Variance-Reduced Manifold Sampling via Polynomial-Maximization Density Estimation

TL;DR

This paper introduces a polynomial-maximization density estimator to improve uniform sampling on manifolds, reducing density estimation errors in motion planning and machine learning applications.

Contribution

It proposes a novel PMM-MASEM module that adaptively replaces the plug-in density rule with a polynomial-maximization estimator based on local spacing statistics.

Findings

Reduces density MSE by 22-36% on certain regimes.

The fallback to MLE is effective on flat homogeneous manifolds.

PMM3 can worsen performance on some spacing distributions.

Abstract

Uniform sampling on implicitly defined manifolds is a core primitive in motion planning, constrained simulation, and probabilistic machine learning. MASEM addresses this problem by entropy-maximizing resampling, but its resampling weights depend on a local k-nearest-neighbour density estimate whose errors can be amplified by aggressive resampling temperatures. We ask whether a polynomial-maximization moment estimator can replace the plug-in density rule without changing the surrounding MASEM architecture. The proposed PMM-MASEM module computes shell spacings from nested k-nearest-neighbour radii, estimates their standardized cumulants, and uses a gated PMM2/PMM3 estimator only when the spacing distribution departs from the flat Exp(1) regime; otherwise it falls back to the plug-in/MLE rule. This fallback is essential: on a flat homogeneous manifold the plug-in estimator is already the MLE, so PMM should not outperform it. A local Known-DGP Monte Carlo experiment confirms this gate: the selector returns MLE on flat Exp(1) spacings and reduces density MSE by 22--36% on asymmetric gamma and boundary-spacing regimes. The evidence is not uniformly positive: PMM3 worsens a platykurtic uniform spacing law, and a lightweight resampling-proxy experiment improves seven-lobes coverage but degrades the sine and swiss-roll proxies. The current evidence therefore supports an applicability-boundary result rather than a general MASEM improvement claim.