Convex Space

A convex space, or barycentric algebra, is a set with a binary operation for each satisfying the barycentric axioms of unit, idempotence, and the medial/entropic law. These are the Eilenberg–Moore algebras of the finitely supported probability monad. The prototype is a convex subset of a vector space, with .

On a smooth manifold with a complete connection (in the metric case, the Levi-Civita connection of a complete metric), the intrinsic interpolation is the geodesic (or Fréchet/Karcher barycentre)

This is defined globally on a Hadamard () space and locally in general. It is a point operation on ; pushing an independent product through it gives the probabilistic binary lift below.

Proposition. Interpolation-preserving maps are affine

A diffeomorphism satisfies for all iff it is affine for , sending geodesics to geodesics while preserving the affine parameter. In the metric case it is then an isometry, such as on a flat space or the congruences on the positive definite cone. This is what makes a structure-preserving flow act affinely on a sufficient statistic.

Proposition. Flatness criterion

For a uniquely geodesic manifold, is a convex space iff the connection is flat (equivalently is affinely a convex subset of a vector space). Nonzero curvature, of either sign, violates the medial law (geodesic parallelograms fail to close). Hence on a curved state space there is no convex-space object, and the sample–parameter homomorphism has no domain, since the curvature obstructs the correspondence before any family is chosen.

The three model classes are the flat case (), the Hadamard case (the positive definite cone with the affine-invariant metric, hyperbolic space), and the positively curved case (a sphere, where is only local).

The binary lift (interpolation of distributions)

Two different operations are called “interpolation”, and separating them is essential. The deterministic ones live on the statistical manifold (the dual-flat - and -geodesics, closed for free as a property of the exponential family). The probabilistic one lives on the state space and is built from . For laws the binary lift is

It pushes the independent product through , a convolution-type operation unlike the deterministic means. Categorically it is the convolution that induces through the commutative structure of the probability monad . With the double strength , . In the Markov category , is deterministic and feeds the independent product into it.

Deformation. Interpolating in a deformed coordinate , with strictly increasing and the normaliser, gives a one-parameter program. The endpoints are the exponential and mixture interpolations, and the preserved families are the -deformed exponential families.

Selection. Turning it around defines the assignment , sending a state-space operation to the family for which sample interpolation equals parameter interpolation. This is the inverse selection question.

References

• Marshall H. Stone, “Postulates for the barycentric calculus”, Annali di Matematica Pura ed Applicata 29 (1949), pp. 25-30
• Tadeusz Świrszcz, “Monadic functors and convexity”, Bulletin de l’Académie Polonaise des Sciences 22 (1974)
• Tobias Fritz, “Convex Spaces I: Definition and Examples”, 2009, arXiv:0903.5522