Near-optimal estimation of linear functionals with log-concave observation errors

Author:

Foucart Simon1,Paouris Grigoris1

Affiliation:

1. Department of Mathematics, Texas A&M University , College Station, TX 77843 , USA

Abstract

Abstract This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex and closed model set containing the object. It complements the article ‘Statistical Estimation and Optimal Recovery’ published in the Annals of Statistics in 1994. There, Donoho showed (among other things) that, for Gaussian noise, linear maps provide near-optimal estimation schemes relatively to a performance measure relevant in Statistical Estimation. Here, we advocate for a different performance measure arguably more relevant in Optimal Recovery. We show that, relatively to this new measure, linear maps still provide near-optimal estimation schemes even if the noise is merely log-concave. Our arguments, which make a connection to the deterministic noise situation and bypass properties specific to the Gaussian case, offer an alternative to parts of Donoho’s proof.

Funder

National Science Foundation

Office of Naval Research

Simons Foundation

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

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5. Mathematical Pictures at a Data Science Exhibition

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