提出量化版卡爾曼定理，实现光滑函数在广义分布下的非渐近逼近控制
Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysis
- 基于复分析构建卡爾曼定理的定量版本，给出多项式逼近速率的精确估计
- 在多变量次高斯/次指数分布下，首次获得统一的 $L^2$ 逼近结果
- 解决学习算法平滑分析中的开放问题，并提供更优的定量改进
卡爾曼的经典结果表明，若某测度 $μ$ 的矩 $\ extstyle \int x^k dμ$ 随 $k \to \infty$ 不增长过快，则多项式在 $L^2(μ)$ 中稠密。本文通过复分析发展了该定理的紧致量化形式，使得对任意在无穷远处具有多项式增长的光滑函数，都能实现多项式逼近的非渐近控制。在诸多情形下，这使我们能对一般分布类（如多变量次高斯或次指数分布）建立此前仅知于特例的 $L^2$ 逼近理论。作为应用，我们证明了巴勒-维纳类函数（带限于 $[-Ω, Ω]$）在所有严格次指数分布上可实现超指数逼近率，从而给出该类的新刻画。此外，我们解决了 Chandrasekaran 等人近期提出的关于学习算法平滑分析的开放问题，并对他们的主结果与应用取得定量改进。
原文摘要 · Abstract (English)
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 $k \to \infty$. 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 $[-Ω,Ω]$ 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.
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