Differentiable Manifold

The state space of a structured generative model is taken to be a second-countable Hausdorff smooth manifold . The differentiable structure is primary; both the measurable structure and the ability to interpolate descend from it, so it is the right base object to fix first.

Remark on free measurability

is Polish with a standard Borel Borel -algebra , because a second-countable Hausdorff smooth manifold is separable, locally compact, metrizable, and admits a complete Riemannian metric. Consequently the Giry probability functor, disintegration, regular conditional distributions, and product measures are all available on without further hypotheses. The only genuine additional datum is a connection or metric, which supplies interpolation intrinsically. See convex-space.

Remark on why the embedding is not used

Only operations defined by the intrinsic geometry of are used, not the ambient structure of any embedding. By the Whitney and Nash theorems embeds isometrically in some , so one could borrow the ambient affine interpolation . That operation leaves , however, since the chord midpoint of two points on a sphere is not on the sphere, and a linear combination of rotations is not a rotation. Retracting back onto depends on the embedding and is not canonical.