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

2025-12-03

AI Summary

This paper addresses the non-asymptotic (L^2) polynomial approximation of smooth functions under measures satisfying the Carleman condition—including multivariate sub-Gaussian and sub-exponential distributions. Motivated by an open problem on smoothed analysis posed by Chandrasekaran et al., we develop a quantitative version of the Denjoy–Carleman theorem, integrating tools from complex analysis, quasianalytic function theory, and joint moment–smoothness analysis to construct a unified approximation framework. Our key contributions are threefold: (i) the first derivation of superexponential approximation rates for Paley–Wiener-type function classes under general sub-exponential measures; (ii) a novel characterization of these rates in terms of moment-based smoothness parameters; and (iii) substantial quantitative improvements over existing (L^2) approximation bounds—specifically, sharper dependence on dimension and smoothness—thereby providing a rigorous theoretical foundation for smoothness modeling in high-dimensional statistical learning.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(μ)$ for any $μ$ such that the moments $int x^k dμ$ do not grow too rapidly as $k o infty$. 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 $[-Ω,Ω]$ 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.

Problem

• Develops quantitative polynomial approximation rates for smooth functions.
• Extends L2 approximation to general sub-Gaussian and sub-exponential distributions.
• Solves open problems in smoothed analysis of learning with improvements.

Innovation

• Quantitative analogue of Denjoy-Carleman theorem via complex analysis
• Nonasymptotic polynomial approximation control for smooth functions
• Superexponential approximation rates for bandlimited functions over distributions