# Local Averaging Accurately Distills Manifold Structure From Noisy Data

arXiv: https://arxiv.org/abs/2506.18761 · HTML: https://mlsanity.com/paper/2506.18761/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 31 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 7 of 7)

- [Manifold Fitting under Unbounded Noise](https://mlsanity.com/paper/1909.10228/) (2019)
- [Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis](https://mlsanity.com/paper/1709.03615/) (2017)
- [On Convergence of Epanechnikov Mean Shift](https://mlsanity.com/paper/1711.07441/) (2017)
- [Convergence Analysis of Mean Shift](https://mlsanity.com/paper/2305.08463/) (2023)
- [Local Regularization of Noisy Point Clouds: Improved Global Geometric Estimates and Data Analysis](https://mlsanity.com/paper/1904.03335/) (2019)
- [TpopT: Efficient Trainable Template Optimization on Low-Dimensional Manifolds](https://mlsanity.com/paper/2310.10039/) (2023)
- [Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization](https://mlsanity.com/paper/2502.16819/) (2025)

## Abstract (arXiv)

High-dimensional data are ubiquitous, with examples ranging from natural images to scientific datasets, and often reside near low-dimensional manifolds. Leveraging this geometric structure is vital for downstream tasks, including signal denoising, reconstruction, and generation. However, in practice, the manifold is typically unknown and only noisy samples are available. A fundamental approach to uncovering the manifold structure is local averaging, which is a cornerstone of state-of-the-art provable methods for manifold fitting and denoising. However, to the best of our knowledge, there are no works that rigorously analyze the accuracy of local averaging in a manifold setting in high-noise regimes. In this work, we provide theoretical analyses of a two-round mini-batch local averaging method applied to noisy samples drawn from a d-dimensional manifold mathcal M subset ℝD, under a relatively high-noise regime where the noise size is comparable to the reach τ. We show that with high probability, the averaged point hatmathbf q achieves the bound d(hatmathbf q, mathcal M) ≤ σ √dleft(1+frackappadiam(mathcal M)log(D)right), where σ, diam(mathcal M),kappa denote the standard deviation of the Gaussian noise, manifold's diameter and a bound on its extrinsic curvature, respectively. This is the first analysis of local averaging accuracy over the manifold in the relatively high noise regime where σ √D ≈ τ. The proposed method can serve as a preprocessing step for a wide range of provable methods designed for lower-noise regimes. Additionally, our framework can provide a theoretical foundation for a broad spectrum of denoising and dimensionality reduction methods that rely on local averaging techniques.