Moving Least Squares without Quasi-Uniformity: A Stochastic Approach
This work bridges the gap between deterministic MLS theory and statistical LPR analysis, offering theoretical guarantees for MLS under random sampling, which is crucial for applications in nonparametric estimation and approximation theory.
The paper provides the first unified stochastic analysis of Moving Least Squares (MLS), showing that classical convergence and smoothness properties persist under random sampling despite the failure of deterministic assumptions. It quantifies the probabilistic behavior of fill distance and separation for i.i.d. samples and proves that approximation error decays at the same rate as in the deterministic case.
Local Polynomial Regression (LPR) and Moving Least Squares (MLS) are closely related nonparametric estimation methods, developed independently in statistics and approximation theory. While statistical LPR analysis focuses on overcoming sampling noise under probabilistic assumptions, the deterministic MLS theory studies smoothness properties and convergence rates with respect to the \textit{fill distance} (a resolution parameter). Despite this similarity, the deterministic assumptions underlying MLS fail to hold under random sampling. We begin by quantifying the probabilistic behavior of the fill distance $h_n$ and \textit{separation} $δ_n$ of an i.i.d. random sample. That is, for a distribution satisfying a mild regularity condition, $h_n\propto n^{-1/d}\log^{1/d} (n)$ and $δ_n \propto n^{-2/d}$ in probability. We then prove that, for MLS of degree $k\!-\!1$, the approximation error associated with a differential operator $Q$ of order $|m|\le k-1$ decays as $h_n^{\,k-|m|}$, establishing stochastic analogues of the classical MLS estimates. Additionally, we show that the MLS approximant is locally smooth with high probability. This work provides the first unified stochastic analysis of MLS, demonstrating that - despite the failure of deterministic sampling assumptions - the classical convergence and smoothness properties persist under natural probabilistic models.