Sharp Debiasing for Smooth Functional Estimation in Banach Spaces

2026-04-01

AI Summary

This study addresses the accurate estimation and inference of smooth functionals of mean parameters in Banach spaces under high-dimensional, unstructured settings. The authors propose a cross-fitting estimator based on a single sample split, which achieves asymptotic normality without requiring structural assumptions such as sparsity, provided the dimension satisfies \(d \log^2(en) = o(n)\). By leveraging non-asymptotic moment bounds and Berry–Esseen-type inequalities, they obtain sharp characterizations of the estimation error. Furthermore, the method enables polynomial-time computation for a broad class of matrix functionals and, in high-dimensional Euclidean settings, simultaneously attains statistical optimality and computational scalability.

Abstract

This paper studies the estimation of smooth functionals $f(θ)$ of a mean parameter $θ= \mathbb{E}_P[W]$ for a distribution $P$ on a general Banach space. We propose a cross-fitted estimator based on a single sample splitting and establish non-asymptotic moment bounds and Berry--Esséen bounds for both $m$-smooth and infinitely smooth functionals under the finite moment assumptions. Our framework is applied to precision matrix estimation and the inference of projection parameters in high-dimensional regression. In these Euclidean settings, the proposed estimators achieve asymptotic normality under the dimension regime $d \log^2(en) = o(n)$ without requiring any structural assumptions (e.g., sparsity). We discuss computational relaxations that enables polynomial-time implementation for a range of matrix functionals.

Problem

• smooth functional estimation
• Banach space
• high-dimensional inference
• precision matrix estimation
• asymptotic normality

Innovation

• cross-fitting
• smooth functional estimation
• debiased inference
• Banach space
• asymptotic normality