Fisher Widths: Local Learning Geometry and Anisotropic Recovery

What the math gives to ML

The paper supplies a pair of Gaussian-complexity measures for anisotropic parameter spaces: the Fisher width expands directions according to local statistical curvature, while the inverse-Fisher width expands statistically flat directions. The transferable asset is the explicit primal-inverse tradeoff: a parameter subset cannot simultaneously have arbitrarily small complexity in both geometries. A practical neural-network use is to choose sparse fine-tuning coordinates or adapter blocks by balancing Fisher sensitivity against inverse-Fisher uncertainty, rather than selecting parameters solely by magnitude or diagonal Fisher score.