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Moving Least Squares without Quasi-Uniformity: A Stochastic Approach

Shir Tapiro-Moshe, Yariv Aizenbud, Barak Sober

TL;DR

This paper addresses the challenge of applying Moving Least Squares under random sampling, where deterministic quasi-uniformity fails. It develops a stochastic MLS theory by deriving probabilistic bounds for the fill distance $h_n$ and separation $\delta_n$, and proves that the MLS derivative approximation error decays as $h_n^{k-|m|}$ up to logarithmic factors, with high-probability smoothness and manifold-estimation guarantees. It extends the deterministic Manifold-MLS framework to stochastic samples, showing that the Hausdorff distance between the true manifold and its MLS reconstruction decays as $h_n^k$ (up to a $\log n$ factor), provided the manifold is $\mathcal{C}^k$. Collectively, the results yield the first unified stochastic analysis of MLS, ensuring convergence and smoothness properties persist under natural probabilistic sampling and enabling reliable manifold reconstruction from random data.

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} $δ_n$ 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 $δ_n \propto n^{-1/d}$. 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

Moving Least Squares without Quasi-Uniformity: A Stochastic Approach

TL;DR

This paper addresses the challenge of applying Moving Least Squares under random sampling, where deterministic quasi-uniformity fails. It develops a stochastic MLS theory by deriving probabilistic bounds for the fill distance and separation , and proves that the MLS derivative approximation error decays as up to logarithmic factors, with high-probability smoothness and manifold-estimation guarantees. It extends the deterministic Manifold-MLS framework to stochastic samples, showing that the Hausdorff distance between the true manifold and its MLS reconstruction decays as (up to a factor), provided the manifold is . Collectively, the results yield the first unified stochastic analysis of MLS, ensuring convergence and smoothness properties persist under natural probabilistic sampling and enabling reliable manifold reconstruction from random data.

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 and \textit{separation} of an i.i.d. random sample. That is, for a distribution satisfying a mild regularity condition, and . We then prove that, for MLS of degree , the approximation error associated with a differential operator of order decays as 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 -times smooth, we show that the Hausdorff distance between the true manifold and its MLS reconstruction decays as , 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
Paper Structure (13 sections, 15 theorems, 182 equations)

This paper contains 13 sections, 15 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Main Results
  5. Bridging the gap between deterministic and stochastic MLS
  6. Proof of Theorem \ref{['main_result']}
  7. Smoothness of the Approximation
  8. Manifold Estimation
  9. Appendix
  10. Stochastic Boundedness
  11. Failure of the deterministic proof in stochastic setup
  12. Detailed computation of the weighted covariance matrix in proof of Theorem \ref{['main_result']}
  13. Acknowledgments

Key Result

Theorem 1

For a distribution $\mathbb{P}$ over $\Omega$ that satisfies Definition NicelyDef, let $X_n=\{x_1,x_2,...,x_n\}$ be a set of $n$$iid$ samples of $\mathbb{P}$. Denote by $h_n$ the fill distance of $X_n$ (with respect to $\Omega$), and by $\delta_n$ the separation of $X_n$. Then,

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Definition 3
  • Remark 2
  • ...and 27 more