Restricted Dynamic Geometric Complexity: Path-Space Reduction and Möbius--Jacobi Response

What the math gives to ML

The paper turns preconditioning from a static endpoint-selection problem into a minimum-motion path problem on the manifold of positive-definite metrics. Its transferable asset is affine-invariant geometry: a metric schedule can be penalized by intrinsic path length or, at fixed horizon, by an equivalent kinetic action, while the endpoint is judged by the generalized Hessian condition ratio. This suggests a dynamic optimizer whose preconditioner changes only as much as necessary to reach a prescribed local curvature target, rather than abruptly switching between poorly conditioned metrics. The most practical first version uses diagonal or block-diagonal metrics, inexpensive curvature estimates, and a receding-horizon controller.