Moving Least Squares without Quasi-Uniformity: A Stochastic Approach

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 $δ_n \propto n^{-2/d}$. 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

💡 Research Summary

The paper “Moving Least Squares without Quasi‑Uniformity: A Stochastic Approach” bridges the gap between two historically parallel non‑parametric approximation frameworks: Local Polynomial Regression (LPR) from statistics and Moving Least Squares (MLS) from approximation theory. While LPR has been analyzed under stochastic sampling assumptions, MLS theory traditionally relies on deterministic geometric conditions—most notably the quasi‑uniformity of the sample set, which requires the fill‑distance (h) and the separation distance (\delta) to be comparable up to a constant factor. The authors point out that this quasi‑uniformity almost never holds for i.i.d. random samples, making a direct transfer of MLS convergence results to the stochastic setting impossible.

Probabilistic Geometry of Random Samples
The authors first formalize a class of “nicely behaving” probability distributions on a compact domain (\Omega\subset\mathbb R^{d}). The distribution must satisfy an interior cone condition on (\Omega) and have a density bounded above and below by positive constants (Definition 4). Under these mild regularity assumptions they prove two fundamental lemmas: (i) the fill‑distance (h_{n}) of an i.i.d. sample of size (n) satisfies (h_{n}=O_{p}\bigl(n^{-1/d}\log^{1/d}n\bigr)); (ii) the separation distance (\delta_{n}) satisfies (\delta_{n}=\Omega_{p}\bigl(n^{-2/d}\bigr)). The proofs combine covering‑number arguments, concentration inequalities, and elementary exponential bounds, showing that the random point cloud becomes increasingly dense while the minimal inter‑point distance shrinks faster than the fill‑distance.

Stochastic MLS Approximation Error
With these geometric rates in hand, the paper revisits the MLS construction. For a weight function (\theta_{h}\in C^{k}) with compact support of size proportional to (h) and for a polynomial degree (k-1), the MLS approximant (s\text{MLS}{F{n},X_{n}}) is defined exactly as in the deterministic literature. The main stochastic analogue of Mirzaei’s deterministic theorem (Theorem 2) states that for any linear differential operator (Q) of order (|m|\le k-1),

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