Local Averaging Accurately Distills Manifold Structure From Noisy Data

TL;DR

This paper provides a theoretical analysis of local averaging for manifold learning from noisy high-dimensional data, demonstrating its accuracy even in high-noise regimes and establishing bounds on the approximation error.

Contribution

It offers the first rigorous analysis of local averaging accuracy on manifolds under high noise, with bounds applicable to practical denoising and dimensionality reduction.

Findings

Achieves a bound on the distance to the manifold proportional to noise level and manifold properties.

First analysis of local averaging accuracy in high-noise regimes where noise magnitude is comparable to manifold reach.

Framework supports preprocessing for manifold-based methods in noisy data scenarios.

Abstract

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 $\mathcal M \subset \mathbb{R}^D$, under a relatively high-noise regime where the noise size is comparable to the reach $\tau$. We show that with high probability, the averaged point $\hat{\mathbf q}$ achieves the bound $d(\hat{\mathbf q}, \mathcal M) \leq \sigma \sqrt{d\left(1+\frac{\kappa\mathrm{diam}(\mathcal {M})}{\log(D)}\right)}$, where $\sigma, \mathrm{diam(\mathcal M)},\kappa$ 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 $\sigma \sqrt{D} \approx \tau$. 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.