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title	source_id	authors	affiliations	date	categories	pages	figures	url
Fisher Width: A Geometric Measure of Complexity on Statistical Manifolds	arXiv:2606.18306v1	Vu Khac Ky	Department of Mathematics, FPT University, Vietnam	2026-06-16	cs.LG stat.ML	48	3	https://arxiv.org/abs/2606.18306v1

Fisher Width: A Geometric Measure of Complexity on Statistical Manifolds

Authors: Vu Khac Ky (FPT University, Vietnam) arXiv: 2606.18306v1 | Date: 2026-06-16 Categories: cs.LG (Machine Learning), stat.ML (Machine Learning) 48 pages, 3 figures

Abstract

Gaussian width is a central geometric complexity measure in high-dimensional probability, compressed sensing, convex optimization, and learning theory. It quantifies the average extent of a set along random directions, thereby capturing the effective dimension of constraint sets, hypothesis classes, and descent cones. However, this notion is intrinsically Euclidean. Statistical models instead carry a natural Riemannian geometry induced by the Fisher information metric, where directions are scaled according to statistical distinguishability rather than ambient Euclidean length.

We introduce Fisher width, a Fisher-geometric analogue of Gaussian width for statistical manifolds. At a parameter point θ, Fisher width replaces the Euclidean identity by the local metric tensor G(θ)^{1/2}, measuring the Gaussian width of the Fisher-rescaled set. This makes the resulting quantity sensitive to local statistical curvature and invariant under smooth reparameterizations.

We develop the basic theory of Fisher width, showing that it retains key structural features of Gaussian width, including concentration, metric perturbation stability, and spectral comparison bounds with the Euclidean baseline, while also capturing anisotropic geometric effects invisible to Euclidean measures. As an application, we prove a generalization bound for Fisher-Lipschitz hypothesis classes and propose computable estimators, which we evaluate empirically on MNIST across three model classes.

Fisher width is to statistical manifolds what Gaussian width is to Euclidean convex bodies. This work lays the foundation for studying complexity and learning on curved statistical manifolds.

Key Contributions

1. Fisher Width Definition: Introduces Fisher width as a local Fisher-geometric analogue of Gaussian width, with the lifting identity w_G(T;θ) = w(G(θ)^{1/2} T) and reparameterization invariance.
2. Structural Theory: Concentration inequalities, algebraic properties, spectral comparison bounds, and stability under metric perturbations.
3. Generalization Bound: For Fisher-Lipschitz hypothesis classes, uniform deviation controlled by w_G(T−T;θ₀)/√n, with tightness proof for exponential-family models.
4. Practical Estimators: Empirical Fisher, randomized low-rank approximation, and score-based sampling, validated on MNIST (logistic/softmax/ridge regression).

Key Concepts

• gaussian-width — Euclidean foundational complexity measure
• statistical-manifold — Riemannian manifold with Fisher metric
• fisher-information-metric — Local metric tensor G(θ)
• fisher-lipschitz — Hypothesis class with Fisher-geometric smoothness
• lifting-identity — w_G(T;θ) = w(G(θ)^{1/2} T)
• empirical-fisher — Score-based computation of Fisher information