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

TL;DR

This paper unifies various mathematical problems across algebraic geometry, approximation theory, and data science, revealing deep connections and developing a framework for near isometry extension and data alignment in high-dimensional spaces.

Contribution

It introduces a novel unified framework for studying near alignment and isometry extension problems, integrating techniques from multiple mathematical disciplines.

Findings

Develops a comprehensive framework linking algebraic geometry and data science.

Surveys connections between manifold learning, clustering, and approximation.

Provides numerous open problems for future research.

Abstract

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.