Near-Optimal Estimation of Linear Functionals with Log-Concave Observation Errors

TL;DR

This paper demonstrates that linear estimation methods remain nearly optimal for linear functionals when observations are corrupted by log-concave noise, extending previous Gaussian-based results to a broader class of noise distributions.

Contribution

It introduces a new performance measure for optimal recovery and proves near-optimality of linear maps under log-concave noise, broadening the applicability of previous Gaussian-focused results.

Findings

Linear maps are nearly optimal under log-concave noise.

The new measure is more relevant for optimal recovery.

Extends Gaussian noise results to log-concave noise.

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.