# Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis

arXiv: https://arxiv.org/abs/1709.03615 · HTML: https://mlsanity.com/paper/1709.03615/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2017.

## Built on by (most cited first, 7 of 7)

- [Manifold learning: what, how, and why](https://mlsanity.com/paper/2311.03757/) (2023)
- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimension Estimation](https://mlsanity.com/paper/2107.03903/) (2021)
- [Turing approximations, toric isometric embeddings & manifold convolutions](https://mlsanity.com/paper/2110.02279/) (2021)
- [Design a Metric Robust to Complicated High Dimensional Noise for Efficient Manifold Denoising](https://mlsanity.com/paper/2401.03921/) (2024)
- [Local Averaging Accurately Distills Manifold Structure From Noisy Data](https://mlsanity.com/paper/2506.18761/) (2025)
- [The Data Manifold under the Microscope](https://mlsanity.com/paper/2606.15760/) (2026)

## Abstract (arXiv)

We consider the problem of recovering a d-dimensional manifold M subset ℝn when provided with noiseless samples from M. There are many algorithms (e.g., Isomap) that are used in practice to fit manifolds and thus reduce the dimensionality of a given data set. Ideally, the estimate Mput of M should be an actual manifold of a certain smoothness; furthermore, Mput should be arbitrarily close to M in Hausdorff distance given a large enough sample. Generally speaking, existing manifold learning algorithms do not meet these criteria. Fefferman, Mitter, and Narayanan (2016) have developed an algorithm whose output is provably a manifold. The key idea is to define an approximate squared-distance function (asdf) to M. Then, Mput is given by the set of points where the gradient of the asdf is orthogonal to the subspace spanned by the largest n - d eigenvectors of the Hessian of the asdf. As long as the asdf meets certain regularity conditions, Mput is a manifold that is arbitrarily close in Hausdorff distance to M. In this paper, we define two asdfs that can be calculated from the data and show that they meet the required regularity conditions. The first asdf is based on kernel density estimation, and the second is based on estimation of tangent spaces using local principal components analysis.