Moving Least Squares without Quasi-Uniformity: A Stochastic Approach

TL;DR

This paper develops a stochastic analysis of Moving Least Squares (MLS), showing that classical approximation and smoothness properties hold under random sampling, and applies this to manifold estimation.

Contribution

It provides the first unified probabilistic analysis of MLS, extending deterministic results to random samples and applying it to manifold reconstruction.

Findings

Approximation error decays as $h_n^{k-|m|}$ with high probability.

MLS approximant is smooth with high probability.

Hausdorff distance between manifold and MLS reconstruction decays as $h_n^k$.

Abstract

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} $\delta_n$ 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 $\delta_n \propto n^{-2/d}$. 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 $h_n^{\,k-|m|}$ up to logarithmic factors, establishing stochastic analogues of the classical MLS estimates. Additionally, We show that the MLS approximant is smooth with high probability. Finally, we apply the stochastic MLS theory to manifold estimation. Assuming that the sampled Manifold is $k$-times smooth, we show that the Hausdorff distance between the true manifold and its MLS reconstruction decays as $h_n^k$, extending the deterministic Manifold-MLS guarantees to random samples. 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