Fisher Width: New geometric measure quantifies complexity on curved statistical manifolds

Vu Khac Ky's new paper proposes Fisher width, a geometric complexity measure designed for statistical manifolds. Traditional Gaussian width, widely used in high-dimensional probability and learning theory, measures the average extent of a set along random Euclidean directions, capturing effective dimensionality. However, statistical models have a natural Riemannian geometry induced by the Fisher information metric, where directions are scaled by statistical distinguishability. Fisher width replaces the Euclidean identity with the local metric tensor G(θ)^(1/2), measuring the Gaussian width of Fisher-rescaled sets. This makes the quantity sensitive to local statistical curvature and invariant under smooth reparameterizations.

The paper develops the theory, showing Fisher width retains key properties like concentration, metric perturbation stability, and spectral comparison bounds with Euclidean baseline, while capturing anisotropic geometric effects. As an application, Ky proves a generalization bound for Fisher-Lipschitz hypothesis classes and provides computable estimators. Empirical tests on MNIST across three model classes demonstrate practical utility. This work lays foundational groundwork for studying complexity and learning on curved statistical manifolds, with potential implications for understanding deep learning's geometry and generalization.