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2020 From Gauss to Kolmogorov: Localized measures of complexity for ellipses
Yuting Wei, Billy Fang, Martin J. Wainwright
Electron. J. Statist. 14(2): 2988-3031 (2020). DOI: 10.1214/20-EJS1739

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. Among other consequences, this connection, when coupled with a previous result due to Chatterjee, leads to a tight characterization of the estimation error of least-squares regression as a function of the true regression vector within the ellipse. This characterization reveals that 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.

Citation

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Yuting Wei. Billy Fang. Martin J. Wainwright. "From Gauss to Kolmogorov: Localized measures of complexity for ellipses." Electron. J. Statist. 14 (2) 2988 - 3031, 2020. https://doi.org/10.1214/20-EJS1739

Information

Received: 1 December 2019; Published: 2020
First available in Project Euclid: 15 August 2020

zbMATH: 1448.62023
MathSciNet: MR4135323
Digital Object Identifier: 10.1214/20-EJS1739

Subjects:
Primary: 62F10, 62F30
Secondary: 62G08

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Vol.14 • No. 2 • 2020
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