The Stability of the First Neumann Laplacian Eigenfunction Under Domain Deformations and Applications

The robustness of manifold learning methods is often predicated on the stability of the Neumann Laplacian eigenfunctions under deformations of the assumed underlying domain. Indeed, many manifold learning methods are based on approximating the Neumann Laplacian eigenfunctions on a manifold that is a...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2018-12
Main Author: Marshall, Nicholas F
Format: Article
Language:English
Subjects:
Approximation
Deformation
Domains
Eigenvectors
Geophysics
Isomorphism
Machine learning
Manifolds (mathematics)
Stability
Teaching methods
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The robustness of manifold learning methods is often predicated on the stability of the Neumann Laplacian eigenfunctions under deformations of the assumed underlying domain. Indeed, many manifold learning methods are based on approximating the Neumann Laplacian eigenfunctions on a manifold that is assumed to underlie data, which is viewed through a source of distortion. In this paper, we study the stability of the first Neumann Laplacian eigenfunction with respect to deformations of a domain by a diffeomorphism. In particular, we are interested in the stability of the first eigenfunction on tall thin domains where, intuitively, the first Neumann Laplacian eigenfunction should only depend on the length along the domain. We prove a rigorous version of this statement and apply it to a machine learning problem in geophysical interpretation.
ISSN:2331-8422
DOI:10.48550/arxiv.1704.02962