---
title: "Fisher Width: A Geometric Measure of Complexity on Statistical Manifolds"
source_id: "arXiv:2606.18306v1"
authors:
  - "Vu Khac Ky"
affiliations: "Department of Mathematics, FPT University, Vietnam"
date: 2026-06-16
categories: ["cs.LG", "stat.ML"]
pages: 48
figures: 3
url: "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|Gaussian Width]] — Euclidean foundational complexity measure
- [[statistical-manifold|Statistical Manifold]] — Riemannian manifold with Fisher metric
- [[fisher-information-metric|Fisher Information Metric]] — Local metric tensor G(θ)
- [[fisher-lipschitz|Fisher-Lipschitz]] — Hypothesis class with Fisher-geometric smoothness
- [[lifting-identity|Lifting Identity]] — w_G(T;θ) = w(G(θ)^{1/2} T)
- [[empirical-fisher|Empirical Fisher]] — Score-based computation of Fisher information
