Fisher Width: New Geometric Measure of Complexity for Statistical Manifolds

Researchers have introduced Fisher width, a novel geometric complexity measure designed specifically for statistical manifolds. The work extends the classical concept of Gaussian width—a fundamental tool in high-dimensional probability, compressed sensing, convex optimization, and learning theory—to settings where data carries a natural Riemannian geometry induced by the Fisher information metric.

Gaussian width quantifies the average extent of a set along random directions and has been central to understanding the effective dimension of constraint sets, hypothesis classes, and descent cones. However, this measure is intrinsically Euclidean and does not account for the statistical geometry of probabilistic models. Fisher width addresses this limitation by replacing the Euclidean identity metric with the local metric tensor at a parameter point, thereby measuring complexity in a way that is sensitive to local statistical curvature and invariant under smooth reparameterizations.

The theoretical development of Fisher width demonstrates that it retains key structural properties of Gaussian width, including concentration bounds, metric perturbation stability, and spectral comparison relationships with the Euclidean baseline. Importantly, it also captures anisotropic geometric effects that are invisible to purely Euclidean measures. The researchers prove a generalization bound for Fisher-Lipschitz hypothesis classes and propose computable estimators for practical application.

The work includes empirical validation on the MNIST dataset across three model classes, demonstrating the feasibility of computing and applying Fisher width in practice. By establishing Fisher width as the manifold-based counterpart to Gaussian width in Euclidean convex bodies, this research lays foundational theory for studying learning and complexity on curved statistical manifolds, with potential applications across machine learning and statistical inference.