From Gauss to Kolmogorov: Localized Measures of Complexity for Ellipses

TL;DR

This paper establishes a connection between localized Gaussian and Kolmogorov widths for ellipses, leading to explicit, location-dependent estimation error bounds in regression, with rates varying based on the true parameter's position.

Contribution

It introduces a novel relationship between localized Gaussian and Kolmogorov widths, enabling precise, location-specific error analysis in regression within ellipses.

Findings

Error rates vary with location within the ellipse.

Explicit formulas for Sobolev ellipses are derived.

Local Kolmogorov width relates to local metric entropy.

Abstract

The Gaussian width is a fundamental quantity in probability, statistics and geometry, known to underlie the intrinsic difficulty of estimation and hypothesis testing. In this work, we show how the Gaussian width, when localized to any given point of an ellipse, can be controlled by the Kolmogorov width of a set similarly localized. This connection leads to an explicit characterization of the estimation error of least-squares regression as a function of the true regression vector within the ellipse. The rate of error decay varies substantially as a function of location: as a concrete example, in Sobolev ellipses of smoothness $\alpha$, we exhibit rates that vary from $(\sigma^2)^{\frac{2 \alpha}{2 \alpha + 1}}$, corresponding to the classical global rate, to the faster rate $(\sigma^2)^{\frac{4 \alpha}{4 \alpha + 1}}$. We also show how the local Kolmogorov width can be related to local metric entropy.