On Statistical Complexity of Learning Kinematic Topologies and Manipulation Maps

Learning kinematic maps requires separating representational constraints from statistical estimation costs. We develop a framework connecting topology, inverse conditioning, and unknown model structure to representation and learning guarantees. For regular kinematic fibrations, minimum numbers of continuous inverse-selection and manipulation domains persist below a geometric error threshold. Sensitivity constraints yield risk and validation bounds, including an exact Lipschitz–risk frontier for circle inversion. A shared-parameter construction yields growing topological complexity with dimension-independent sample requirements. Chart-dependent budgets and calibration risk quantify the effects of conditioning and parameter sharing. We compute invariants for robot maps and general torus homomorphisms, and formulate protein/RNA kinematics on constrained torsion spaces. A molecular decoder bound transfers angular error to Cartesian error, with observed-coordinate learning distinguished from latent inference. Planar and spatial seven-joint studies, including sixty retained spatial regressors and independent Jacobian checks, isolate representation effects and redundant-solution ambiguity.