# Variance-Reduced Manifold Sampling via Polynomial-Maximization Density Estimation

arXiv: https://arxiv.org/abs/2605.19938 · HTML: https://mlsanity.com/paper/2605.19938/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 18 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 4 of 4)

- [Analysis of KNN Density Estimation](https://mlsanity.com/paper/2010.00438/) (2020)
- [Applying the Polynomial Maximization Method to Estimate ARIMA Models with Asymmetric Non-Gaussian Innovations](https://mlsanity.com/paper/2511.07059/) (2025)
- [EstemPMM: Polynomial Maximization Method for Non-Gaussian Regression and Time Series in R](https://mlsanity.com/paper/2605.02673/) (2026)
- [Manifold Sampling via Entropy Maximization](https://mlsanity.com/paper/2605.12338/) (2026)

## Abstract (arXiv)

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.