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

Abstract

This paper is an exposition of work of the author et al. detailing fascinating connections between several mathematical problems which lie on the intersection of several mathematics subjects, namely algebraic-differential geometry, analysis on manifolds, complex-harmonic analysis, data science, partial differential equations, optimization and probability. A significant portion of the work is based on joint research with Charles Fefferman in the papers [39,40,41,42]. The topics of this work include (a) The space of maps of bounded mean oscillation (BMO) in R D , D ≥ 2. (b) The labeled and unlabeled near alignment and Procrustes problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (c) The Whitney near isometry extension problem for point sets with certain geometries and for not too thin compact sets both in R D , D ≥ 2. (d) Partitions and clustering of compact sets and point sets with certain geometries in R D , D ≥ 2 and analysis on certain manifolds in R D , D ≥ 2. Many open problems for future research are given.

On a realization of motion and similarity group equivalence classes of labeled points in $\mathbb R^k$ with applications to computer vision

“…Our goal is to realize the space of all equivalence classes of these n pointed images as a metric space with a computable metric. We do this as follows: Recall that The assumption n ≥ k is not ideal for several data applications, see for example [2]. To this end, we have our next main result dealing with the motion group case.…”

“…We refer the reader to the following references which give a good perspective of Section (1.1) in various ways. [2,1,3,5,6,7,8,9,10,11,12,13,14,16,15].…”