Manifold Fitting in Ambient Space

TL;DR

This paper introduces a novel manifold fitting method in ambient space that combines statistical and geometric techniques, using MLS and eigenvalue analysis to accurately estimate complex underlying structures.

Contribution

It proposes a new approach for manifold fitting in high-dimensional ambient space, extending subsampling and MLS techniques with theoretical bounds and empirical validation.

Findings

The method accurately estimates underlying manifolds.

Theoretical bounds support the approach.

Simulation results demonstrate superiority over existing methods.

Abstract

Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure, namely a manifold. Manifold learning is an emerging field for learning the underlying structure. The study of manifold learning can be split into two main branches: dimension reduction and manifold fitting. With the aim of combining statistics and geometry, we address the problem of manifold fitting in the ambient space. Inspired by the relation between the eigenvalues of the Laplace-Beltrami operator and the geometry of a manifold, we aim to find a small set of points that preserve the geometry of the underlying manifold. From this relationship, we extend the idea of subsampling to sample points in high-dimensional space and employ the Moving Least Squares (MLS) approach to approximate the underlying manifold. We analyze the two core steps in our proposed method theoretically and also provide the bounds for the MLS approach. Our simulation results and theoretical analysis demonstrate the superiority of our method in estimating the underlying manifold.