Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysis

TL;DR

This paper develops a quantitative analogue of Carleman's theorem to control polynomial approximation rates for smooth functions under various distributions, with applications to smoothed analysis and learning theory.

Contribution

It introduces a nonasymptotic approximation framework based on complex analysis, extending classical results to broader distribution classes and solving an open problem in smoothed analysis.

Findings

Polynomial approximation rates are established for functions over sub-Gaussian and sub-exponential distributions.

Superexponential approximation rates are achieved for bandlimited functions over sub-exponential distributions.

The work provides quantitative improvements to existing smoothed analysis results in learning theory.

Abstract

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(\mu)$ for any $\mu$ such that the moments $\int x^k d\mu$ do not grow too rapidly as $k \to \infty$. In this work, we develop a fairly tight quantitative analogue of the underlying Denjoy-Carleman theorem via complex analysis, and show that this allows for nonasymptotic control of the rate of approximation by polynomials for any smooth function with polynomial growth at infinity. In many cases, this allows us to establish $L^2$ approximation-theoretic results for functions over general classes of distributions (e.g., multivariate sub-Gaussian or sub-exponential distributions) which were previously known only in special cases. As one application, we show that the Paley--Wiener class of functions bandlimited to $[-\Omega,\Omega]$ admits superexponential rates of approximation over all strictly sub-exponential distributions, which leads to a new characterization of the class. As another application, we solve an open problem recently posed by Chandrasekaran, Klivans, Kontonis, Meka and Stavropoulos on the smoothed analysis of learning, and also obtain quantitative improvements to their main results and applications.