# Fisher Width: A Geometric Measure of Complexity on Statistical Manifolds

arXiv: https://arxiv.org/abs/2606.18306 · HTML: https://mlsanity.com/paper/2606.18306/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Builds on (most cited first, 11 of 11)

- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima](https://mlsanity.com/paper/1609.04836/) (2016)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [New insights and perspectives on the natural gradient method](https://mlsanity.com/paper/1412.1193/) (2014)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Limitations of the Empirical Fisher Approximation for Natural Gradient Descent](https://mlsanity.com/paper/1905.12558/) (2019)
- [Fisher-Rao Metric, Geometry, and Complexity of Neural Networks](https://mlsanity.com/paper/1711.01530/) (2017)
- [Universal Statistics of Fisher Information in Deep Neural Networks: Mean Field Approach](https://mlsanity.com/paper/1806.01316/) (2018)
- [Riemannian metrics for neural networks I: feedforward networks](https://mlsanity.com/paper/1303.0818/) (2013)

## Built on by (most cited first, 2 of 2)

- [Fisher Widths: Local Learning Geometry and Anisotropic Recovery](https://mlsanity.com/paper/2607.20578/) (2026)
- [Loss-Parameterized Fisher Width Along Learning Trajectories](https://mlsanity.com/paper/2608.21561/) (2026)

## Abstract (arXiv)

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.