Exponential Families from Group Orbits

A group representation builds an exponential-family out of an orbit. The family’s minimality, meaning that distinct parameters give distinct laws, becomes a clean geometric condition on the orbit map. (Tojo–Yoshino, arXiv:1907.04212, strengthening arXiv:1811.01394.)

Fix a group with a finite-dimensional representation on and a base point . The orbit map has image the orbit , and pulling the natural pairing back along it produces a family of densities indexed by the parameter carried by . The question is when this parametrisation is faithful.

Minimality is an orbit condition

• Injectivity (Thm 1). The map is injective under an explicit condition, so the family is a non-redundant parametrisation. Prop 1 translates that condition into geometry. Injectivity holds iff the orbit is not contained in any proper affine subspace of , equivalently the cyclic-vector condition on . Minimal sufficiency is thus not an analytic accident but a statement that the orbit spans its ambient space affinely.
• Classification (Lemma 2). Equivalence classes of pointed representations are in bijection with finite-dimensional -invariant subspaces of , realised through the space of matrix coefficients. This bijection collapses redundant parametrisations to a canonical one, since the family depends only on the invariant subspace and not on the presentation.

Example

Taking with a suitable representation, the construction yields the generalized inverse Gaussian family (§3). The orbit-map recipe thus reproduces a standard exponential family from pure representation data.

Why it matters here

The injectivity of is exactly the precondition for putting a well-defined vector field on the parameter space , which is the object a flow-matching path integrates. The orbit condition therefore sits upstream of structured generative models on exponential families, since it certifies that the family is a manifold worth flowing on in the sense of information-geometry. A residual freedom remains in the base-measure reparametrisation , which plausibly corresponds to the choice within a reproductive family and links this classification to the dispersion axis of exponential-dispersion-models.