# Non-Parametric Estimation of Manifolds from Noisy Data

arXiv: https://arxiv.org/abs/2105.04754 · HTML: https://mlsanity.com/paper/2105.04754/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 49 references that are in the map.
- First posted on arXiv 11 May 2021.

## Builds on (most cited first, 5 of 5)

- [Manifold Approximation by Moving Least-Squares Projection (MMLS)](https://mlsanity.com/paper/1606.07104/) (2016)
- [Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis](https://mlsanity.com/paper/1709.03615/) (2017)
- [Approximation of Functions over Manifolds: A Moving Least-Squares Approach](https://mlsanity.com/paper/1711.00765/) (2017)
- [PCA-Based Out-of-Sample Extension for Dimensionality Reduction](https://mlsanity.com/paper/1511.00831/) (2015)
- [Approximating the Riemannian Metric from Point Clouds via Manifold Moving Least Squares](https://mlsanity.com/paper/2007.09885/) (2020)

## Built on by (most cited first, 13 of 13)

- [Tangent Space and Dimension Estimation with the Wasserstein Distance](https://mlsanity.com/paper/2110.06357/) (2021)
- [Semi-Supervised Manifold Learning with Complexity Decoupled Chart Autoencoders](https://mlsanity.com/paper/2208.10570/) (2022)
- [Probabilistic Robust Autoencoders for Outlier Detection](https://mlsanity.com/paper/2110.00494/) (2021)
- [Quadratic Matrix Factorization with Applications to Manifold Learning](https://mlsanity.com/paper/2301.12965/) (2023)
- [Manifold Free Riemannian Optimization](https://mlsanity.com/paper/2209.03269/) (2022)
- [CA-PCA: Manifold Dimension Estimation, Adapted for Curvature](https://mlsanity.com/paper/2309.13478/) (2023)
- [Design a Metric Robust to Complicated High Dimensional Noise for Efficient Manifold Denoising](https://mlsanity.com/paper/2401.03921/) (2024)
- [Subspace-Constrained Quadratic Matrix Factorization: Algorithm and Applications](https://mlsanity.com/paper/2411.04717/) (2024)
- [A Mean Curvature Approach to Boundary Detection: Geometric Insights for Unsupervised Learning](https://mlsanity.com/paper/2605.04274/) (2026)
- [High-Dimensional Statistical Process Control via Manifold Fitting and Learning](https://mlsanity.com/paper/2509.19820/) (2025)
- [Robust Tangent Space Estimation via Laplacian Eigenvector Gradient Orthogonalization](https://mlsanity.com/paper/2510.02308/) (2025)
- [Manifold Dimension Estimation via Local Graph Structure](https://mlsanity.com/paper/2510.15141/) (2025)
- [Denoising distances beyond the volumetric barrier](https://mlsanity.com/paper/2604.00432/) (2026)

## Abstract (arXiv)

A common observation in data-driven applications is that high dimensional data has a low intrinsic dimension, at least locally. In this work, we consider the problem of estimating a d dimensional sub-manifold of ℝD from a finite set of noisy samples. Assuming that the data was sampled uniformly from a tubular neighborhood of M∈ Ck, a compact manifold without boundary, we present an algorithm that takes a point r from the tubular neighborhood and outputs hat pn∈ ℝD, and widehatThat pnM an element in the Grassmanian Gr(d, D). We prove that as the number of samples n→∞ the point hat pn converges to p∈ M and widehatThat pnM converges to TpM (the tangent space at that point) with high probability. Furthermore, we show that the estimation yields asymptotic rates of convergence of n^-frack2k + d for the point estimation and n^-frack-12k + d for the estimation of the tangent space. These rates are known to be optimal for the case of function estimation.