Lower Complexity Adaptation (LCA) for estimating OT maps and couplings

Prove the lower complexity adaptation principle for estimating optimal transport maps and optimal couplings between measures on \mathbb{R}^d: establish that minimax estimation rates depend on the intrinsic dimension of the least complex of the two input measures (e.g., when one measure is supported on an s-dimensional set), rather than on the ambient dimension d.

Background

The LCA principle, rigorously established for Wasserstein distance estimation, asserts that statistical rates can depend on the intrinsic complexity (dimension) of the simpler measure rather than on ambient dimension. Extending this principle to OT map and coupling estimation would significantly broaden the regimes where nonasymptotic accuracy is achievable.

Semi-discrete OT provides evidence of dimension-free behavior, suggesting that an LCA-type theory may hold more generally for maps and couplings. A formal proof would unify and extend current special-case results, providing adaptive guarantees and practical guidance in high-dimensional applications with low-dimensional structure.