  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.
