On the Whitney near extension problem, BMO, alignment of data, best approximation in algebraic geometry, manifold learning and their beautiful connections: A modern treatment

This paper provides fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic geometry, approximation theory, complex-harmonic analysis and high dimensional data science. Modern techniques in algebraic geometry, appr...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Author Damelin, Steven B
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 09.02.2023
Subjects
Clustering
Differential geometry
Fourier analysis
Harmonic analysis
Manifolds (mathematics)
Mathematical problems
Optimization
Partial differential equations
Online AccessGet full text

Cover

More Information
Summary:This paper provides fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic geometry, approximation theory, complex-harmonic analysis and high dimensional data science. Modern techniques in algebraic geometry, approximation theory, computational harmonic analysis and extensions develop the first of its kind, a unified framework which allows for a simultaneous study of labeled and unlabeled near alignment data problems in of \(\mathbb R^D\) with the near isometry extension problem for discrete and non-discrete subsets of \(\mathbb R^D\) with certain geometries. In addition, the paper surveys related work on clustering, dimension reduction, manifold learning, vision as well as minimal energy partitions, discrepancy and min-max optimization. Numerous open problems are given.
ISSN:2331-8422