	Lévy index
heavy tails	deformed	-Gaussian / Student-	-affine	n/a

The positive definite cone carries three classical matrix means at once. Arithmetic and harmonic are the two -intrinsic dual-flat interpolations, whereas geometric is the -intrinsic geodesic. Dual flatness is therefore distinct from the Riemannian geometry of the state space.

Map of the framework

Construction (the original one, with its theorems and properties inline): convex-flow-matching, coordinate-free flow matching whose transport and interpolation are both the intrinsic convex-space operation.

Definitions (textbook notes, standard facts folded in): differentiable-manifold · convex-space · exponential-family · flow · interpolation · conditional-flow-matching (the last folds in flow closure)

Equation card

A compressed reference, in the order of the sections above.

Setup

smooth manifold Polish standard Borel (measurability is free); the only datum is a connection/metric giving intrinsically (geodesic / Fréchet); flows are isometries. Exponential family on a measurable space with base measure : , , ; , , ; dually flat. (The generating velocity is upstream flow-matching background: fixed only up to divergence-free fields, minimal representative , unique tangent on ; → flow.)

The auxiliary closures. Transport (unary) makes the flow act affinely on the sufficient statistic. Interpolation (binary) lifts to the binary , which preserves only for stable / reproducible families. With and :

solves to stable (, Gaussian at ) and reproducible (, index adds ). Unary and binary are one affineness, read on and on respectively.

The correspondence. Homomorphism ; curvature obstructs the object ( flat, so a curved has no domain). In both charts → convex-space, convex-flow-matching:

operation	curvature	family with a sample–parameter homomorphism
affine barycentre (flat)		Gaussian, and isometric images (logistic-normal, log-normal)
convolution (additive group)	n/a	infinitely divisible / Tweedie, on the index axis
geodesic barycentre (curved)		none; the homomorphism breaks with the curvature

Extensions. Deformation , endpoints . Dispersion: continuous index Lévy, closed Tweedie (), transport iff . Synthesis: bilinear the - and -axes commute.

Thesis

Design choosing the operation the path should close under; the family is . Gaussianity is the value at the flat, self-dual point ( affine, ); it loses its privilege under convolution ( infinitely divisible) or curvature ( none).

References

The ingredients are classical; the contribution is their synthesis for the flow-matching setting and the reading of the selection map .

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