Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis

Abstract: We consider the problem of recovering a $d-$dimensional manifold $\mathcal{M} \subset \mathbb{R}^n$ when provided with noiseless samples from $\mathcal{M}$. 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 $\mathcal{M}\mathrm{put}$ of $\mathcal{M}$ should be an actual manifold of a certain smoothness; furthermore, $\mathcal{M}\mathrm{put}$ should be arbitrarily close to $\mathcal{M}$ 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 $\mathcal{M}$. Then, $\mathcal{M}\mathrm{put}$ 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, $\mathcal{M}\mathrm{put}$ is a manifold that is arbitrarily close in Hausdorff distance to $\mathcal{M}$. 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.

• The paper introduces two novel asdf methods based on kernel density estimation and local PCA for recovering underlying manifolds.
• The KDE approach derives the asdf from -log estimated density while the local PCA method uses tangent space approximations to define cylinder packets.
• Simulations reveal that the PCA-based method yields more precise manifold estimations with reduced RMS error and guarantees on Hausdorff proximity.

Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis

Introduction

The paper "Manifold Learning Using Kernel Density Estimation and Local Principal Components Analysis" (1709.03615) addresses the challenge of accurately recovering a $d$-dimensional manifold $\mathcal{M}$ from noiseless samples. While traditional manifold learning algorithms like Isomap and locally linear embedding have theoretical support, they often fail to output a manifold near the original in Hausdorff distance. This research aims to overcome such limitations by employing an approximate squared-distance function (asdf) to define the estimated manifold $M$. The authors propose two distinct methods of calculating asdfs based on kernel density estimation (KDE) and local principal components analysis (PCA).

Methodology

Kernel Density Estimation

The KDE-based approach utilizes the estimated kernel density $\widetilde{p}_N(x)$ to formulate the first asdf. The approximate squared-distance function is derived from $-\log \widetilde{p}_N(x)$ plus a normalization constant. The selection of kernel bandwidth $\sigma$ is crucial; it is adjusted to ensure concentration around the true density, facilitating the formation of an asdf meeting specific smoothness and curvature criteria. Lemmas provided in the paper demonstrate that the KDE-based asdf meets the requirements for recovery accuracy under Hausdorff distance.

Local Principal Components Analysis

The second approach involves estimating tangent spaces using local PCA, where sample points within a neighborhood characterize the local geometry near each point on the manifold. The authors construct cylinder packets based on these tangent plane estimations, which adhere to geometric constraints allowing the definition of a valid asdf. The use of PCA implies extracting eigenvectors corresponding to the largest eigenvalues of the covariance matrix of local samples. The Davis-Kahan theorem is used to demonstrate the proximity between estimated and true tangent spaces, ensuring the local PCA asdf is suitably precise.

Results

Simulations showcase the performance of both KDE and local PCA in approximating three types of manifolds: a circle, a curve, and a sphere. Results indicate a significant decrease in RMS error relative to sample size, underscoring the efficacy of the proposed methods. Comparisons between the KDE and local PCA approaches reveal that PCA tends to offer more precise manifold estimations due to better handling of geometric roughness and convergence properties.

Implications and Future Work

The research provides robust techniques for manifold estimation that yield provable guarantees of the output's manifold properties, including bounded reach and Hausdorff proximity. Practically, these techniques enhance manifold learning outcomes in scenarios with complete data, thereby expanding the applicability of manifold techniques in statistical data analysis and machine learning domains.

Looking ahead, addressing samples with noise introduces a challenging yet insightful research direction. Moreover, refining numerical constants within the bounds provided—including those derived from existing theoretical foundations—could further tighten the precision of manifold recovery.

Conclusion

The paper effectively demonstrates that careful application of KDE and local PCA can define asdfs meeting the strict conditions for manifold recovery from sampled data. These techniques advance current understanding in manifold learning by ensuring output manifolds with guaranteed smoothness and proximity attributes, paving the way for further exploration into noisy data contexts and refined theoretical formulations.