Local Galois groups for arbitrary version spaces

Construct a notion of a local Galois group for version spaces of arbitrary hypothesis classes such that noncommutativity controls the gap between their Natarajan and DS dimensions.

Background

Group classes have homogeneous version spaces: version spaces arising from samples on the same points are translates of one another. For arbitrary classes, the paper suggests attaching local symmetry data to a version space, possibly through the monodromy groups of the pseudo-cubes it contains.

The desired construction would extend the paper’s phenomenon that noncommutativity governs the Natarajan–DS gap for two-fibered pseudo-cubes.