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 foundational work bridges multiple mathematical disciplines to address data alignment and extension problems, with potential applications in high-dimensional data science and manifold learning.
This paper develops a unified mathematical framework connecting problems in algebraic geometry, approximation theory, harmonic analysis, and data science, specifically addressing near alignment of data and near isometry extension in ℝᴰ. It surveys related work in clustering, manifold learning, and optimization while presenting numerous open problems.
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.